You are aware there evidence of wrongdoing., right? — Relativist

Denying what one is aware of is nothing other than lying.

In general, philosophers especially have sought imperishable knowledge. — jjAmEs

That is what we call truth.

Philosophy offers us the pleasure of stepping out of time with all its risks and rottenness. — jjAmEs

This I disagree with. As philosophers we might seek that eternal truth, but when all we find is the deficiencies of human knowledge we are deprived of that pleasure. Philosophy doesn't offer us that pleasure, it dispels the illusion that we might obtain it.

'Will someday be falsified', is not the same as 'has been falsified'. The falsification is what determines the faults, demonstrating the weaknesses of the theory, showing us where improvement is needed. There is no point in dismissing theories which have not yet been falsified, because we would not know what needs to be improved. That's the scientific method, observations which are inconsistent with what the theory predicts reveal the faults in the theory. But until those inconsistent observations come about, we don't know where the weaknesses of the theory lie.

That explains a lot. Why should I (or anyone else) accept the constraints of your peculiar language? — aletheist

The constraints of my language are the fundamental laws of logic, identity, non-contradiction, excluded middle. These are what facilitate discourse.

How could you ever make such a determination, given your admission that you are unwilling even to try to understand my (or others') usage of the terms, simply because it is different from yours? — aletheist

You have demonstrated that you change the meanings of the words that you use, at will, as you use them, which results in the appearance of contradiction. You justify this with the claim that the same word has a different meaning in a different context. Yet there is only one context here, discussion between you and I. So all this does is lessen my charge against you from contradiction to equivocation, if it happens to be that the apparent contradictions are actually just ambiguous use, and not intentional deception.

Since you seem incapable of getting beyond this way of speaking, I see no point in continuing. I'm sorry, but I truly tried to understand your usage of terms, but I am incapable because your usage is extremely undisciplined. Let me make it clear though, it is just your usage, not others on this thread which I have this difficulty with. I disagree with others, but I understand their usage of language, not yours though.


You made the following statement:

By all our known theories of physics, galaxies should have flown apart long ago.

If you cannot see that the truth of this statement indicates that the theories have been falsified, then I'm afraid your denial is beyond hope.

How do you measure how much we need to know about something before we can name it? — fishfry

You missed the point. You still do not seem to see the difference between describing something and measuring something. I said that if you're at the point of naming something, you ought to be able to say something about that thing.

On the contrary, the day they discovered that the galaxies are spinning too fast to hold together, they named the cause "dark matter" while having no idea what it is or whether it exists at all. — fishfry

You said: "By all our known theories of physics, galaxies should have flown apart long ago. Why didn't they?"

I think the answer is very obvious, the theories are wrong. If the galaxies don't behave the way that the theories say they should behave, then the theories are wrong. We know what "dark matter" means, it means that the theories are wrong. But instead of facing this fact, that the theories are wrong, someone has dreamed up a name "dark matter", and they attribute the fact that the theories are wrong to this mysterious thing, "dark matter". Why not just call it like it is, "the theories are wrong", dump the theories, and the "dark matter" which the theories necessitate because they're wrong, and get on with producing a new theory which doesn't make this mistake?

I cannot communicate with someone who doesn't speak my language. My apologies, as I am not inclined to learn yours. It strikes me that you have disregard for the fundamental rules of logic, and that's why I am simply not motivated toward wasting the effort.

They do represent objects--abstractions, not existents. — aletheist

Therefore the dualism of Platonic realism.

On the contrary, this is Semeiotic 101--in a proposition, the subjects denote objects, and the predicate signifies the interpretant. — aletheist

And symbols represent subjects, so there's a double layer of representation, exactly what Plato warned us against, what he called "narrative", which allows falsity into logic, sophistry.

Incommensurability does not preclude (mathematical) existence. Our inability to measure two different objects (abstractions) relative to the same arbitrary unit with infinite precision does not entail that one of them is (logically) impossible. — aletheist

I'm really tired of your unsupported assertions. As I explained It is not a case of imprecision in practise, it is a case of something being logically impossible within the theory. The theory dictates it as impossible, just like a square circle is impossible, by definition. It has nothing to do with our inability to measure with "infinite precision" (whatever that might mean), as it has been demonstrated that no degree of precision can give us that measurement. This is a defect of the theory, it gives us a so-called theoretical "object" which cannot be measured. Why would we produce a theory which presents us with an object that cannot be measured, when the theory is created for the purpose of measuring objects? It's self-defeating.

Only according to your peculiar theory, not the well-known and well-established theory in question. — aletheist

It's not my "peculiar theory", it's the "Pythagorean theorem". This issue has been known for thousands of years, and I'm shocked by the level of denial in this thread. Accusing me of coming up with my own idiosyncratic theory, that's just ridiculous.

Pythagoras demonstrated how we can construct an abstract mathematical object, using accepted mathematical principles, which is impossible to measure. That is the diagonal of a square. Creating "impossible" abstract objects is nothing new, it is easily done through the use of contradiction. The problem here is that it is done through the use of accepted mathematical principles.

If you view mathematics as explanative device for natural phenomena, I can certainly understand your concern. However, I see mathematics first and foremost as an approximate number crunching and inference theory. I do not see it as a first-principles theory of the space-time continuum or the world in general. I see physics and natural sciences as taking on that burden and having to decide when and what part of mathematics to promote to that role. If necessary, physics can motivate new axiomatic systems. But whether Euclidean geometry remains in daily use will not depend on how accurately it integrates with a physical first-principles theory. Unless the accuracy of the improved model of space is necessary for our daily operations or has remarkable computational or measurement complexity tradeoff, it will impact only scientific computing and pedagogy. Which, as I said, isn't the primary function of mathematics in my opinion. Mathematics to me is the study of data processing applications, not the study of nature's internal dialogue. The latter is reserved for physics, through the use of appropriate parts of mathematics. — simeonz

To me, this is a new and refreshing perspective of mathematics, much more realistic than the Platonic realism defended by many mathematicians. From this perspective, we can see that mathematics is not the primary tool required for an understanding of "nature's internal dialogue". To continue this analogy we could say that nature speaks a different language, and if we want to translate nature's language into mathematics, we need to first recognize that the language is completely different from mathematics, then proceed to determine the differences and similarities to produce some principles for translation. I'm afraid that most physicists would not see things this way though. The prevalent theme here is Platonic realism, according to which, the physical, sensible universe, is a direct representation of the mathematical objects. Under the precept of Platonic realism therefore, learning mathematics directly enables one to understand the universe.

I meant applications where the grain is indeed uniform, such as the atomic structure of certain materials. — simeonz

I don't think that this is a realistic perspective. I don't know how often, if ever, there are individual atoms naturally existing, as they tend to come in molecules. And the "same" atom has a different structure in a different molecule. The first principle we understand about physical reality is the law of identity, and this recognizes that each particular thing is unique. To adopt a principle of grain uniformity would mean dismissing the law of identity, and I don't believe that would be consistent with our experience of the uniqueness of particular individuals.

The particular is unique, with a unique identity. In abstraction, we look beyond the uniqueness, and class unique things together as "the same" according to some principle of categorization. This allows that we might have "5", or "8", apples. Notice that "apples" is the qualifier, the principle of sameness, by which we class the things together as "the same", thus allowing for the abstraction to take place. If we allow that two distinct instances of particular objects are "the same" in an absolute way, like two distinct grains in "grain uniformity" or two distinct occurrences of the number 5, then we violate the law of identity. We would allow that we might have a group of those things classed together without any qualifier (principle of sameness), because we have already assumed that they are the same in an absolute way. There is nothing which makes them the same except the assumption that they are the same. Now we have entered into an extremely confused and contradictory conception within which distinct things are said to be distinct particulars, and they are treated by the application of the theory as distinct particulars, yet they are stipulated by the assumptions of that same theory to be the same in an absolute way. That's the kind of mess which "grain uniformity" might give us.

I do agree that the use of mathematics in real applications is frequently naive. And that further analysis of its approximation power for specific use cases is necessary. In particular, we need more rigorous treatment that explains how accuracy of approximation is affected by discrepancies between the idealized assumptions of the theory and the underlying real world conditions. I have been interested in the existence of such theories myself, but it appears that this kind of analysis is mostly relegated to engineering instincts. Even if so - if mathematics already works in practice for some applications, and the mathematical ideals currently in use can be computed efficiently, this is sufficient argument to continue their investigation. Such is the case of square root of 2. Whether this is a physical phenomena or not, anything more accurate will probably require more accurate/more exhaustive measurements, or more processing. Thus its use will remain justified. And whether incommensurability can exist for physical objects at any scale, I consider topic for natural sciences. — simeonz

I pretty much agree with what you say here. I think that there is no problem with making approximations in practise. This is common, and as an engineer one would know the acceptable limits of such approximations, established by convention. When I use pi for example I use 3.14, and this is my personal convention. When I use the Pythagorean theorem to lay out right angles, I might round off to about a quarter inch, more or less depending on the length of the perpendicular sides.

But approximation in practise is not the same as approximation in theory. Approximations within theory are employed when the theory cannot provide accuracy due to some deficiency of the theory. The approximations are used, and the theory proceeds from them. However, the approximations are covering over the original deficiencies, and as the theory extends and extrapolates, the effects of the deficiencies compound and magnify. If we refuse to recognize that the approximations are a manifestation of deficiencies in the theories, and address those deficiencies, we will never overcome the problems which inevitably result.

Do you happen to know what dark matter is? Don't worry if you don't, because nobody knows what dark matter is. It's a name given to something we can not understand but wish to study. — fishfry

That's not an example at all. We know a lot about dark matter, that's why we can name it. It's not at all like naming something which we have not apprehended. If we have apprehended it, it has an appearance to us, and we can describe it. I describe it as a manifestation of the deficiencies of the general theory of relativity. Maybe you recognize it as this as well, but there seems to be a convention amongst physicists and cosmologists making it taboo to mention deficiencies of general relativity.

By all our known theories of physics, galaxies should have flown apart long ago. Why didn't they? — fishfry

Yes! You do recognize it as a deficiency of the theory. Why hide this? Why not call it what we already know it is, rather than the mysterious "dark matter"?

I have so many pennies in this hand, that many in that hand. How many do I have in all. If X is my left hand and Y is my right hand, then I have X+Y pennies. — tim wood

This is not a demonstration of algebra. So your example is irrelevant. Refutation complete.

What I dispute is that a symbol represents a number which is an object. In your example, X represents how many pennies in your left hand, and Y represents how many pennies in your right hand. Through abstraction we might reduce this to simple numbers, "6" and "8" for example. But those abstracted numbers is not what X and Y represent, as stated in your example. The numbers could only be produced through an abstraction from what is stated.as represented. You did not even state any numbers.

It aimed to illustrate that solving world-space problems imperfectly (due to efficiency constraints) results in the adoption of a modus operandi solution, whose own structure exists only in concept-space. — simeonz

The problem I've been discussing is that whatever it is which is expressed as "a square" does not actually exist in "concept-space" because the perpendicular sides are incommensurable. There is a deficiency in the concept which makes it impossible that there is a diagonal line between the two opposing corners, when there is supposed to be according to theory. The figure is impossible, just like the irrational nature of pi tells us that a circle is impossible.

This impossibility is not a case of us not being able to do in practise what can be done in theory, due to a lack of precision. It is the opposite of this, it is a defect inherent within the theory. The figures defined by the theory are impossible, according to the theory, just like a square circle is impossible.

In other words, we conceptualized indeterminacy, not because of its objective existence (aleatoric uncertainty), but due to our lack of specific knowledge in many circumstances and because introducing indeterminacy as a model was the most fitting solution to our problems. — simeonz

The problem in this situation is that the indeterminacy is created by the deficient theory. It is not some sort of indeterminacy which is inherent in the natural world, it is an indeterminacy created by the theory. Because this indeterminacy exists within the theory, it may appear in application of theory, creating the illusion of indeterminacy in the natural thing which the theory is being applied to, in modeling that natural thing, when in reality the indeterminacy is artificial, created by the deficient theory.

We can allow the indeterminacy to remain, if this form of "concept-space" is the only possible form. But if our goal truly is knowledge, then it cannot be "the most fitting solution" to our problems. When we proceed under an MO, which is essentially a habit, and we recognize that it is not the best, there is a certain laziness associated with "it works", which leaves us uninspired to seek a better operation. The usual way "works", in some instances because it is layered with a multitude of complexities, piled one on top of the other, exceptions to the rules etc.. And, regardless of these massive complexities we continue with the usual, extremely complex, way, because it works. In reality though, taking the time to analyze the fundamental problems at the base of the usual way might produce a much simpler, more efficient, and better way for revealing truth, by removing the indeterminacy from the theory.

As I said, I may misunderstand the topic of the discussion altogether, which is fine. But just wanted to be sure that the intention of my example was clear. (That is - that the space tessellation was not intended as a mathematical structure, but as representation of some unknown coarseness of the physical structure, being ignored for efficiency reasons.) — simeonz

I'm not sure I actually understood your example. Maybe we can say that Euclidian geometry came into existence because it worked for the practises employed at the time. People were creating right angles, surveying plots of land with parallel lines derived from the right angles, and laying foundations for buildings, etc.. The right angle was created from practise, it was practical, just like the circle. Then theorists like Pythagoras demonstrated the problems of indeterminacy involved with that practise.

Since the figures maintained their practicality despite their theoretical instability, use of them continued. However, as the practise of applying the theory expanded, first toward the furthest reaches of the solar system, galaxy, and universe, and now toward the tiniest "grains" of space, the indeterminacy became a factor, and so methods for dealing with the indeterminacy also had to be expanded.

Now, to revisit your example, why do you assume "grain uniformity"? Spatial existence, as evident to us through our sense experience consists of objects of many different shapes and sizes. Wouldn't "grain uniformity" seriously limit the possibility for differing forms of objects, in a way inconsistent with what we observe?

Again, I do not hold than there is such a thing as "an abstraction existing as an object." — aletheist

Then your beliefs are irrelevant to my concerns with algebra and set theory, which hold that the symbols represent objects.

No, a symbol in logic is itself either a subject or the predicate within a proposition. If it is a subject, then it denotes an object, which can be an abstraction or an existent. If it is the predicate, then it signifies the interpretant, which is a relation among the objects denoted by the subjects. — aletheist

I'm afraid you have things backward. The symbol is itself an object. The symbol may signify a subject, and it may signify a predicate. It is impossible that the symbol "is" the subject, or "is" the predicate because then there would be no way to determine whether any given symbol is a subject or a predicate. It is only by the means of representing something (subject or predicate) that the distinction is made.

Again, your peculiar metaphysical terminology is not binding on the rest of us. — aletheist

The challenge is open. All you do is assert without any justification. Where is your demonstration of an abstraction existing as an object, which is not a demonstration of Platonism?.

Apparently not--an object is whatever a logical subject denotes, which can be an abstraction or a concrete existent. — aletheist

Sorry if you misunderstood, but I was talking about what is represented by the symbol in logic, and that is a subject, not an object. Whether the subject denotes an object, a number of objects, or no object at all, is irrelevant to the point.

Platonism is by no means the only philosophy of mathematics that employs the well-established term "existence" when referring to abstract objects. As I have clearly and repeatedly stated, for those of us who are not mathematical platonists, ontology has nothing whatsoever to do with the "existence" of such objects. — aletheist

Yes, you can state that all you want, assert and insist until the cows come home, and then continue to assert some more. The challenge is yours, describe how abstractions, concepts exist as "objects", without invoking Platonism. To call the abstraction an "object" is already using a Platonic term. How are you going to show that an abstraction is an object, in any sense other than a Platonic object.

If we say that abstractions, and conceptions "exist", I have no problem with this. I very much agree that they exist. But "object" refers to a very specific type of thing, a unique individual, a particular, having an identity as described by the law of identity. Because an abstraction is not this type of thing, an object, mathematical axioms which assume that abstractions are objects, are simply wrong.

You reject science. In science we DON'T know what something is, so we give it a symbolic name, write down the symbol's properties, and reason about it in order to learn about nature. — fishfry

I'm not rejecting science. When we name something which we don't know what it is, the name still represents a thing, it's just the case that we didn't know "what" that thing was at that time so we name it.. Once it is named, we know what it is, the thing called by that name. This is not the same as having a symbol which we do not know whether it refers to a thing or not.

So if you have a symbol "2", and you have apprehended a thing and assigned that symbol to this thing, then tell me something about this thing, so that I may use the symbol correctly, to refer to the thing named by that symbol.. It doesn't make sense that you would have assigned the symbol "2" to something and you know absolutely nothing about this thing which you have assigned the symbol to.

When Newton wrote F=maF=ma those were made up terms. Nobody knew (or knows!) exactly what force or mass is. Acceleration's not hard to define. But even then Newton had to invent calculus to define acceleration as the second derivative of the position function.

You reject all that.

Nihilism. — fishfry

Force, mass, and acceleration are not things. They are not objects. They are concepts which describe properties. Properties are not things, and that's why you cannot point to the things which are referred to by these terms. You might provide a definition of the term, but having a definition does not make the term refer to an object. We might call it a logical subject then, the definition being what is predicated of the subject.

That I reject the notion that properties which are described by concepts like "force" "mass" and "acceleration" are themselves objects, doesn't make me nihilist. It just means that I understand the difference between an object and a logical subject.

Perhaps in metaphysics/ontology, but definitely not in mathematics. — aletheist

Yes, I agree, in mathematics some people make the unsubstantiated claim that the symbols represent existent objects. This is called Platonic realism

You keep imposing your peculiar metaphysical terminology, as if everyone else is obliged to conform to it regardless of the context. In this case, you seem to be insisting that only an ontological existent can be the object of a symbol. In mathematics, and even in ordinary language, an abstraction can also be the object of a symbol, as long as the universe of discourse is established. — aletheist

That's right, in Platonic realism the abstraction is an object, and it is believed to exist as an object. In the "universe of discourse" called "Platonism", an abstraction is an object. You seem to believe that there is some other form of ontology, some other universe of discourse, which allows that abstractions have "mathematical existence", as objects, which is not Platonism. So I am waiting for you to produce the principles which distinguish this universe of discourse from Platonism. All you have done is stated Platonist principles and lied in asserting that no one is assuming Platonism.

Alright man. It's not set theory you object to, it's 10th grade algebra. It's not abstraction you object to, it's the very concept of using the symbol '2'. — fishfry

I don't object to using the symbol "2". But like any other language I might use, I want to know what the symbols are being used for. If you assert that the symbol "2" represents an object, I want a clear description of that object, so that I can recognize it when I apprehend it, and use the symbol correctly. If you are simply claiming that the symbol represents an object when you know full well that it doesn't, then you are engaged in deception.

How is algebra faulty? — tim wood

I went through this already, it assumes that the symbol represents an object. This is Platonic realism which is a faulty ontology.

Far as I know, and in my limited experience, it is a tool that works and does and accomplishes its proper tasks. But you say no. Make your case. — tim wood

It's very clear that it works, I never disputed this fact. However, "works", and "it is designed to help us determine theh truth" are two distinct things. That is the problem with pragmatism, if the purpose is anything other than to bring us truth, then the premises employed will reflect that other purpose instead of the goal of truth. "It works" has no necessary relationship with truth, as deception clearly demonstrates.

I'm not looking for arcane nonsense. The sense I am interested in is analogous to your saying that knives don't cut. I have knives and used as knives, they cut. Algebra, used as algebra, "cuts." So in implying that algebra is faulty, in what sense of its proper use, when used properly, does it not "cut"? — tim wood

If it is being used as a system of logic employed toward determining the truth, it is faulty because it has a false premise. Platonic realism is false. Mathematical symbols do not represent objects. Aletheist seems to believe that the existence of mathematical objects can be supported by something other than Platonism, something called "mathematical existence". Perhaps you can assist and demonstrate how mathematical symbols represent some sort of objects which are other than the objects assumed by Platonic realism. As I told aletheist, I would be highly interested in this new ontology.

I am trying to honestly understand, but why do you propose that sets should only include apriori existing entities, and not ones defined by the processes of inference and computation themselves. That is - logic is an algorithm and our application of that algorithm manifests the imperatives in the axiomatic system. The algorithm is inaccurate in almost all practical cases, and therefore is not exactly representative of apriori existing objects. — simeonz

Either the symbols represent objects or they do not. We might say that they may or may not represent objects, but then we would need to know whether or not they do, in order for the symbols to be useful. If we assume that there is an object represented by the symbol then the symbol is useful. But if there really is not an object represented by the symbol and we use it under the assumption that there is, then the use is deceptive.

Sorry simeonz, but your example seems to be lost on me. The question was whether whatever it is which is represented by √2 can be properly called "an object". You seem to have turned this around to show how there can be an object which represents √2, but that's not the question. The question is whether √2 represents an object.

I suggest, based on our conversation, that you may be highly expert on what the great philosophers say about abstraction; but your actual experience and knowledge about how abstraction works is virtually nil. At least when it comes to mathematical abstractions. And what's more abstract than mathematical abstractions? — fishfry

Using abstractions (concepts) is not the same things as the act of abstraction. To conflate these two is equivocation. To define abstraction as "giving a name to something you can't kick with your foot" is woefully inadequate, for someone accusing me of having no knowledge of how "abstraction works".

Abstraction is the act which creates that supposed thing which you cannot kick with your foot. And, you cannot create something by giving a name to nothing. Therefore the "something" which is created by abstraction, must exist prior to the act of giving it a name. The name is what you kick around (with logic), but "abstraction" (verb) refers to the creation of that thing which the name "abstraction" (noun) signifies.

But let's try it the other way around. Let's assume that we make a name which refers to nothing, and manipulating that name, with logical processes (kicking it around), creates a thing which that name represents. Let's say that this is the act of "abstraction", we take a name which refers to nothing, we manipulate that name using logical processes until the name refers to something, and this creates a concept, an abstraction. If this is the case, then all we have done is given meaning to the name. There never was anything which the name referred to, and there still isn't anything which the name refers to, but kicking the name around with logical proceedings has given the name meaning.

Either way you look at it, it's false to assume that we can create something which the name refers to, by starting with a name which refers to nothing. Either there is something immaterial there (Platonic object), which the name refers to from the start, or there is nothing which the name refers to, but the name is given meaning through use. These are two distinct ways of looking at this issue. To confuse these two, and say that we start with a name which refers to nothing, and then by using the name we create "something", and this something is an immaterial object, is to say that we create something from nothing. The point being that if the name starts out as referring to nothing, and logical processes are applied, it must end up as referring to nothing, or the logic would be invalid. therefore the symbol must have meaning from the time it is presented.

In college you look at derivatives and integrals, more abstractions. — fishfry

See, you're using "abstraction" as a noun here. Please do not equivocate in your demonstrations. If a name refers to something (has a definition) that definition must be upheld or else the logic is invalid due to equivocation.

I see no reason to abandon my casual definition, paired with my experience of grappling with mathematical abstractions. — fishfry

Sure, your "casual definition" allows you to equivocate. Without equivocation you'd have no argument. Therefore you see no reason to abandon your casual definition.

Aren't you conflating book learnin' with actual experience? How can you tell a math person they don't know abstraction? That's like telling a pizza chef he doesn't know marinara sauce. — fishfry

Are you familiar with Socrates? You have precisely described Socrates' MO. The person who knows how to do something, does not necessarily know what is being done. Such is my example of a 3,4,5 triangle. I know numerous people who can construct a right angle using the 3,4,5, formula, who have never even heard of the Pythagorean theorem. We can do the thing which the theory describes without knowing the theory. The pizza chef can very easily be making very good pizzas using marinara sauce, without knowing how to make marinara sauce, or even knowing the ingredients of the sauce. The mathematician, engineer, or physicist is most often using abstractions (concepts) without knowing what "an abstraction" is, because this is only studied in the field of philosophy. You should allow the truth of this matter. The philosopher who is trained in this, is much more likely to know what "an abstraction" is than the mathematician who uses abstractions, just like the person who is trained in making marinara sauce is much more likely to know what marinara sauce is than the person who makes pizza.

When a person (such as a mathematician) who doesn't know what abstraction is, not being trained in philosophy, starts to produce logical arguments based in unsound premises concerning the nature of "abstractions", this is called "sophistry". I call them mathemagicians, because their most famous trick is to make (mathematical) objects appear from nothing.

If you think the mathematical existence of the square root of two is a "weakness" or defect in mathematics, it is because you are so ignorant of mathematics, that you haven't got enough good data to reason soundly about mathematics. I would think someone in your position would be desirous of expanding their mathematical understanding. Think of it as "opposition research." Learn more so you can find more sophisticated ways to poke holes. — fishfry

Well, it's very clear, that it's a deficiency in spatial representation, just like the irrational nature of pi indicates a deficiency in that spatial representation. You are simply in denial. And as I said, covering up these deficiencies with complicated mathematics doesn't make them go away. That is where your denial leads you astray. Since you deny that these irrational numbers are the manifestation of a faulty spatial representation, you produce extremely complex numerical structures in a sophistic effort to cover up the truth of this fundamental flaw.

Well, a field is typically defined as a type of set; but the definition really has nothing to do with set theory. It's about what algebraic operations are allowed. — fishfry

You don't seem to understand the criticism. Algebra makes the same mistake as set theory, assuming that a symbol represents an object. Using algebra instead of set theory doesn't get you past my objection. From this premise, the square root of two is not a problem at all. We have a symbol, √2 it represents an object, and the problem is solved. The real question though is whether the "object" supposedly represented by √2 is a valid object. If you assume as a premise, that every symbol represents an object, then of course it is. But then that premise must be demonstrated as sound.

If you will grant me the existence of the rational numbers; I'll build you a square root of 2. — fishfry

Go ahead, but no algebra or other faulty premises.

Symbols don't necessarily need to represent anything. If I have a symbol that behaves a certain way; that's just as good as a thing that behaves that way. At some level one can take the symbol for the thing.

That's abstraction. — fishfry

My OED: symbol, "a thing conventionally regarded as typifying, representing, or recalling something..."

If your symbol does not represent something, it simply "behaves" in a particular way, then the symbol simply has meaning, due to its behaviour. We can say that it is used as "recalling something". In this way, the symbol is the thing, as you say, and as I described above. But if we define "symbol" in this way, then we cannot use algebra or set theory, which require that a symbol represents something. We'd have equivocation.

Why, pray tell, may I not type that symbol on the page? And say that it stands for a green thing? Why can't I do that? It's the foundation of civilization. — fishfry

You can say that the symbol stands for whatever you want. It is "the symbol stands for nothing" which is problematic.

So I ask you? Is Ahab the captain of the Pequod? Or its cabin boy? Do you really claim to be unable to answer on the grounds that Ahab's a fictional character? Nihilism. — fishfry

Do you understand that predication is made of a subject, not an object. Whether or not there is a named object is irrelevant to the act of predication. There is no problem making predications of fictional characters, these characters are known as subjects. The problem is in claiming that the fictional character is an existent object.

2) In order to do (1) I require only one premise from you. You must grant me the existence, in whatever way you define it, of the rational numbers. If you'll do that I'll whip up sqrt 2 in no time flat, no set theory needed, and no cheating on that point. I will use no set theoretic principles. — fishfry

I will grant you the existence of symbols, and you define what the symbols mean.

3) We have a sticking point, which is that you don't accept a symbol unless it comes with a meaning attached. If you truly believe that then you can't solve the snaggle problem, which requires you to reason logically about symbols whose meaning is not defined. — fishfry

I have no idea what the "snaggle problem" is, but if you ask me to use symbols which represent nothing, then we must dismiss any mathematical premise which assumes that a symbol represents an object. Even if we allow that a symbol may or may not represent an object, we must dismiss such premises. If we allow that a symbol has meaning, rather than that it is representative of an object, then that meaning must be defined, or else the symbol will be dismissed as having no meaning and irrelevant.

Here's the issue I see. We can get past "the symbol must have meaning attached", by assuming that the symbol represents an object, and the object represented is unknown. However, if the object represented is unknown, then we also cannot know whether the symbol actually represents an object or not. Then we might say that a symbol may or may not represent an object, but that premise is useless as an epistemological principle.

4) We have stumbled on an interesting point. You say that a symbol can never be conflated with the thing it's supposed to symbolize. But in math, we often do exactly that. We don't know "really" what the number 2 is. Instead, we write down the rules for syntactically manipulating a collection of symbols; we use those rules to artificially construct a symbol that acts like the number 2. Then we just use that as a proxy for the number 2. — fishfry

Are you confessing to the use of equivocation in mathematics?

This is the modern viewpoint of math. We don't care what numbers are; as long as we have a symbol system that behaves exactly as numbers should. — fishfry

Symbols are passive entities, objects. The logician manipulates, moves the symbol. The "behaviour" of the symbol is a direct result of, a representation of, the logician's actions, so what is described here is the nature of the rules, and whether the logician follows the rules. Therefore "a symbol system that behaves exactly as numbers should", expresses nothing more than a judgement of the rules of the system. What is left unknown, is "as numbers should", and this might be an arbitrary criterion.

In order to figure out things that we don't understand, we give them names. We write down the properties we want the names to have. We apply logical and mathematical reasoning to the names and the properties to learn more about the things. That's how science works. That's how everything works. — fishfry

Right, but there is something here, which the name has been given to. We do not assign the name to nothing, unless it is stated that this name signifies nothing. If it is stated that the name signifies nothing, it cannot change and evolve towards signifying something, that would break the stated rule. If, instead, we allow that the name has meaning, instead of representing something, then we can allow that logic enables the meaning to "grow". It cannot contradict the earlier meaning, just expand on that. But if this is the case, then the name must have some meaning from the beginning, and a symbol with no meaning is invalid, as providing nothing to grow.

So we are ALWAYS writing down and using symbols without much if any understanding of the things we're representing. It's exactly through the process of reasoning about the symbols and properties that we LEARN about the things we're interested in. That's Newton writing that force = mass times acceleration. At the time nobody knew what force, mass, and acceleration were. Newton defined those things, which may or may not "really" exist; then he applied mathematical reasoning to his made-up symbols and terms; and he thereby learned how the universe works to a fine degree of approximation. — fishfry

Clearly, you premise that symbols have meaning, and that the meaning may expand and grow. If this is truly what you believe, how can you make this consistent with the premises of algebra and set theory which dictate that a symbol represents a thing?

You reject all of this. If Newton had said F = ma to you, you'd have said, what's force? And Newton would describe it to you, and you'd say, well that's not real, it's only a symbol. You reject all science, all human progress, rationality itself. If I tell you all x's are y's and all y's are z's, and you REFUSE TO CONCLUDE that all x's are z's because I haven't told you what x, y, and z are, you are an absolute nihilist. You believe in nothing that can help you get out of the cave of your mind. — fishfry

What I reject is inconsistency and contradictory premises, not "all of this".

Wow, this keeps getting more and more ridiculous. No one is claiming that mathematical existence has anything to do with "existing substance." In mathematics--again, except for platonism--the term "existence" does not imply anything ontological whatsoever. — aletheist

Any claim of "existence" is validated (substantiated) with substance. If you think that there is a type of existence which is not substantial then please explain.

Something exists mathematically if it is logically possible in accordance with an established set of definitions and axioms. The natural numbers, integers, rational numbers, real numbers (including the square root of two), and complex numbers all exist mathematically, in this context-specific sense. — aletheist

Sorry, but "possible" does not necessitate existence. Do you not recognize that "possible" refers to what may or may not be, so it is contradictory to say that possible things are existing things.

Nonsense, prediction is just as much a significant aspect of the scientific method as observation. — aletheist

I didn't deny that. Both are essential parts of the scientific method. What I deny is that prediction is the goal of the scientific method. It is simply a part of it. Pragmatists allow that prediction is the goal of science.

One more time: No one is claiming otherwise. — aletheist

Then how do you explain set theory, which proceeds from the assumption that a mathematical symbol represents a mathematical object? It's easy to assert "no one is claiming that an abstraction is an existent object", yet everyone backs up set theory which clearly assumes that the abstraction which a symbol represents, is an object. If these objects are not supposed to be "existent", then why attempt to back up their existence with the idea of "mathematical existence"? Due to this behaviour of yours, I can find nothing else to say other than you are boldly lying.

, it is the study of being, which is not necessarily synonymous with existence. For example, one view is that ontological existence (i.e., actuality) is a subset of reality (which also encompasses some possibilities and some necessities), which is a subset of being (which also encompasses fictions). — aletheist

OK, I'll assume for the sake of argument that there is a type of existence, "mathematical existence", which is a different type of existence from "ontological existence". I'll assume two different types of existing substance, like substance dualism.

By defining "existence" in another context-specific way, obviously. There are plenty of other terms that mean something different in mathematics than in metaphysics or in other sciences. — aletheist

So can you tell me what fishfry didn't seem to be able to tell me. How would I define "mathematical existence"? Do all fictional things (like fishfry's example) have mathematical existence, or is it only mathematical fictions which have mathematical existence?

That is not just pragmatism, it is the scientific method. How else would you propose that we evaluate our hypotheses to ascertain whether they accurately represent reality? — aletheist

That should be obvious to you, we ought to evaluate through the criteria of truth and falsity. Do you not see a difference between "accurately represent reality", and "facilitate prediction"? A significant aspect of the "scientific method" involves "observation", and observation is meant to be objective. The goal of "prediction" introduces a bias into observation.

As I have explained to you several times now, no one except a platonist would claim that mathematical existence conforms to "the rigorous philosophical definition" of (ontological) existence. Everyone else understands this, so please stop belaboring your terminological objection. — aletheist

I don't understand what you are saying. How do you propose that "mathematical objects" could have existence, except through some form of Platonic realism? It's very clear, that imaginary, or fictitious objects, do not have existence as objects. If you want to assign some sort of existence to imaginary, or fictitious scenarios, it would be rather strange to say that these exist as objects.

That is a misinterpretation, and you know it by now. — aletheist

I've never seen the existence of "mathematical objects" justified by any ontology other than Platonism. So there is no "misinterpretation". If you think that you can justify the existence of these so-called objects in some way other than Platonism, then I'd really appreciate the demonstration. I've actually been looking for this for years, to no avail. And, since I do not agree with the principles of Platonic realism, I've come to the conclusion that abstractions are not existent objects.

Yes, but the problem is that (for example) particles are always detected as little spots (such as a point on a photographic plate) and wave functions are spread all over the space, or on a space much larger than the observed spot. Nobody has never seen an elementary particle that looks like a wave function! — Mephist

But the issue is that they are not categorically distinct, as if one refers to a physical object and the other a mathematical object. It's different forms of the same thing.

Special relativity allows us to represent time as discontinuous?? Why? — Mephist

Special relativity is based in a discontinuous representation of time. Each frame of reference has a unique, and discrete "time" of its own. Therefore "time" in general consists of all these different unique and particular, "times", making it discontinuous.

On the contrary, in special (and general) relativity space and time have to be "of the same kind", because you can transform the one into the other with a geometrical "rotation" ( https://en.wikipedia.org/wiki/Lorentz_transformation ) simply changing the point of view of the observer. — Mephist

But that "rotation" is the faulty geometry we've been discussing. The circle is not real, pi is not real. Nor is the so called "continuity" assumed by this geometry real, because the idea that we can put a point anywhere in space is faulty. So general relativity has a double mistake. It starts from a discontinuous time (which ought to be represented as continuous), then it applies to this discontinuity the principles of a continuous space (which ought to be represented as discontinuous), to create the illusion of a continuous "space-time". In reality the concept of space-time consists of a faulty representation of time, with an attempt to fix it using a faulty representation of space.

Do you agree that the volume is an attribute of an object? — Mephist

The "volume" is what we assign to the thing, a judgement we make based on our principles of measurement. So we cannot really say that it is something intrinsic to the object, it's a judgement, and that's why we have different standards to measure volume.

I know, this explanation is not very "philosophical"... and, to say the truth, I don't really understand why is this such a philosophical problem :yikes: But what's the problem with this interpretation? — Mephist

Saying that we can assign the same volume to two distinct things, or that we can assign "5" to two distinct groups of objects, is really no different from saying that we can call two distinct things "a chair". This does not mean that there is an immaterial object which "chair" refers to, nor is there an object that "5" refers to, or an object which "volume" refers to.

But I think you are on the right track here, towards understanding what mathematics is incapable of helping us with, and that is description. Take your example of the two different pictures, where we use "5" to describe what's in the picture. From looking at one picture, and not knowing what five means, no one could ever know which aspect of the picture 5 is describing. So we look at two, and compare similarities, and perhaps deduce what "five" means. If "5" only gets meaning from referring to two or more different pictures, how is it describing anything in any of the pictures?

But I am afraid that's all what physics (at least contemporary physics) does: prediction. Nothing else! — Mephist

Do you not see this as misguided, and not proper science? If the goal of science is to predict, and not knowledge, understanding, then observations will be made under that bias, therefore not truly objective. The observations will be made with the goal of being able to predict, rather than the goal of knowledge and understanding. As I explained in the other post, knowledge consists of a lot more than just being able to predict. Thales predicted the solar eclipse with very little knowledge of the solar system, and that capacity to predict did not give him a comprehensive understanding of the solar system.

Maybe that is a problem, but it is a problem of physics since the beginning: Newton didn't know how to make sense of a "force" that acts from thousands of kilometers of distance. — Mephist

Perhaps this problem of pragmatism has always pervaded science to some exist, but I would argue that it has gotten much worse. There is a difference between establishing a principle, with the goal of applying that principle toward further understanding, and establishing a principle for the purpose of predicting.

The reality is that there are equations that work, and you can apply a mathematical theory made of imaginary things with imaginary rules that happen to give the right results. The real "ontological" reason why this system is able to "emulate" the experiments of the real world, nobody is able to explain. And it's not only about the use the square root of 2. — Mephist

This is the same illusion propagated by the sophists of ancient Greece. The equations work, Thales predicted the eclipse. Nobody was able to explain why they work, the "real ontological reason". But that doesn't mean they all should have buried their heads in the sand, and not proceeded to investigate the real ontological reason. There's a matter of completion. Equations that work for prediction is only a part of a complete understanding.

Mathematicians and philosophers of mathematics, with the presumed exception of platonists, reject the premiss that all "existence" is ontological existence. Specifically, they acknowledge that mathematical existence does not entail ontological existence. — aletheist

Ontology is the study of existence. Isn't it? How could there be a form of existence which isn't ontological existence? That sounds very contradictory to me.

Yeah well, let's say so... The way QM is formulated is: there are "observables" that represent the objects (or better: the results of experiments), and then there are other mathematical "objects" (such as wave functions- https://en.wikipedia.org/wiki/Wave_function) that are not meant to represent something that normally we could call "objects" in physics. — Mephist

Actually, if you analyze this situation closely, "wave functions" are produced from observations, so they are still mathematical representations of the movements of objects. The wave function is a use of mathematics to represent observable objects. There is no such separation between the representation of a physical particle and the wave functions, the wave functions represent the particles. They are of the same category, and I think the physicist treats the particle as a feature of the wave functions. Wave functions are used because such "particles" are known to have imprecise locations which they can only represent as wave functions. With observed occurrences (interactions) the particles are given precise locations. Wave functions represent the existence of particles when they are not being observed.

Well, the grid of points represents only the topology of space-time, not the metric.
Meaning: there are 4 integer indexes for each point (the grid is 4-dimensional), and then the ordering of the points, only defines which are the pieces of space-time that are adjacent (attached to each-other), not their size or the orientation of the edges. — Mephist

I don't quite get this. "Space-time" here is conceptual only, like "the square" we've been talking about, or, "the circle". Therefore, the positioning of the points is what creates the "object" called space-time, just like we could position points in a Euclidian system, to outline a line, circle, or square. What is at issue, is the nature of the medium which is supposed to be between the points, which accounts for continuity. The continuity might be "the real", what exists independently of our creations of points according to some geometrical principles.

So, we create a grid of points according to a geometry of space-time, like we might position points according to Euclidian geometry. There is an assumed figure produced by this positioning like a square etc., and we might assume that the figure could exist as a real physical object. If the geometry we use to position those points, is not consistent with what is allowed for by the real positioning of objects, because the medium is not continuous, or if the nature of the continuity is completely misunderstood, then I would say that this is a problem. This is what we see in Euclidian geometry. The geometry allows that we can place a point virtually anywhere. But when we create the figures which connect the points, like a square or a circle, we see that there is a problem with this assumption, that we can put the point anywhere we want. Notice that the problem is with the conception itself, it has nothing to do with "the real". The idea that we can conceive a point anywhere is false, as demonstrated by the square root problem. Conceiving of continuity in this way, such that it allows us to put a point anywhere is self-defeating. Therefore we need to change our concept of the continuity of "space".

Now. let's add time to the mix. We already have a faulty conception of space assumed as continuous in a strange way which allows us to create irrational figures. Special relativity allows us to break up time, and represent it as discontinuous, layering the discontinuous thing, time, on top of the continuous, space. Doesn't this seem backward to you? Time is what we experience as continuous, an object has temporal continuity, while space is discontinuous, broken up by the variety of different objects.

So, basically, every "object" that is composed of well-identifiable parts can be considered to be a natural number, if you specify how to perform the arithmetical operations with the parts. — Mephist

Your analogy is faulty, because what you have presented is incidents of something representing what is meant by the symbol "5". So what you have done is replaced the numeral "5" with all sorts of other things which might have the same meaning as that symbol, but you do not really get to the meaning of that symbol, which is what we call "the number 5". The point being, that for simplicity sake, we say that the symbol "5" represents the number 5. But this is only supported by Platonic realism. If we accept that Platonic realism is an over simplification, and that the symbol "5" doesn't really represent a Platonic object called "five", we see instead, that the symbol "5" has meaning. Then we can look closely at all the different things, in all those different contexts, which you said could replace the symbol "5", and see that those different things have differences of meaning, dependent on the context. Furthermore, we can also learn that even the symbol "5" has differences of meaning dependent on the context, different systems for example. Then the whole concept of "a number" falls apart as a faulty concept, irrational and illogical. That's why you can easily say, anything can be a number, because there is no logical concept of what a number is.

Well, the problem is that what you say doesn't seem to be in any way "compatible" with current physical theories. And current theories are VERY good at predicting the results of a lot of experiments.
To me, it seems VERY VERY unlikely that a simple physical theory based on a simple mathematical model can be compatible with current physics at least in a first approximation. The physical world seems to be much more complex than we are able to imagine... — Mephist

Prediction is not a good indicator of understanding. Remember, Thales predicted a solar eclipse without an understanding of the solar system. All that is required for prediction is an underlying continuity, and perhaps some basic math. I can predict that the sun will rise tomorrow morning without even any mathematics, so the math is not even prerequisite, it just adds complexity, and the "wow' factor to the mathemagician's prediction. So, continuity and induction is all that is required for prediction. Mathematics facilitates the induction, but it doesn't deal with the continuity. Real understanding is produced from analyzing the continuity. This is an activity based in description, and as I mentioned, is beyond the scope of mathematics.

Again, we encounter the problem of pragmatism. If prediction is all that is required, then we gear our epistemology toward giving us just that, predictability. If this is easiest done using false premises like Platonic realism, then so be it. But we do this at the expense of a real understanding.

That's what abstraction is! It's giving a name to something immaterial in order to manipulate it. — fishfry

I believe you do not have a very thorough education in philosophy, or you would not characterize "abstraction" in this way. Abstraction is a process. That process is sometimes described as producing a thing which might be called "a concept", or "an abstraction". There might be a further process of manipulating that thing called "an abstraction", but notice the separation between the process which is abstraction, creating the immaterial thing called an abstraction, and the process which is fixing a name to the supposed "immaterial thing" (an abstraction) and manipulating it.

To begin with, we need to analyze that process of abstraction, and justify the claim that an immaterial object is produced from this process. If there is no immaterial object produced, then the name which is supposedly given to an immaterial object, simply has meaning, and there is nothing being manipulated except meaning. But if you are manipulating meaning you stand open to the charge of creating ambiguity and equivocation. This is why we separate logic, which is manipulating symbols, from the process of abstraction which is giving meaning to those symbols. So it is very good to uphold this principle. In logic we manipulate symbols, we do not manipulate "something immaterial" (meaning) which the symbols represent. What the symbols represent is determined by the premises. The "something immaterial" (meaning) precedes the logic as premises, and extensions to this, as new understanding, may be produced from the logical conclusions, but what is manipulated is the symbols, not the immaterial thing (meaning).

You say you have found a loophole that allows you to accept the world yet reject ... something. The square root of 2. — fishfry

I don't say that I've found a "loophole", I say that there is weakness. And, it's not me who found this weakness, which is a deficiency, it's been known about for ages. You look at this deficiency as if it is a loophole, and insist that the loophole has been satisfactorily covered up. But covering a loophole is not satisfactory to me, I think that the law which has that deficiency, that weakness, must be changed so that the loophole no longer exists.

* If we believe in the rationals, we can build a totally ordered field containing the rationals in which there is a square root of 2. — fishfry

Until you provide me with a definition of "field" for this premise, your efforts are futile. If a field requires set theory, I'll reject it for the same reason I rejected your other demonstration. If you can construct a field with square root two, without set theory, then I'm ready for your demonstration. If you produce it I'll make the effort to try and understand, because I already believe that you would need to smuggle in some other invalid action, because that's what's occurred in all your other attempts.

Yes I know you already believe that. The question is whether you're willing to believe that it has mathematical existence. You ask me what that is but I've given you many demonstrations of the mathematical existence of 2–√2; as the limit of a sequence, as an extension field of the rationals, as a formal symbol adjoined to the rational numbers. All these things are part of mathematical existence. You will either have to take my word for it, or work with me to work through a proof of the mathematical existence of 2–√2 . — fishfry

You never explained to me what you mean by "mathematical existence" that remains an undefined expression.

You showed me that you have a psychological block in dealing with symbology; leading to a massive area of ignorance of math; leading to making large errors in your philosophy. That's my diagnosis. — fishfry

It's not the case that I have a block in dealing with symbology, but what I need is to know what the symbol represents. Until it is explained to me what the symbol represents I will not follow the process which that symbol is involved in. I believe that whatever it is that is represented by the symbol, places restrictions on the logical processes which the symbol might be involved in. Supposedly, you could have a symbol which represents nothing (though I consider this contradiction, as a symbol must represent something to be a symbol), and that symbol might be involved in absolutely any logical process. However, once the symbol is given meaning, the logical processes which it might be involved in are limited. So if you start with the premise that a symbol might represent nothing, I'll reject your argument as contradictory.

Like Captain Ahab, who only has fictional existence in a novel. Nevertheless statements about him can can truth values, such as whether he's the captain of the Pequod or the cabin boy. So there's fictional existence. — fishfry

"Fictional existence" is contradiction plain and simple. To be fictional is to be imaginary, and to exist is to be a part of a reality independent of the imagination. If you are handing to "existence" a definition which allows that an imaginary, fictional thing, exists, then it's not the rigorous philosophical definition which I am used to. I think that if you cross this line, you have put yourself onto a very slippery slope, denying the principles whereby we distinguish truth from falsity.

I can indeed specify 2–√2; but when I do so I am merely "expressing my epistemic stance" toward 2–√2; yet not necessarily saying anything about 2–√2 itself, whatever that means. I think this is Metaphysician Undercover's point perhaps.

And then "expressing my epistemic stance" towards a mathematical object is what I mean by endowing that object with mathematical existence. Perhaps this is the distinction being made. — fishfry

I interpret this as your "epistemic stance" requires Platonic realism as a support, a foundation. I deny Platonic realism, so I think your epistemic stance is ungrounded, unsound.

The result is obtained by purely mathematical considerations on objects made of complex number functions (the states are the eigenvalues of the system's wave function), but the effects predicted using a purely mathematical abstract model generate real physical predictions in the form of measurable quantities. That seems very strange if mathematical objects are only symbols subject to arbitrary rules. In some way, the rules that we invented for the symbols correspond exactly to some of the "rules" of the physical (real) world. — Mephist

The mathematical system being employed premises that a symbol represents an object, and that each time the symbol appears within an expression, like an equation, it represents the very same object. Any conclusions produced must uphold this premise.

f you consider geometric spatial figures as real physical objects, there are a lot of "problems" with them: first of all, they are 2-dimensional (or 1-dimensional, if you don't consider the internal surface), and all real physical objects are 3-dimensional. — Mephist

Right, clearly there are "problems" if we represent mathematical figures as real objects. Notice I removed your qualifier, "physical" objects. If we begin with a statement as to the nature of an "object", a definition, such as the law of identity, then we must uphold this definition. If the claim is that a "mathematical object" is fundamentally different from a "physical object", such that the same definition of "object" cannot apply to both, then we need to lay out the principles of this difference so that equivocation can be avoided.

They are not real objects, and there is no problem with the distinction between finite or infinitesimal distances: it works even if you consider space-time as discrete. In fact, in practice it's very common in GR simulations to approximate space-time as a 4-dimensional discrete grid of points. — Mephist

The problem is that any such "grid of points" is laid out on a spatial model. If a square is an invalid spatial model, then so is the Cartesian coordinate system Then "space-time" itself is improperly represented. If the claim is that "space-time" is not supposed to be an object, then we have nothing being represented except mathematical objects, and no grounding principle, defining what a mathematical object is.

The main point to keep in mind with physical models is that they don't have to be considered the real thing: they simply have to WORK as the real thing. — Mephist

Pragmatism is not the answer, it is the road to deception. Human objectives often stray from the objective of truth. When we replace "the truth" with "they simply have to work", we allow the deception of sophism, because "what it works for" may be something other than leading us toward the truth..

Now, if you think that the distinction between measures expressed with rational or with real numbers is essential in your theory (represents some important characteristics of the real physical space), I don't see any other way other than making lengths become discrete at the microscopical level. — Mephist

Yes, as I explained, I agree that this is the way to go. Once we recognize that this is what's needed, we can proceed towards a proper analysis of space and time, to establish some principles.

Why should this background of mathematics remain a secret? And is it merely aesthetic in nature (a consideration of mathematical beauty alone)? — fdrake

The secret background is the intent of the author. So long as the intent remains a secret, pragmaticism remains unacceptable because we do not understand the end. "It works" has no meaning when the end remains a secret.

OK, division and multiplication are not symmetrical for integers, because integers are "quantized": you can't give one candy to three children, because candies are "quantized". But physical space is not quantized, or is it? The mathematical description of continuous measures is not inconsistent: there are several ways to make them at least as consistent as natural numbers are.
So, if integers (quantized) objects exist in nature, why shouldn't continuous objects exist? — Mephist

This is good, a very good start. Suppose there actually is this distinction in "objects". Suppose there are both "quantized objects", and "continuous objects". We would need different principles to apply to each one. Then we would need to establish some principles of application, are we working with a continuous object, or a quantized object. Thirdly, we'd need some principles to relate the continuous to the quantized. For example, to me time appears to be continuous, and space appears to be quantized. If this is the case, then we need different principles for modeling time than we do for space, and some principles to relate these two systems to each other.

So is the 3,4,5 triangle really straight or not? I don't understand... — Mephist

The 3,4,5, triangle is just as faulty as the square, because it is validated by the faulty Pythagorean theorem. The point was that I can make a 3,4,5 triangle without any knowledge of the Pythagorean theorem. So let's say I am in the habit of doing this, and I know absolutely nothing about the theory. I am producing right angles at will, and I believe that the right angle is a perfectly natural thing. Then I learn about the square, and realize that there is a problem with the right angle, and therefore it is not a natural thing. Likewise, I could tie a string to a stake, and make a simple compass, and go around creating circles at will, thinking that the circle is a natural object, until I realize the irrationality of pi. This tells me that these are not natural objects, they cannot exist.

So now I want principles to explain why I can make a figure which is theoretically impossible to make. I make excuses, I rationalize that I am not making a "perfect" square, or a "perfect" circle, and the theory says that such "perfect" figures are impossible. But there's something fundamentally wrong with this rationalizing. The theory is supposed to give me the "ideal", the perfect geometrical figure, and my inability to construct it ought to be due to my imperfect procedure. But here, what is indicated by the mathematics is that this supposed "ideal" is actually less than perfect, so that the more perfect the procedure is, all it does is demonstrate how much less than ideal the ideal is. Therefore I can conclude that the "ideal" is not the ideal at all, and there is a fundamental contradiction here which tells me that we need is a better, more ideal "ideal".

OK, so what can I do with identities?

If I cannot refer to them with names, I would say that it's impossible to speak about identities. So, they surely cannot be used in logic arguments. Logic is basically manipulation (operations) of language, isn't it? — Mephist

Aristotle established and used the law of identity as a fundamental tool against the deception of sophism. So let's assume as you say, that logic is the manipulation of symbols, and it doesn't say anything about any real things. To say something about a thing is an act of description, and this is distinct from logic which works with symbols. Therefore if a logician is claiming to say something about real things, we can charge that person with sophism, deception. This is what we do with "identity" then, use it to demonstrate that someone is falsely claiming identity. So how do we approach set theory, doesn't it look like sophism, a false claim of identity, to you?

But Einstein's relativity is based on differential calculus and real numbers. How can it be correct, if the whole system is wrong? — Mephist

Good question, I think the jury is out still on that decision.

OK, continuous change cannot be identified by a finite number of steps. But does this prove that continuous change cannot exist? — Mephist

No, it does not prove that continuous change does not exist. But it proves that numbers, which represent change in finite steps, are the wrong tool for representing continuous change. This is where mathematicians demonstrate their stubbornness. They want numbers to be capable of representing everything, so they twist and turn the systems, adding layer upon layer of sophistication, mixed with deceptive sophism, and voila, numbers represent the fundamental reality, or even more extreme, numbers are the fundamental reality. It's like physicalism, or scientism, but it's more properly called mathematicism. and it appears to be the root of the other two. People think that mathematics is all we need to describe reality, when in reality mathematics cannot describe anything.

So let's revisit this root problem. Numbers, which demonstrate finite increments of difference, cannot properly represent continuous change. Either we assume that continuity is not something real, therefore numbers can be applied to all of reality, or we assume that continuity is real, and numbers cannot be applied to all aspects of reality. I suggest the latter is what is really the case. But then mathematicism looms menacingly in front of me, and I feel it my duty to demonstrate the sophistry of the mathemagician.

As I said I find it nihilistic because you must then reject all of the modern world that sprung from that basic act of abstraction. — fishfry

This is not the case. To reject that "the abstraction" exists as an object does not require that I reject abstraction. What I reject is any instance where an abstraction is presented as an object.

Now that you mention it, that makes perfect sense relative to your neo-Pythagoreanism. By that I mean that you still profess to be "Shocked, shocked, I tell you!" at the fact that the square root of 2 is irrational. The rest of the world got over that a long time ago. — fishfry

I'm not "shocked" at the fact that the square root of two is irrational, what shocks me is that the rest of the world got over this.

Question: I get that you do not believe in the ontological existence, however you personally define that, of 2–√2. My question is:

Do you believe in the mathematical existence of 2–√2?

If you say yes, then our disagreement is over whether mathematical existence is sufficient for ontological existence.

If you say no, then our disagreement is whether 2–√2 has mathematical existence.

So, do you think 2–√2 has mathematical existence; even though you maintain that's not sufficient justification to grant it ontological existence as you define it? — fishfry

I'd answer this, but I really don't know what you would mean by "mathematical existence". Many things can be expressed mathematically, but what type of existence is that? I suppose the short answer is no. The symbol √2 does not stand for anything with real "existence". I agree that is has a large amount of mathematical significance, and it is quite important mathematically, so the symbol definitely has meaning, but I don't think I'd agree that the symbol stands for anything which has "existence", in any proper sense of the word. All existence is "ontological existence" so it makes no sense to try and separate "mathematical existence" from "ontological existence".

Do I strike you as a person who expects people to do what I suggest?

[q

I'm a peacenik and we never should have gone in. I think we should leave tomorrow morning. — fishfry

The problem with this perspective is that once you've carried out an action, and determined that it was a mistaken action, you cannot simply undo the mistake. So "we never should have gone in" does not justify "we should leave". When you put yourself into such a position, as having made a mistake, you are forced to live with the consequences, and the consequences of such a mistake are retributive requirements. It is utterly wrong to walk away from a mistaken action pretending that it didn't happen.

You are saying that there are 2 books and two fish and 2 schools of thought; but there is no 2 in the abstract.

Well, imagining or mentally conjuring up a "thing" that is 2, by itself, is one of the greatest intellectual leaps of humanity. As I've noted before, you appear to reject the very concept of abstraction. — fishfry

Right, my argument is that there is no such thing as an abstract object represented by "2". I replace this supposed object with something closer to the truth, "what 2 means". The symbol "2" has meaning. The key point I make, which some argue against, is that the meaning of "2" varies according to circumstances, context. This variance, or difference, indicates that it is impossible that "2" signifies an object. The Platonic realist argues that this is a difference which doesn't make a difference, but in making this argument, the realist has already invoked contradiction. This contradiction supports the realist's category mistake of failing to distinguish between a particular object, and a universal (meaning).

Well, imagining or mentally conjuring up a "thing" that is 2, by itself, is one of the greatest intellectual leaps of humanity. As I've noted before, you appear to reject the very concept of abstraction.

The invention of the concept of number was a great leap forward for mathematics and also for civilization. — fishfry

I would say that you call this a "leap forward" as determined in relation to a pragmatist perspective. This move serves a purpose. But in relation to a true ontology, it is a blatant falsity put forward for a purpose. Therefore it is a move of deception. The pragmatist, from the perspective of the metaphysician, is a deceiver, a sophist.

That let us study 2 + 2 without having to say 2 fish plus 2 fish and then having to re-calculate 2 elephants plus 2 elephants, and then still not being sure about 2 birds plus 2 birds. — fishfry

Such a study is the study of meaning, it is not a study of objects. When it is presented as a study of objects it is a deceptive presentation.

t's the power of abstraction that allows us to handle all these cases at once.

You reject abstraction. You're not wrong. It's just a nihilistic philosophy of math and of civilization. Everything about our lives is abstraction. We can't live without abstraction. How do you live without abstraction? How do you function, not believing in numbers? — fishfry

I do not reject abstraction, I take it for what it is, and that is not a process of creating objects, it is a process of generalization. You don't seem to understand what "abstraction" means. Fundamentally, abstraction replaces particulars with generalizations, universals. If the universal (the product of abstraction) is presented as a particular (object), what is proposed is clearly a false proposition.

So 3 x 3 = 2. That is, 3 is a number that, when squared, results in 2. So in the integers mod 7, 3 is the square root of 2. Just to startle people I'd go as far as to write — fishfry

This is not true. It's false to claim that 3 is the square root of 2, just because you've taken seven away, just like it would be false to claim that one o'clock in the afternoon is the same as one o'clock in the morning, just because you've taken twelve hours away. You've just presented a mathemagician's trick, pure sophistry.

Like every statement in math, its truth value depends on the context. In the context of the integers, the statement is false. In the context of the integers mod 7, the statement is true. — fishfry

Yes, context is key, as I stated above, and the importance of context is reason why a numeral cannot refer to a number which is an object. The numeral has a meaning which is context dependent. The numeral "2" has a different meaning in mod 7 from the meaning it has in the rational numbers. In your example, you are conflating two distinct concepts. Your use of "the square root of two" is right out of context, because "2" has a different meaning in mod 7. Therefore your use of "2" is irrelevant to any common use of "the square root of two", because "2" has a different meaning in mod 7, like "one" has a different meaning as "one o'clock" from the meaning it has as a rational number. If you deny this, you arguing by equivocation.

This is what I have always felt...but as someone who does not believe in any gods, that may be more normal. — ZhouBoTong

The way I laid it out, I omitted some key points which add complexity, that I will now try to elucidate. As demonstrated by Plato and Aristotle, we define "good" as the thing wanted, the good is what is wanted. In human beings, this manifests as a situation described by, "the thing is said to be good because it is what is wanted". If in God, we invert that to say, "God wants the thing because it is good", and this would mean that its goodness is independent of God, we need to be able to account for how the goodness of a thing makes God want the thing. This is explained by rationality. The intellect apprehends the goodness of the thing, and this is why it is the thing is wanted. The thing moves the rational intellect towards it, because it is good.

Now we still have to deal with the complexity that free will adds. In human beings, the intellect can apprehend something as good, yet the will might still move us in a different direction. So when we say that "the good" is what moves the will, as the thing wanted, and cause of human action, we call this the "apparent good". The "real good" is the thing which the omniscient intellect would apprehend as good. But the intellect doesn't have the capacity to necessarily move the body toward that thing, because the will, which is free, is what moves the body. If I understand the Thomistic argument correctly, if an intellect apprehends the real good (and this might require an omniscient intellect independent from a body), it would also be apprehended as the apparent good, and the individual would be incapable of acting otherwise.

Assuming morality is external to god then we would not need god to objectify morality, but how could we ever define morality objectively? I would also think that the Christian/Muslim version of god would be objective in relation to the goal of entering heaven (although why is that goal desired? seems to become subjective). I guess any version where god=nature creates objectivity as god is no longer a subject (sort of)...but those versions of god rarely mandate morality??

I get I am a bit wishy-washy on this, but I think I am generally in agreement with what you are saying.

Maybe I was a bit flippant with my use of "objective"? I agree it is complicated. — ZhouBoTong

I think the issue is that it would require an omniscient intellect to accurately determine the "real good". Any time that a human being, or human beings attempt to determine the real good, they are actually only determining the apparent good, what is wanted by them. It is determined as "good", because it is what they have decided that they want, and a good which is determined in this way, good because it is what is wanted, rather than wanted because it is good, is an apparent good. Human beings haven't got the capacity to determine what is good independent of what they want, because this would require separating themselves from their desires. That's why Aquinas argues that only an independent intellect, one separate from, and not influenced by the body, could make such determinations.

So, we can say, and assume that there is a real good, independent from human wants and desires, and try to use this as the basis for an objective morality, but it doesn't do us any good. We haven't got the capacity to separate ourselves from our wants and desires, so we haven't got the capacity to determine the real good. As different human beings attempt to dictate this real good, it would rapidly become corrupted by these individuals' wants and desires. Therefore we would have to determine a "God's perspective", which we could agree on, and attempt to determine the real good from this "God's perspective". But isn't assuming "God's perspective" the same thing as assuming God?

All that the sanctions do is to keep oil prices higher. — ssu

Actually sanctions can have the reverse effect. If the sanctioned country can establish a black market, the price might be lower and the oil could flood the world market as an uncontrolled source, lowering prices. This may have actually happened when Iraq was sanctioned; notice that Bush was very anxious to get rid of Saddam, and oil prices soared afterwards. The problem is that the black market puts money into the wrong hands while the average person of the sanctioned country suffers.

OK, but I don't understand how all this can be related to irrational numbers. — Mephist

The problem of irrational numbers arose from the construction of spatial figures. That indicates a problem with our understanding of the nature of spatial extension. So I suggested a more "real" way of looking at spatial extension, one which incorporates activity, therefore time, into spatial representations. Consider that Einsteinian relativity is already inconsistent with Euclidian geometry. If parallel lines are not really "parallel lines", then a right angle is not really a "right angle", and the square root of two is simply a faulty concept.

Division between integers is repeated subtraction ( A/B you count how many times you have to subtract
B from A to reach 0 ); multiplication between integers is repeated addition ( A*B you add A B times starting from 0 ). — Mephist

Ok, we can look at division as a matter of asking how many times we can subtracting B from A, as you say. The issue is that in many cases one does not reach 0, and this is what we call the remainder. So the problem is, how do we deal with the remainder. If we are dividing ten by three, we get a remainder of one. In this case, you might divide the unit into three. But in most practical circumstances, if you were dividing a group of objects, it would be unfeasible to split up one of the objects, rendering it useless. So the remainder is very often a problem in division.

The definitions are quite symmetric between each-other. What do you mean by "division presupposes no such base units"? OK, A/B is not an integer ( there is a reminder ) if A is not a multiple of B. Again: what does this have to do with physical space-time? — Mephist

No, division and multiplication are not at all symmetrical, because you never have a remainder in multiplication. In multiplication, you take a designated number as the "base unit", a designated number of times, and you never end up with a remainder. You have no such "base unit" in division, you have a large unit which you are trying to divide down to determine the base unit, but you often end up with a remainder.

Evidence of this difference is the existence of prime numbers. These are numbers which we cannot produce through multiplication. We can still divide them, knowing there will be a remainder, but that doesn't matter, because there's often a remainder when we divide, even if the dividend is not prime.

By using compass and straightedge (as described by Euclides) you can build all the lengths that can be obtained from the integers with a finite sequence of operations of addition, subtraction, multiplication, division, and taking square roots (https://en.wikipedia.org/wiki/Straightedge_and_compass_construction). Square roots are not so special from this point of view. — Mephist

On paper you produce "a representation" of the Euclidean ideals. That representation is something completely different from the square root, which is part of the formula behind the representation which you draw on paper. When I want to lay out a square corner, a right angle, on the ground, I might use a 3,4,5, triangle. In this exercise I am not using a square root at all. I could make this square corner without even knowing the Pythagorean theorem, just knowing the lengths of 3,4,5. But if one side of the right angle is to be 5, and the other side 6, I'll need to know the Pythagorean theorem, and then figure the diagonal as the square root of 61 if I am going to make my right angle.

OK, I translate this as: you can always distinguish a thing (meaning: physical entity) from all the other things. Not quite true in quantum mechanics, but let's assume it is. — Mephist

That's not quite right. We, as human beings, cannot necessarily distinguish two distinct things, due to our limited capacities of perception and apprehension. So it's not quite right to say that you can always distinguish a thing from all other things. A thing is distinct from other things, but we cannot necessarily distinguish it as such. And that difference may be a factor in quantum mechanics.

OK, but when you give a name to a concrete object, the name is a reference that identifies always the same concrete object, isn't it? — Mephist

Right, but to perceive a thing, name it "X", and then claim that it has the "identity" of X, is to use "identity" in a way inconsistent with the law of identity. You are saying that the thing's identity is X, when the law of identity says that a thing's identity is itself, not the name we give it. The law says a thing is the same as itself, not that it is the same as its name.

Consider that human beings are sometimes mistaken, so it is incorrect to say "the name is a reference that identifies always the same concrete object". The meaning of the name is dependent on the use, so when someone mistakenly identifies an object as "X", when it isn't the same object which was originally named "X", then the name doesn't always identify the same concrete object. And, there are numerous other types of mistakes and acts of deception which human beings do, which demonstrate that the name really doesn't always identify the same concrete object, even when we believe that it does.

Anyway, my main objection to what you say is that you don't explain how to use the fact that square roots are irrational (some of them) to deduce something about physical space-time. A physical theory in my opinion (even if limited) should be falsifiable in some way (meaning: should be usable to predict that something should happen, or that something else can't happen). And if it's not physics but only mathematics, then there should be some kind of logical "proof". Don't you agree? — Mephist

Do you recognize that Einstein's relativity is inconsistent with Euclidian geometry? Parallel lines, and right angles do not provide us with spatial representations that are consistent with what we now know about space, when understood as coexisting with time. My claim is that the fact that the square root of two is irrational is an indication that the way we apply numbers toward measuring space is fundamentally flawed. I think we need to start from the bottom and refigure the whole mathematical structure.

Consider that any number represents a discrete unit, value, or some such thing, and it's discrete because a different number represents a different value. On the other hand, we always wanted to represent space as continuous, so this presents us with infinite numbers between any two (rational) numbers. This is the same problem Aristotle demonstrated as the difference between being and becoming. If we represent "what is" as a described state, and later "what is" is something different, changed, then we need to account for the change (becoming), which happened between these two states. If we describe another, different state, between these original two, then we have to account for what happens between those states, and so on. If we try to describe change in this way we have an infinite regress, in the very same way that there is an infinite number of numbers between two numbers.

If modern (quantum) physics demonstrates to us that spatial existence consists of discrete units, then we ought to rid ourselves of the continuous spatial representations. This will allow compatibility between the number system and the spatial representation. Then we can proceed to analyze the further problem, the change, becoming, which happens between the discrete units of spatial existence; this is the continuity which appears to be incompatible with the numerical system.





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If we remove any a moment or time interval if you prefer, then all subsequent moments or time intervals become undefined. Time with no start means no initial moment/interval, so the basic argument therefore still holds. — Devans99

How would you suggest removing a moment from time?

We can arbitrarily designate instants to mark off intervals of time with fixed and finite duration, but we cannot "remove" those, either. — aletheist

We cannot even mark off intervals of time by designating instants, because as special relativity indicates such designations would differ depending on your frame of reference. So to use Einstein's example; what you might call "the instant" that the lightning strikes the embankment is different depending on whether you are on the train or whether you are standing on the platform. But it really doesn't make any sense to talk about instants in time with the premise of special relativity.

Aristotle and Aquinas don't accept that a person can sin intentionally. — frank

Of course they accepted that. It was well described by Plato, accepted by Aristotle, and described even more thoroughly by Augustine. The solution to this issue involves free will. We can freely choose to do what we know is wrong.

t's not about will, it's about how Aristotle defined good and evil. Strip your concept of goodness down to something mundane, mechanical, and naturalistic. A good thing is an effective thing or a beneficial thing. Any time you act, you're trying to benefit yourself in some way. It's really more than that, though. It's close to this: for Aristotle, it's like you're a vector and "good" is a name for the direction you're trying to go in. "Evil" is what you're trying to leave behind. — frank

Aristotle provided a distinction between real good and apparent good, which Aquinas developed further.

This sounds like an attempt at objective morality...? . — ZhouBoTong

If you believe in God, there is an objective morality, objectified by God. This is the basis for the "real good". But this brings up Plato's Euthyphro question. Is the good a real good because it's what God wants, or does God want it because it's good? In monotheism this is not a significant issue, but for Plato it was, because the different gods might want different things, resulting in incompatible goods, if "good" is defined by what a particular god wants. So to maintain a truly objective good we must say that God wants it because it's good. This places "the good" as external to God. But if the real good is necessarily external to God, why do we need "God" to objectify morality?

1. Assume time has no start
2. Then there is no first moment
3. If there is no nth moment there is no nth+1 moment
4. But we have moments (contradiction)
5. So time must have a start — Devans99

Your argument assumes that time consists of distinct and individual moments. But we experience time as continuous. These two have not been shown to be compatible, so to proceed with your argument you need to demonstrate the real, distinct, individual moments that time consists of, to support your premise. This is the matter which aletheist has brought up.

. He never wanted a war with Iran, and in fact wants to negotiate a better nuclear deal. — NOS4A2

That appears to be what he wants. But his diplomacy is terrible. The Iranian say it's that of a terrorist. Should we expect them to negotiate with terrorists? We go to war with terrorists.

You're losing it, MU — tim wood

You never thought I ever had it, that's why you'd tell things like get back on your meds. Is the fact that you keep addressing my post with such nonsense evidence that you're losing it?

So there cannot exist any fundamental minimal length of physical space (kind of a microscopic indivisible stick) that can be oriented in any direction. If there is such a thing, every physical object at the microscopic scale should be made of tetrahedrons, or something similar. So circles and squares are really only approximations of the real "physical" shapes. Is your idea something of this kind? If not, in what other way can you make all the lengths be rational numbers? — Mephist

I do not think that "there cannot exist any fundamental minimal length of physical space" is a reasonable starting point. If space has physical existence, then it has limitations just like any other physical things. So we ought to assume that space must have some fundamental "shapes" just like you suggest. Once it was believed that space is an aether, so the fundamental shapes were waves.

A wave is active, so it requires the passing of time, for its activity. So let's assume "space" is an active medium. Now suppose we try to make something static, like a circle or a square, within this medium which is active. The shape won't actually be the way it is supposed to be, because the medium is actively changing from one moment to the next. So if we want to make our shape, (circle or square), maintain its proper shape while it exists in an active medium, we need to determine the activity of the medium, so that we can adjust the shape accordingly. Understanding this activity would establish a true relationship between space and time, because defining this activity of space would provide us with a true measure of time.

You mean that there is no defined physical procedure to divide a generic geometrical segment by another? — Mephist

What I am talking about specifically, is dividing numbers. Divide ten by three, and you have a remainder of one. It is the remainder which is a problem. When we multiply numbers we never get remainders, yet we tend to treat division as the inversion of multiplication. It's actually quite different from multiplication because multiplication starts from premises of fundamental base units, whereas division presupposes no such base units.. So I think we need to pay close attention to this fact, that constructing a magnitude through multiplication is really a completely different process from destroying a magnitude through division.

If you take two generic segments of whatever length, you can always build a third segment that is proportional to their ratio (whatever it is, even irrational). — Mephist

I don't understand how you would build an irrational length segment.

Sorry for the intrusion, but I am curious of this issue (only one premise: I didn't study philosophy :yikes:, so, for example, I don't really understand why this "law of identity" is so important...).
However, that's my question: how do you refer to an object instead of to it's value? I mean: if every symbol refers to a different object, even if the symbol is the same as the one that you used before, you can never refer to the same object twice, can you? — Mephist

What the law of identity says is that a thing is the same as itself. This puts the identity of the thing within the thing itself, not as what we say about the thing, or even the name we give it. This is a fundamental ontological statement about what it means to be a thing. First, to be a thing is to have an identity (but this is irrelevant to the identity we give the thing, it is the identity that the things has by virtue of being the thing that it is). Second, a thing is unique, and no two things are alike, and this is the principle Leibniz draws on. So the law of identity is not concerned with how we refer to objects, it is a statement concerning the real existence of objects, as the objects that they are, independent of what we say about them.

It's a good thing we all have a sense of humour here. LOLOL!

Do abstractions exist at all? I suggest they don't. A number line "exists" only as an abstraction, but this is not true existence. It's just a concept, in which a set of logical/mathematical properties are considered abstractly. The same is true of numbers, whether rational or irrational. "3" doesn't exist, but collections of 3 objects exist - so we can think abstractly about 3-ness. Neither does Pi exist; nevertheless we can abstractly consider the fact that all "circles" (another abstraction) have Pi as the ratio between their circumference and diameter. — Relativist

That's right, we covered this earlier in the thread. We need to distinguish between three things, the symbol, what the symbol means (abstract concept, what you called "3-ness"), and what the symbol is being used to refer to (physical object, and groups of physical objects). There appears to be an inclination in this thread, to conflate the latter two things, and claim that what the symbol means, and what it refers to, are one and the same thing, i.e. the symbol simply refers to a concept. This is an ontological error, meaning is not a thing.

LOLOL. — fishfry

Alethiest and I go way back. We're not too far apart metaphysically, only disagreeing on some finer points. But in relating ontology to mathematics, altheist employs intentional vaguery and ambiguity in terms, as well as outright contradiction to support unreasonable mathematical principles. So we part here.

I am curious to know: do you have an answer to this question? — Mephist

I think there are two issues becoming evident. One is that we do not know how to properly represent space. The irrational nature of the "square", and the "circle", as well as the incompatibility between the "point" and the "line" indicate deficiencies in our spatial representations.

The other is that we do not know how to properly divide something. There is no satisfactory, overall "law of division", which can be consistently, and successfully used to divide a magnitude. We tend to look at division as the inversion of multiplication, "how many times" the divisor goes into the dividend. Because there is often a remainder, division really cannot be done in this way. The "square root of two" is amore complex example of this simple problem of division, the issue of the remainder.

You want to make a mathematical claim (sqrt 2 doesn't exist) but you won't accept a mathematical response. Makes for pointless conversation. — fishfry

Your solution involves a violation of the fundamental laws of logic, the law of identity (as explained on the other thread), therefore I reject it. My argument is that the problem is fundamentally an ontological problem, and the objective ought to be to resolve the problem with principles which are ontologically sound.

I have already explained to you at length that set theory is based on the law of identity; and that the mathematical equals sign expresses identity between two expressions. — fishfry

As I demonstrated in the other thread, the "identity" expressed here as "equals", is not consistent with "identity" as expressed by the law of identity. Therefore despite your claim that set theory is based in "identity", it uses a form of identity which is in violation of the law of identity.

To state the problem succinctly, set theory allows that two distinct things have the same identity, in the same way that we might say two distinct things are equal. The faulty premise is that things with the same value "2" for example, are the very same thing. In other words set theory premises that, "2" refers to an object, rather than a value assigned to an object. It is a category mistake to treat what "2" refers to, as a particular object, rather than as a universal principle.

Define "valid operation." You should have been around to make your current argument about 1700BC when the Sumerians were calculating the square root of two (and its reciprocal) on cuneiform tablets. They would have appreciated your perspective. — jgill

A valid operation is one carried out with consistency according to consistent laws of a system. The example of imaginary numbers, as well as the various different attempts to prove the square root of two, demonstrate that there is a lack of consistency to the square root procedure.

I can't relate to your analogy. We haven't established that there is such a thing as a "numberline". I think the other thread demonstrated that there is afundamental inconsistency between numbers, which mark non-dimensional points, and a line which has spatial dimension. The issue is how is the incremental increase (discrete quantity) between two numbers, accurately represented by a line?

My argument is that both of these problems, the incompatibility between points and a line, and the irrational nature of the square root of two, are each a part of the same problem. That problem is the way that we represent space, in dimensions. The spatial representation is incompatible with the numeric representation. The trend in mathematics, at least from Descartes onward, has been to adapt the numerical system to account for the problems encountered by this incompatibility. I am arguing that this is the wrong approach, what is really at fault here, is our spatial representation, and this is what needs to changed.

* Now the set {a+b2–√:a,b∈Q}{a+b2:a,b∈Q} happens to have a mathematical structure identical to that of the rationals; and in addition, it contains a square root of 2. — fishfry

It appears like we need to go back over the law of identity, and the difference between identical and equal. Remember, I don't accept set theory on the basis that it violates the law of identity, so why give me a proof based in a set?

Fact: If you believe in the rationals, you must believe in the rationals augmented by the square root of 2. — fishfry

Why do you believe this? "The square root of two" has no valid meaning in the rational number system. This means that taking a square root is not a valid operation. So your claim is like saying if you believe in the rationals then you believe in the rationals augmented by a tree. You can't augment a system by something which is inconsistent with the system, that creates a contradiction, or at best, meaninglessness.

Square roots are a problem in mathematics, as is demonstrated by "imaginary numbers". At first glance, it appears like a square root is simply the inversion of the power of two. But the power of two is a valid procedure whereas the square root is not. If we define "square root" as the inversion of the power of two, then we'll find many numbers which simply do not have a square root. Why not accept this as a natural fact of numbers, rather than trying to force a square root onto these numbers?

If you are going to make a mathematical claim you need to be able to accept a mathematical disagreement. — fishfry

So long as you spell out your premises, and they are acceptable, I'm good to go. But it's already been proven that the square root of two is not a rational number. Why flog a dead horse? Why not move on, and inquiry what this principle tells us about numbers and spatial relations, instead of trying to disprove it.

You can literally build a square root of 2 out of the rational numbers. — fishfry

The construction procedure you described is never ending, just like the never ending digits. How is that any different? Without reaching the end, you have no definite solution, an approximation of something unresolved.

My favorite part of the day is my morning coffee. — fishfry

Seems we have something in common.

Ignorance as a debating point. "Your argument stands refuted because I'm incapable of understanding it." I can't top that. — fishfry

The problem is that you do not address the substance of the argument. You go off on some tangent using mathematical jargon, without addressing the issue.

You keep clinging to your mistaken belief, thinking that the rational numbers are good and the irrationals bad. This is a personal psychological condition that can be remedied with mathematical knowledge. If you so desire. — fishfry

No, I most definitely would not want that, and I've already explained why. I don't think it's a good idea to do a second bad thing to cover up an original bad thing. So you'd have to demonstrate to me first that the original thing which I consider to be bad (irrational numbers), is not really bad, with reference to solid ontological principles, rather than referring to what I called the second bad, which is just a cover up of the first bad. I have no inclination to learn the cover up, call it a psychological condition if that makes you happy.

You keep repeating this without engaging with the fact that math says you're wrong. — fishfry

I am arguing against accepted mathematical principles. How is "math says you're wrong" any sort of a counter argument? Of course math says I'm wrong, that's a given.

I am not responsible for what people type into Wikipedia. Some math articles are very good, some are highly misleading. — fishfry

OK, math says I'm wrong, but Wikipedia says you're wrong. Now we're even, both wrong.

I'm speaking sophisticated math to you and you just want to cling to what they told you in high school. That's your choice. — fishfry

Fishfry! Get with the program, wake up and smell the coffee! We've engaged in these discussions for weeks now, you know it's pointless to speak sophisticated math at me. You're wasting your time, we're discussing philosophy on this forum, not sophistic math.

I did. The passage of time. — fishfry

Ok, so as time passes, a "funny gadget" is magically converted into a "mathematical object". Tell me another one.

It's something you made up. — fishfry

No, I got it from Wikipedia, someone else made it up. But how is that any different from your "funny gadgets", which someone makes up, and through the passage of time magically turn into mathematical objects?

You are psychologically stuck to your wrong ideas and you're incapable of engaging substantively with any point that anyone makes. — fishfry

Actually, it's you who has not engaged in any of the substance of my post, and has regressed to ad hom, and repeated insistence of "your wrong".

ou are really good at writing stuff that sort of sounds intelligent, but doesn't hold up to scrutiny. That's why I was initially interested in your posts and why I took the trouble the read them. Now I've scrutinized them. You're ignorant about math and wrong on the philosphy. — fishfry

Huh, I don't see any evidence of that scrutiny, only repeated assertions, "you're wrong", "you're wrong", you're wrong".

That's right! Rational numbers are just as fictional or just as real as irrational numbers. — fishfry

Right, we've been through this already they are fictions, like your "funny gadget". But in logic we look for consistency, along with the capacity to fulfil the purpose. Why would a geometrical system produce a distance which is impossible to measure? How is this consistent with the purpose of geometry?

he bad thing is some misinformation you got stuck with in high school or earlier. You have to let go of things you think you learned that don't happen to be true. — fishfry

I thought we were talking about fictions. How is truth relevant?

learned a lot talking with you. Mostly I learned that I know a lot more about the philosophy of math than at least one person on this site who claims to know a lot about the philosophy of math. It's taken me years to get to this point. Thank you. — fishfry

I never claimed to know a lot about philosophy of math. I didn't even know there is such a thing. I've been arguing ontology. No wonder we're on different pages.

You apparently take "rational number" to mean the same thing as "number." And you apparently think that rational numbers possess some quality that those irrational thingies don't and can't have. What quality might that be? What quality can you name, identify, describe that rational numbers share but that the irrationals cannot have - other than, of course, being rational. — tim wood

That sums it up pretty well. Irrational ideas are incoherent, so there is good reason to rid our conceptions of such things. What we like in our conceptions is the quality of being rational and we are wise not to accept irrational ones conceptions.

And if you allow for there being such a thing as the square root of two, never mind what it might be beyond being the square root of two, then you're obliged to acknowledge that it has a lot of brothers and sisters, in fact a very large family. So large that in comparison, the rationals are next to just no size at all. — tim wood

No, I do not allow that there is such a thing as the square root of two, that's the whole point, and it is what I just explained to you in the last post. Saying "the square root of two" is really a matter of saying something, which represents nothing real, just like "pi" is a matter of saying something which represents nothing real. The mathematics clearly demonstrates that there is nothing real represented by these expressions. So it really doesn't make sense to say that there is "such a thing as the square root of two", just like it doesn't make sense to say that there is such a thing as pi. These are symbols which we use because they are extremely useful, but we ought to respect the fact that there really is nothing which is represented by them. You might imagine an ideal square with a diagonal line bisecting it, or an ideal circle with a line bisecting it through the middle, but the mathematics demonstrates that these are incoherent images.

Perhaps it might help if you tell us what you suppose a number, or number itself, to be. No doubt the difficulties here originate in the foundations of your thinking. — tim wood

A number is a definite unit of measurement, a definite quantity. The so-called "irrational number" is deficient in the criteria of definiteness. The idea of an indefinite number is incoherent, irrational and contradictory. How would an indefinite number work, it might be 2, 3; 4, or something around there? We can speak of an indefinite quantity as a quantity to be measured, when we assume that the thing is measurable, but to say that a quantity has a number is to say that it has been measured and is no longer indefinite. To say that a quantity has a number, but that the number is indefinite, is contradiction pure and simple.

Are you saying there is no square root of two? — tim wood

There is no rational number which is equivalent to what is represented by "square root of two". So the problem is not as you say, one of numerical representation, it is that there is no number for what is represented. There is a numerical representation, commonly used, but no number which is represented by it.

Now back to your quote. Of course we measure mathemtical objects. A unit square has side 1, area 1, and its diagonal ... well you know what its diagonal is. In fact it falls out of the Euclidean distance formula as the distance between the origin (0,0)(0,0)and (1,1)(1,1). — fishfry

OK, so we agree that if so-called "mathematical objects" are things which can be measured, Euclidian geometry creates distances which cannot be measured by that system. That agreement is a good starting point.

As a philosopher, doesn't the question, or wonderment, occur to you, of why we would create a geometrical system which does such a thing? That geometrical system is causing us problems, inability to measure things, by creating distances which it cannot measure.

No, I'm pointing out that there only seems to be a difference depending on what age one lives in. If you live in the age of integers you don't believe in rationals. You're stuck in the age of rationals and don't believe in irrationals. Matter of history and psychology.

…

We're in agreement then. But that's what a mathematical object is. A made-up gadget that, by virtue of repetition, gains mindshare. — fishfry

Maybe we can take this as another point of agreement. A "mathematical object" is nothing other than what you called a "funny gadget". Let's simplify this and call it a "mental tool". Do you agree that tools are not judged for truth or falsity, they are judged as "good" in relation to many different things like usefulness and efficiency, and they are judged as "bad" in relation to many different things, including the problems which they create. So a "good" tool might be very useful and efficient, but it might still be "bad" according to other concerns, accidental issues, or side effects. Bad is not necessarily the opposite of good, because these two may be determined according to different criteria.

Let's look at the Euclidian geometry now. In relation to the fact that this system produces distances which cannot be measured within the system, can we say that it is bad, despite the fact that it is good in many ways? How should we proceed to rid ourselves of this badness? Should we produce another system, designed to measure these distances, which would necessarily be incompatible with the first system? Having two incompatible systems is another form of badness. Why not just redesign the first system to get rid of that initial badness, instead of creating another form of badness, and layering it on top of the initial badness, in an attempt to compensate for that badness? Two bads do not produce a good.

I've studied set theory and read a number of set theory texts. I've never read or heard of any such thing. Set theory in fact is the study of whatever obeys the axioms for sets. If you ask a set theorist what a set is, they'll say they have no idea; only that it's something that obeys whatever axiom system you're studying.

You're making stuff up to fill in gaps in your mathematical knowledge. Set theory doesn't assume anything at all. It doesn't assume it's "about" anything other than sets; which are things that obey some collections of set-theoretic axioms. — fishfry

Come on, get real fishfry. Check Wikipedia on set theory, the first sentence states that it deals with collections of "objects". Then it goes on and on discussing how set theory deals with objects. Clearly set theory assumes the existence of objects, if it deals with collections of objects.

This is why it is so frustrating having a conversation with you. You are inclined to deny the obvious, common knowledge, because that is what is required to support your position. In the other thread, you consistently denied the difference between "equality" and "identity", day after day, week after week, despite me repeatedly explaining the difference to you.

But I already did. From the naturals to the integers to the rationals to the reals to the complex numbers to the quaternions and beyond. At each stage people didn't believe in the new kinds of number and though it was only a kind of "calculating device." Then over time the funny numbers became accepted. This is a very well known aspect of math history. — fishfry

You have not explained how acceptance of a mathematical tool, through convention, converts it from a funny gadget, to an object. If you cannot demonstrate this conversion, then either the tool is always an object, or never an object. Then an extremely bad tool is just as much an object as an extremely good tool, and acceptance through convention is irrelevant to the question of whether the mental tool is an object.

But do you mean how can sets be used to model mathematical objects like numbers, functions, matrices, topological spaces, and the like? Easy. We can model the natural numbers in set theory via the axiom of infinity. Then we make the integers out of the natural, the rationals out of pairs of integers, the reals out of Cauchy sequences of reals, the complex numbers out of pairs of reals, and so forth. If you grant me the empty set and the rules of ZF I can build up the whole thing one step at a time. — fishfry

Until you recognize that an "element", or "member" of a set is an "object", you are simply in denial of the truth, denying fundamental brute facts because they are contrary to the position you are trying to justify.

You haven't made any such case. — fishfry

The case I made is very clear, so let me restate it concisely. You appear to agree with me that mathematical tools are not objects, they are "mind" gadgets, yet you defend set theory which treats them as objects.

No, I took pains to make a distinction. Driving laws are completely arbitrary. But many mathematical ideas are forced on us somehow, such as the fact that 5 is prime. — fishfry

Mathematical ideas such as "5 is prime" are forced on us by the rules (laws) of the mathematical system, the definition of "prime" and "5" with deductive logic. So there is no difference. We create mathematical rules arbitrarily, as they are required for our purposes, just like we create driving laws arbitrarily as required for our purposes.

Here's an example. Take 1/3 = .333333.... Would you say that 1/3 is not resolved or requires an infinite amount of information? But it doesn't. I could just as easily say, "A decimal point followed by all 3's." That completely characterizes the decimal representation of 1/3. I don't have to physically be able to carry out the entire computation. It's sufficient that I can produce, via an algorithm, as many decimal digits as you challenge me to. — fishfry

This is nonsense. I can very easily say "the highest number". Just because I say it doesn't mean that what I've said "completely characterizes" it. We can say all sorts of things, including contradiction. Saying something doesn't completely characterize it.

It has been completely resolved. — fishfry

Unjustified, and false assertion.

It has been completely resolved. You can't write down infinitely many digits any more than you can write down all the digits of .333... But in the case of 1/3, there's a finite-length description that tells you how to get as many digits as you want. And with square root of 2, there is ALSO such a finite-length description. Would you like me to post one? — fishfry

if you switch to a different number system, one which is incompatible with the first from which the irrational number is derived, like switching from rational numbers to real numbers, this does not qualify as a resolution, if the two systems remain incompatible.

For instance, if there is infinite rational numbers between any two rational numbers, and we take another number system which uses infinitesimals or some such thing to limit that infinity, we cannot claim to have resolved the problem. The problem remains as the inconsistency between "infinite" in the rational system, and "infinitesimal" in the proposed system.

I really hope you'll consider the example of 1/3 and the fact that we can predict or determine every single one of its decimal digits with a FINITE description, even though there are infinitely many digits. Square root of 2 and pi are exactly the same. They are computable real numbers. There is a finite-length Turing machine that cranks out their digits. — fishfry

This has no relevant significance. To say "the square root of two", or "the ratio of the circumference of a circle to its diameter" is to give a 'finite description". We've already had the "finite description" for thousands of years. And, this finite description determines that the decimal digits will follow a specific order, just like your example of 1/3 determines .333.... The issue is that there is no number which corresponds to the finite description, as is implied by the infinite procedure required to determine that number.

So my analogy of "the highest number" is very relevant indeed. Highest number is a "finite description". And, the specific order by which the digits will be "computed" is predetermined. However, there is no number which matches that description, "highest number", just like there is no number which matches the description of "the square root of two", or "the ratio of the circumference of a circle to its diameter", or even "one third".

This demonstrates that there is a problem we have with dividing magnitudes, which has not yet been resolved.

It's just a representation, imperfect from the start. — fishfry

Let me return your attention to this remark. If you agree with me, that the representations are "imperfect" from the start, then why not agree that we ought to revisit those representations. Constructing layer after layer of complex systems, with the goal of covering over those imperfections, doing something bad to cover up an existing bad, is not a solution.

For the side of the square, take a straightedge, lay it along the side and mark the straightedge at the ends of the side. That's the length of the side — tim wood

The issue though, is that we are talking about measuring the ideal square, just like set theory talks about measuring the ideal numbers. We are not talking about measuring a representation of the ideal square, which is written on the paper, just like set theory is not concerned with measuring the numerals, it is concerned with "the numbers" represented by the numerals.

As pointed out repeatedly by others, the problem with irrationals is not with their number, but with some of their numeric representations. — tim wood

This is absolutely false. The reason why the "numeric representation" of irrational numbers is a problem is because there is no number to be represented. How it is proven that the square root of two is irrational is a demonstration that it breaks the rules of "rational numbers". Pythagoras proved that the square root of two is not a rational number. This means that if we're using the rational number system, the square root of two falls outside of that system, there is no number for it. In the other thread, fishfry called this "a hole" in the rational numbers, but I disagreed with that term.

In reality therefore, an irrational number, has a numeric representation, we clearly have a numeric representation of pi, and square root two, but there is no corresponding number for these representations. As fishfry indicated in the other thread, we might create a new number system (the real numbers) and try to include what is represented by irrationals, within that system, as numbers. I do not understand the construction of the real numbers, but I am willing to argue that this is a flawed approach. Instead of addressing the real problem, which is the fact that we can produce spatial representations (circles and squares) where numbers do not apply, and adjusting these representations accordingly, the mathematicians have created an extremely complex number system, which simply veils this problem.. In other words, instead of addressing the real problem, which is a feature of our faulty spatial representations, and trying to solve that problem, the mathematicians have just hidden it under layers of complexities.

But your stance here is literally pre-Pythagorean. The Pythagoreans threw some guy overboard for making the discovery, but they accepted the fact of the irrationality of 2–√2. You refuse even to do that. You're entitled to your own ideas, but to me that is philosophical nihilism. To reject literally everything about the modern world that stems from the Pythagorean theorem. You must either live in a cave; or else not live at all according to your beliefs. You must reject all of modern physics, all of modern science and technology. You can't use a computer or any digital media. You are back to the stone age without the use of simple algebraic numbers like 2–√2. — fishfry

I accept the fact that the square root of two is irrational. That's not the issue. And I actually use the Pythagorean theorem on a regular basis working in construction, laying out foundations. The issue is that I am inclined to ask why is it the case that the square root of two is irrational, and in doing this I need to consider what it means for a number to be irrational.

To simply say as you are saying, that some numbers are rational and some numbers are irrational, and that's a brute fact, does not express an understanding of what a "number" is. But then, to ask why is it that some numbers have the property of being rational and other numbers have the property of being irrational requires asking what it means to be an irrational ratio, and one might be faced with the prospect that what we call an "irrational number" ought not even be called a "number". Perhaps the Pythagoreans threw the baby out with the bathwater, saying we can't resolve this problem so let's just call them all "numbers" anyway, and get on with the project.

So, let me state clearly and concisely what the situation is. We have a very simple looking problem of division which cannot be solved because there is no "number" which can represent the solution. You say, the problem can be solved, the resolution is an "irrational number", so just forget about that problem, it's not a problem at all. And, you say it's just "a fact of life" that some numbers are irrational. I say it's a simple fact that a so-called "irrational number" is not a number at all, because it's quite obvious that there is not a definite number which expresses the resolution of the irrational ratio. See, the very simple looking problem of division has not been resolved, and it is a pretense to claim that it has been resolved to an "irrational number".

No of course not. It tells me something interesting about abstract, idealized mathematical space. It tells me nothing about actual space in the world. — fishfry

If idealized mathematical space tells us nothing about space in the world, then physics has a big problem. But of course this is nonsense. That the square root of two is irrational, and that pi is irrational tells us something about idealized mathematical space, and that is that there is a problem with commensurability in idealized mathematical space. And, since idealized mathematical space is the tool by which we make measurements in real space, the problem of idealized space is simply ignored in application

There are no irrational distances in physical space for the simple reason that there are no exact distances at all that we can measure. So it's not meaningful to talk about them except in idealized terms. — fishfry

OK, let's talk about "irrational distances" in idealized terms then. Lets take a point A. Lets make a point B at a specific distance from point A, and a point C at the very same distance from point A, but in a direction at a right angle to the direction of point B. Do you agree that there is no definite distance between B and C? If you disagree then you are simply denying the fact. if you agree, then you might be inclined, like I am, to ask why this is the case. And so it appears to me, like there is a very real problem with "idealized mathematical space", making it less than ideal.

For all these reasons, 2–√2 is essentially a finite mathematical object. You're simply wrong that it "introduces infinity," because you have only seen some bad high school teaching about the real numbers. Decimal representation is only one of many ways to characterize 2–√2, and all the other ways are perfectly finite. — fishfry

You might say "√2" is a finite mathematical object, but until you define what a mathematical object is, it's you who's just typing word salad. In reality "√2" is an unresolved mathematical problem. That you call it a "mathematical object" doesn't mean that it is a "number", nor does it mean that it actually is an object. And, when one attempts to represent this so-called object as a number, infinity is introduced

However, I didn't say that it "introduces infinity" on this thread. If I mentioned that, it was another thread in another context. Perhaps I said that in a thread on infinity. What I am focused on here is simply the meaning of an irrational ratio, and whether it is appropriate to claim that the ratio has been resolved to a "number", called an "irrational number".

But to simply say that you don't like 2–√2; that's just a hopelessly naive viewpoint. — fishfry

It's not "√2" that I dislike, it's what it represents that is what I dislike. And it's not that I am simply saying this, I am giving you the reasons for my dislike. But you seem to be good at ignoring reasoning.

It's just a representation, imperfect from the start. — fishfry

Right! That's why we ought to seek a better one! That's exactly what I'm arguing. Don't you agree?