I imagine a unicorn by picturing a unicorn. — RussellA

Similar, but I wouldn't call it "picturing". Anyway, the point is that this "picturing" does not require a "concrete instantiation", which I assume implies a physical object being sensed.

"1 = 1" is a mathematical expression. — RussellA

When you say "=" signifies identity, "1=1" is not a mathematical expression. Think about it. If "1=1" means that the quantitative value signified by the first "1" is equivalent to the quantitative value signified by the second, then this is a mathematical expression. And in the case of ordinals, if "1" signifies "first", and the expression means that the first is equivalent with the first, as first, then this is also a mathematical expression. But if "1=1" is meant to signify that the thing identified by "1" on the right side is the very same as the thing identified by the "1" on the left side, then it is not a mathematical expression. It is an expression of identity. And, the fact that it is analogous with Twain = Clemens, which is clearly not a mathematical expression is evidence that it is not a mathematical expression.



I could address your examples, but I do not see how they are relevant really. In set theory it is stated that the elements of a set are objects, and "mathematical realism" is concerned with whether or not the things said to be "objects" in set theory are, or are not, objects.

To play chess you must accept the reality of the pieces as objects in order to move them, therefore you must accept "chess reality" to play chess. Since it may not be stated in the rules that the pieces are "objects" the acceptance is only implicit, unlike set theory in which case the rule is explicit, therefore acceptance is explicit.

It seems to me that you do not understand "realism". Do you agree, that to be able to take "an object", manipulate it, move it, do whatever you please with it, or move it according to some set of rules, you need to accept that the object which you are doing this with is "real"? And this implies that believing the things which you are manipulating to be "objects", implies some sort of realism. Or, do you separate "realism" from "objects", so that realism has nothing to do with objects? In which case, what would you base "realism", and consequently "antirealism" in?

I already did above. The axioms of some given set theory are just rules that you must follow when using that set theory. Different set theories have different axioms and so different rules. Given that there's no connection between using some set theory and believing in the mind-independent existence of abstract mathematical objects, there's no hypocrisy in using some set theory and being a mathematical antirealist. — Michael

"Mathematical antirealist" is also a "rule". It states an ontological principle, or rule. It is a rejection of mathematical objects. The rules of set theory are inconsistent with this rejection of mathematical objects, because set theory assumes mathematical objects, as a foundational premise. Therefore you must assume mathematical objects, as a fundamental premise, to be able to follow the rules of set theory. This activity is contrary to the ontological belief stated as "mathematical antirealist", and is therefore hypocrisy for anyone claiming to be a mathematical antirealist.

Your position is like arguing that it's hypocritical to play chess if I do not believe that the rules of chess correspond to some mind-independent fact about the world. — Michael

I don't see the relevance. You do not need to accept the premise of "mathematical objects" to play chess. You do need to accept the premise of "mathematical objects" to follow the rules of set theory.

And how would you justify that claim?

You don't need to believe in Platonic realism to use set theory. — Michael

I agree. I didn't say you need to believe in the truth of the principles you employ. However, it's hypocrisy to say "I'm a mathematical antirealist" and then go ahead and use set theory. But that sort of hypocrisy is extremely commonplace in our world, it's actually become the norm now. Very few people make the effort to understand the metaphysics which they claim to believe in, and whether it is consistent with the metaphysics which supports the theories which they employ in practise.

Why? — Michael

Set theory begins with the assumption of mathematical objects, hence it is based in Platonic realism.

I wonder if mathematical realists and mathematical antirealists have different views about mathematical infinity. I'm a mathematical antirealist. I have no problem with mathematical infinity. The "existence" of infinite sets does not entail the existence of infinities in nature (whether material or Platonic). — Michael

The issue is with the premises (axioms) of set theory. If you are "mathematical antirealist" you ought to reject set theory on the basis of the axioms it employs. Your view on "infinity" would be irrelevant at this point. So how set theory treats "infinite" would not even enter into the reasons for your rejection of it.

Set theory is based in the assumption of "mathematical objects". And, the mathematical objects as elements of the sets, are allowed to have relations which physical objects, according to our knowledge of them, cannot have. So a "set" by set theory is a bunch of "objects", but since they are mathematical objects instead of physical objects, what can be 'truthfully' said (what is acceptable by the axioms) about that bunch of objects, doesn't have to be consistent with our knowledge of physical objects. So for example, there can be an empty set (a bunch of objects with no objects), and sets do not necessarily have an order (a bunch of objects without any order to them).

Exactly, you understand the concept using images. — RussellA

The key point here, is that imagination does not require sensation of whatever it is that is imagined. But you claim understanding a concept requires "concrete instantiation" and I assume that means something which is sensed. If I'm wrong here, and "concrete instantiation" means the production of an image in the mind, without the requirement of sensing it, then we might have something to discuss.

There are two different cases.

The first a case of identity where the two 1's refer to the same thing. The second a case of equality where the two 1's refer to different things.

The practical advantage of using identity rather than equality is to distinguish two very different cases. — RussellA

Yeah sure, you've indicated that in the first case "=" signifies identity. I agree, that's what you've stipulated. The point though, is that in the case where you used "=" to signify identity, it is not a mathematical usage. The "practical advantage" you refer to is rhetorical only, intended to persuade me. The usage is not mathematical, because in the application of mathematics "=" is not used to represent identity. That's the issue with Tones' example of Twain = Clemens, it just demonstrates that it's possible to use "=" in this way. However, it doesn't at all represent how people applying mathematics actually use "=" in the formulation of equations. So it's nothing but a rhetorical example, produced solely for the purpose of trying to persuade, in the mode of sophistry.

"A mythical animal typically represented as a horse with a single straight horn projecting from its forehead" describes an object, even through the object is fictional. — RussellA

I would definitely disagree with this. There is a big difference between seeing, hearing, touching, or otherwise sensing an "object", thereby describing what i sensed, and creating an imaginary "object". The latter does not involve an object, nor does it involve a "description" ( in the proper sense of the word) because it is an imaginary creation an invention rather than a description. A "fictional object" is not an object, that's actually what "fictional" means. OED #1 definition of object "a material thing that can be seen or touched". "Fictional", on the other hand, means exactly the opposite, invented by the imagination, therefore not able to be seen or touched.

In fact, from my position of Neutral Monism, all objects, whether house, London, mountain, government, the Eiffel Tower, unicorn or Sherlock Holmes are fictional, in that no object is able to exist outside the mind and independently of the mind. — RussellA

I suggest that your "position" is not consistent with common understanding. It might benefit you to give up on the monism.

My belief is that the mind cannot understand an abstract concept in isolation from concrete instantiations of it, in that, if I am learning a new word, such as "ngoe", it would be impossible to learn its meaning in isolation from concrete instantiations of it. — RussellA

There is no such thing as a "concrete instantiation" of a concept. Concepts are categorically different from concrete objects. To take your example, show me where I can find a concrete instantiation of beauty, 6, or the square root of 2. It is one thing to assert that there is a concrete instantiation of a six out there somewhere, but quite another thing to prove this. And if it is true, it ought to be easy to prove. Just point out this 6 to me, so i can go see it with my own eyes, or otherwise sense it.

If I wanted to teach you the meaning of the symbol "ngoe", which I know is a concept, how is it possible for you to learn its meaning without your first being shown particular concrete instantiations of it? — RussellA

This seems to be completely inconsistent with what you've already argued. You've already made the claim that you can make a fictious description, so why couldn't you also define a concept, thereby providing the means for someone else to understand it, without showing a concrete instance of that type of thing? I mean, you presented me with "a horse with a single straight horn projecting from its forehead", and i understand this image without seeing a concrete instantiation, so why take the opposite position now, and say that a person cannot understand the meaning of a concept without being shown a concrete instantiation of it.?

Given 1 and 1, if the second use of 1 refers to the same thing as the first use of 1, then the proper equation should be 1 = 1. The symbol "=" means identity

Given 1 and 1, if the second use of 1 refers to a different thing as the first use of 1, then the proper equation should be 1 + 1 = 2. The symbol "=" means equality. — RussellA

If this is the case, then what you have shown is logical inconsistency in the use of "1". In the first case, the two instances of 1 must refer to the very same thing, and in the second case, the two 1's must refer to two different things. If we simply say "=" means equality, then there is consistency between your two examples. Furthermore, there is no practical advantage to designating "=" as meaning identical in the case of "1=1", so you're just proposing logical inconsistency for no reason. That is simply illogical, therefore not a fair representation of the logic of mathematics.

Can there be a description without an object being described? — RussellA

Of course, that's known as fiction.

However we can think of the numbers 1. 6 and 10 as not only abstract mathematical objects but also as natural concrete objects. — RussellA

This evades me. How do you think of a number as a natural concrete object? Are you talking about the numeral, or the group of objects which the numeral is used to designate, or what?

That raises the question as to how we are able to think of something that is abstract, disassociated from any specific instance (Merriam Webster – Abstract). For example, independence, beauty, love, anger, Monday, ∞
∞
, 2–√
2
and the number 6.

George Lakoff and Mark Johnson in their book Metaphors We Live By propose that we can only understand abstract concepts metaphorically, in that we understand the concept of gravity by thinking about a heavy ball on a rubber sheet.

Thereby, we understand the concept of independence by remembering the feeling of leaving a job we didn't like. We understand the concept of beauty by looking at a Monet painting of water-lilies. We understand the concept of infinity by thinking about continually adding to an existing set of objects. We understand the concept of 2–√
2
by thinking about the number 1.414 etc etc. We understand the concept of 6 by picturing 6 apples.

IE, we can only understand an abstract concept metaphorically, whereby a word or phrase literally denoting one kind of object or idea is used in place of another to suggest a likeness or analogy between them (Merriam Webster – Metaphor). — RussellA

So why would we label an abstract concept an "object", as in "mathematical object", and speak of it as if it had an identity in the same way that a natural concrete object has an identity? If we only know abstract concepts through analogy, or suggestions of likeness, isn't it completely wrong to suggest that anything which only exists in this way, i.e. through metaphor, could have an "identity"?

This is the problem which @TonesInDeepFreeze is stuck on. Tones seems to think that just because one can show how "=" can be used to to show a relationship of identity between two distinct names for the same natural concrete object (]Mark Twain = Samuel Clemens), we can conclude that when "=" is used in mathematics, it's being used in that same way.

But of course in mathematics this is not true. There is no such natural concrete object which the symbols refer to, in theory. only abstract concepts. Natural concrete objects are only referred to through application. And in application the concrete situation referred to by the right side of the equation is never the same as the concrete situation referred to by the left side. So all that Tones has indicated is that there is two very different ways to use "=", the mathematical way, and the way which signifies a relation between two different names for the same natural concrete object. Therefore, we must be careful not to confuse the two different ways, or equivocate between them, because that would be misleading.

It is exactly the point that it is not a mathematical expression, so mathematics is not called on to account for its intensionality. More generally that ordinary mathematics is extensional, and we don't require that it also accommodate intensioncality. That is how it is relevant. — TonesInDeepFreeze

I interpret that as 'mathematics is extensional and that's how intensionality is relevant'. Whatever it is you are trying to say here, it appears to be just as irrelevant as your analogy was.

Later, hopefully, I'll have time and motivation to dispel a number of misconceptions in a catalog of them you've posted lately. — TonesInDeepFreeze

I'll be looking forward to that.

Doesn't this problem, soluble by set theory, assume "objects", such as the object "a person who can speak English"?

If the number "1" does not refer to an object, what does it refer to? — RussellA

The issue is a little more complex than how you represent here, but this is a good indication of why "set theory" is not applicable to mathematics. In your first question, "a person who can speak English" is a description, not an object. It represents a category by which we could sort objects. In the second sentence, the numeral "1" represents a specific concept, which can be described as a quantitative value. It is not a true representation of how we use numbers, to think of a number as itself an object. Set theory may represent a number as an object, but that's the false premise of set theory.

The issue is not whether or not some mathematicians define "=" as meaning 'is identical to', as a premise for a mathematical theory, or some other purpose, like debate or discussion. We've seen very much evidence here that some actually do this. So there is no question concerning that.

The question is how "=" is actually used in the application of mathematics. And anyone who takes a critical look at an equation in the application of mathematics will see that the right side never signifies the very same thing as the left side. In fact, it's quite obvious that if the right side did signify the same thing as the left, the equation would be completely useless. That is why many philosophers will argue that the law of identity is a useless tautology.

Since this is the case, we can clearly see that those mathematicians who define "=" as meaning 'is identical to' do not properly represent the meaning of "=" with that definition. Therefore we can say that they are wrong with that definition.

For example:

Mark Twain = Samuel Clemens — TonesInDeepFreeze

This is not a mathematical equation, so I do not see how it is relevant. You are trying to compare apples with oranges, as if they are the same thing, but the requirement that "Mark Twain = Samuel Clemens" is a representation of a mathematical equation renders your analogy as useless.

Please consider a real mathematical equation as an example, like how the circumference of a circle "is equal to" the diameter times pi, or the square of the hypotenuse of a right triangle "is equal to" the sum of the squares of the two perpendicular sides, for example. Be my guest, pick an equation, any equation, and we'll see if the right side signifies the very same thing as the left side. I think that an intelligent mathematician such as yourself, ought to know better than to argue the ridiculous claim that you have taken up.


The principal problem with set theory, as I indicated in my reply to @Banno above, which is evident from Chat GPT's statement, is that set theory is derived from a faulty Platonist premise, which assumes "mathematical objects". If we recognize as fact, that mathematics does not consist of objects, we must reject the whole enterprise of set theory, along with its fantastic representation of "infinite" and "transfinite", as completely unsound, i.e. based in a false premise.

Are you serious? — TonesInDeepFreeze

Lying requires intent, which GPT lacks.

In mathematics, equality and identity are the same. — TonesInDeepFreeze

Here's the example I gave Banno in the other thread. You and I are each one. Together we are two. We can symbolize this as 1+1=2. The two 1's here each represent something different, one represents you, the other I. Because the two each represent something different, the two together as 1+1 can make 2, meaning two distinct things. Also, we can say 1=1. But if the two 1's here both represent the same thing, then 1+1 could not make 2, because we'd still just have two different representations of the very same thing.

While "=" is commonly understood to denote equality in basic arithmetic and algebra, its use to signify identity in formal logic or set theory arises from the need to express relationships between objects or sets in a precise and rigorous manner. — Banno

That mathematics consists of "objects" with identity is Platonist metaphysics. In this metaphysical theory, mathematical ideas like numbers are objects, rather than quantitative values. Set theory is nothing but Platonist based mathematical theory. Notice that it is "theory", not mathematics in practise.

In the actual application of mathematics, values are assigned, and the left side of an equation must represent something different from the right side, or the equation would be useless, as I explained.

The conclusion we can make is that set theory does not represent mathematics, as mathematics is actually used. That's the problem, We can define terms, or in this case symbols, for theory, in a way which doesn't actually represent how they are used in practise. That's an idealist folly. I think Wittgenstein made a similar point.

However, in more advanced mathematical contexts like set theory, "=" is sometimes used to signify identity, — ChatGPT

As I argue, there is much inconsistency in mathematics. The use of "sometimes" here is very telling.

I mean that all makes sense, although my understanding was that the question of whether or not space-time is infinitely divisible was an open one. — Count Timothy von Icarus

It appears to me, like the quantum nature of energy demonstrates quite conclusively that the reality of space and time cannot be infinitely divisible. You see, the wave-function represents a continuity, but what it represents is not an observable aspect of reality. Observations indicate discrete occurrences of so-called particles (quanta), with not necessary continuity between the occurrences. If there is a true continuity, it is not represented by the wave-function, which represents possibilities. And, it is not the continuity of space-time, which fails at the quantum level. So it hasn't yet been determined.

Ok. So we still have no explanation of how you came to misapprehend "=". — Banno

Did you not understand the example I gave you in the other thread? I suggest you go back and read that post you made for me when you fed that example to Chat GPT. It totally agreed with me. It said, in much arithmetic and mathematics "=" signifies equality, not identity. Chat GPT does not lie you know. The simple fact, as my example shows, an "equation" would be completely useless if the left side signified the very same thing as the right side.

Is Metaphysician Undercover a product of the New Maths? :wink: — Banno

I do not think I was ever subjected to new math. I simply learned at a very young age not to follow rules without a reason for doing so. I was not interested in the things which mathematics was useful for, so it was dismissed from my curriculum, as soon as possible, until the need was developed. So my education in mathematics was done in an 'as required' way, rather than a force-feeding of conventional 'fact' to memorize, like history.

Infinity pools can indeed be awesome — ssu

You get the same effect when you take a boat on a reservoir, up toward the dam, the higher the dam the better. It's like empirical proof that the earth is flat, and you're at the edge of the world.

I am not sure if this is so much a problem with mathematics though as it is with how it gets applied to the sciences and philosophy. It seems to me that infinite divisibility might be worth investigating even if it doesn't accurately reflect "how things are." — Count Timothy von Icarus

I think there is a very close relationship between "mathematics" as the principles, rules etc., and the application of those principles. As Plato said, the people who use the tools ought to have a say in the design of the tool. And in reality they do, because the ones using the tools choose and buy the ones they like, therefore design and production is tailored for the market of application.

So in the case of "infinite" for example, the principles of calculus allow for the representation of an operation which is carried out without a limit. The limit is infinite, which essentially means there is no limit, and the operation proceeds endlessly. This representation proved to be very useful in application.

The issue we can look at, as philosophers, is what exactly is the effect of such an untruthful representation. First, we need to accept the fact that it is untruthful. To allow into any logical "conclusion", that an operation has been carried out without end is a false premise. In reality, the need to carry out the operation endlessly would deny the possibility of a conclusion.

The next step I believe, is to apprehend the level of ignorance which this untruthful representation propagates. There are some very specific problems produced from our conceptions of the continuity of space and time, which were demonstrated by Zeno. The mathematical representation (or more properly misrepresentation) as a premise in calculus, creates the illusion that these problems have been resolved, and so there is denial and ignorance concerning the reality of the problem amongst many people.

Finally, we can see how allowing this untruthful representation actually magnifies the problem rather than resolving it. When the usefulness of the misrepresentation is apprehended and recognized, it, and similar forms are allowed to pervade throughout the logical system (we can call this the propagation of self-deception). This creates the issue pointed to by the op, the need for different types of infinities, infinities of infinities, and the transfinite in general. The issue with the transfinite being, that some applications require the truth, a finite number, while others require the impossible, or false representation of a conclusion drawn from an endless operation, so some applications require a relationship between the two, hence the "transfinite". We can look at it as a bridge between the untruthful, and the truthful, a bridge which enables the self-deception.

You have mentioned, for example, that the limit concept is flawed, although it works well most of the time. But I don't recall your argument beyond that point. A more complete knowledge of space and time and points and continuity? Oh yes, something about the Fourier transform and the Uncertainty principle. What are your suggestions to fix that up? Intuitive mathematics? Remind me where doing something specific makes it better. — jgill

I think the obvious point to start with is divisibility. Generally, mathematics provides that a quantity, any quantity, can be divided in any way. We can call that "infinite divisibility". In reality, there is very clearly many division proposals which simply cannot be done. Because of this fact, that there are real restrictions on divisibility, there is a very big difference between dividing a group of things, and dividing a single object. Each of these two types of division projects has a different type of restrictions or limitations on it.

For example, to divide a group of seven human beings into two equal groups is a project that cannot be done, even though common math would say seven divided by two is three and a half. So we'd have to chop a person in half. But then we'd have eight objects instead of seven, because we'd have have two halves, which are two objects, but unequal to the other six objects. So we have to conclude that the way we divide a group, or quantity of things is seriously restricted.

Further, the way that we quantify something dictates the way that the quantity can be quantized. So if we use weight for example, to measure the volume of a group of grains of sand, we do not count the grains and divide the number of grains evenly, we look at the sand as one thing, with one weight, and divide that weight however we will. But there will still be a issue with precise division, when we get to the point of needing to divide individual grains of sand.

This leads into the problem of dividing single objects. An object is a unit, and this is fundamentally a unity of parts. If there is an object which is not composed of parts, like the ancient atomists proposed for the "atom", this object would be indivisible, and provide the basis for the rules of all division projects. However, such an object has not been found, so the guidelines for dividing a unit must follow the natural restrictions provided by the divisibility of the type of object. Different types require different rules, so mathematics provides for all possibilities (infinite divisibility). What physicists have found, is that the true restrictions to divisibility of all things, are based in mass and wave action, rather than composite "parts".

This means that in order to provide the proper rules or guidelines for the division of units, unities, we need to understand the real nature of space and time. Mass is a feature of temporal extension at a point in space, and waves are a feature of spatial extension at a point in time. Where the common principles of mathematics mislead us is the assumption of "continuity", and this is closely related to the simplistic notion of "infinite divisibility".

Now we have two closely related, but faulty principles of mathematics, infinite divisibility and continuity. They are applied by physicists, and people believe they provide a true representation of reality, when physicists know that the evidence indicates the presence of discrete quanta rather than an infinitely divisible continuity. Therefore our representations of spatial and temporal features need to be completely reworked. To begin with, as I've argued in other threads, representing space with distinct continuous dimensions (Euclidian geometry) is fundamentally flawed. The separations within space indicated by quantum physics, must indicate distinct incommensurable parts. These distinct parts are the parts which may be represented dimensionally, and the parts which cannot be represented that way. However, they must be incorporated together in a way which adequately represents what's real. At the current time, we have a dimensional, continuous line (numberline), with non-dimensional points (real numbers) which may divide the line infinitely, but this is just an unprincipled imaginary concept which in no way represents the real divisibility of space, and it becomes completely inapplicable when physicists approach the real divisibility of space.

Except incompleteness (in the sense of the incompleteness theorem). — TonesInDeepFreeze

That's a specific, restricted definition of "incompleteness". The term is slightly different in physics for example. So this is an example of what I am talking about. Mathematics also uses a specific, restricted definition of "infinite", a meaning exclusive to mathematics, determined by the axioms. The mathematicians designing the axioms tailor the meaning of the term, to suit their purposes.

Here's actually some advice to all non-mathematicians (from a non-mathematician):

If you really can ask an interesting foundational question that isn't illogical or doesn't lacks basic understanding, you actually won't get an answer... because it really is an interesting foundational question!

Yet if the answer is, please start from reading "Elementary Set Theory" or something similar then yes, you do have faulty reasoning. — ssu

I disagree. The "math boys" here at the forum tend to respond with 'go read some math texts' to anyone who disagrees with them on fundamental principles. In that case, the issue is not a matter of better learning the mathematical representation of the fundamental principles, and how to apply them mathematically, as a math text will demonstrate, it is a matter of disagreement with those mathematical representations. Therefore the reply of "please start from reading 'Elementary Set Theory' or something similar", is usually just a copout, a refusal to engage with the philosophical matter at hand as if further reading of the mathematics will change a person's mind, who already disagrees with it. That's like telling an atheist to go read some theology, as if this is the way to turn the person around.

The point I was making is that concepts like infinity, incompleteness, and even computability, extend beyond mathematics. So, the mathematical approach is only one approach to such concepts. The philosophical approach, specifically the dialectical approach, is to consider the way that such concepts appear in all the different fields. The way that each field deals with the concepts demonstrates how that field fits, or does not fit, within a consistent whole philosophy. To say that such concepts are the domain of mathematics, therefore mathematicians ought to define them, is to make a statement not consistent with the world we live in.

And sometimes people post questions about mathematical subjects that have bearing on philosophy, such as about infinities, incompleteness and computability. — TonesInDeepFreeze

You have this inverted. These are actually philosophical issues which have a bearing on mathematics. The way that a particular mathematician deals with these issues exposes their philosophical inclinations, or lack thereof.

It's interesting that you title the thread "the art of thinking". I believe the key to bringing anxiety into the category of beneficial is to provide it with direction. Notice what is stated in your first sentence, your inclination is to ask the person in distress, what do they want. But if the distress is anxiety related this is a useless question, because anxiety only creates distress if the anxiety in undirected. Rather than the fear of the unknown, anxiety is the unknown of fear.

Anxiety can be said to "create" distress, and the purpose for anxiety in general is to motivate the creative act. But if there is nothing substantial being created, the anxiety is undirected and the result is distress. The key to making your anxiety beneficial (and this means to progress beyond simply coping with anxiety) is to be creative. Anxiety is a reflection of your living disposition toward the future. When you are fearful of the future and the reason for being fearful is unknown, that thought, the reason why you are fearful is replaced by anxiety. When you are actively doing something, being creative, you bring the future into your control, by knowing what you are doing. If a reason for fear arises, it is in relation to what you are doing, and you know the reason for it.

So my answer to "Does my anxiety help me in something?" is yes, it most certainly does. It helps you to be creative. Without any anxiety you would not do anything. However, it is very important that when you feel anxiety you actually do something. So the anxiety is first, it is fundamental to the living being, and it is from the subconscious, the drive to be active. If you are actively doing something your fears will be revealed to your thoughts, and your anxiety will continue to seek more unknowns (as curiousity), to be revealed through action. If you are inactive, the unknowns will overwhelm you.

What is the proper interpretation of the cosmological constant Λ? I understand that it corresponds to a vacuum energy density, pervading all reality. Such energy is called dark energy, I gather. Since I'm sketchy on field theory, I don't know how this goes, but somehow this energy density produces a repulsive force beween any two objects in spacetime (within each other's lightcones?). Matter remains cohesive because Λ is very small compared to other forces, so that its effects really only show at an intergalactical scale (megaparsec). — DanCoimbra

I would say that is one interpretation of "dark energy", but here would be a number of possible interpretations.

ow, somehow this leads to the expansion of the Universe even in the case where the Universe is finite and bounded, which is a possibility considered by cosmologists. In this case, the Universe is increasing in total size, but not increasing *into* anywhere, so it becomes bigger because it has more internal spatial structure. This is what I meant. Why do you think this is incorrect? — DanCoimbra

What I meant is that "more internal spatial structure" is not consistent with Einsteinian relativity, because that would render a whole lot of predictions about the motions of things as inaccurate. We can posit "dark energy" as the reason why the predictions are inaccurate, but then where is this dark energy, and what is it doing other than making the predictions inaccurate,
.

I just think if mathematical axioms are to be selected, they have to be such that they do not lead to what is contradictory to Existence/Truth (or just semantics in general). — Philosopher19

The problem inherent within pragmaticism is that whatever is the purpose at the time (the flavour of the day), the axioms chosen will support that purpose. As time goes by, and needs change, other axioms will be produced to satisfy the evolving needs. At this point, the new and the old are not necessarily consistent, so there may be a degree of contradiction between different logical structures, depending on the purpose which they each serve,

If a mathematician or a philosopher decides on an axiom or theory that requires belief in the following (or at least logically implies it or leads to it): Nothing can be the set of all things (which logically implies Existence is not the set of all existents), or one infinity is a different bigger than another (or is a different quantity than another), I believe that axiom or theory should be disregarded or at least viewed as contradictory to Existence/Truth (or at least contradictory to the semantic of infinity). — Philosopher19

I agree that it is appropriate to set a standard of "truth" for mathematical axioms.

What the Universe's expansion means, whether it is infinite or not, is that its local energy density is decreasing. In other words, there is more spatial structure between each of its internal field excitations (particles, energy). — DanCoimbra

Hi Dan, I see you're new here, so welcome to this space.

I don't think it's proper to say that expansion means "more spatial structure" between internal field excitations, unless you are speaking of a "spatial structure" which is other than Einsteinian space-time.

So true. The OPs lay out belief systems in one form or another, and sometimes they don't budge. Which I find acceptable in Metaphysician Undercover's pronouncements, for he dwells with the ancients as they ponder space, time, and points and curves - although he balks at 1+4=5 and has little patience with Weierstrass and his limit ideas: admittedly useful, but fundamentally flawed. But I see where he is coming from there. Others, like this thread, are more or less unmovable in their opinions, which clash with standard mathematics. How you deal with the frustration of offering knowledge to those unwilling to accept it is admirable. — jgill

Now, now, let's have fair representation. I balk at the claim that "1+4=5" implies that "1+4" refers to the exact same thing as "5" does. And, I have no patience for people like fishfry who simply assert over and over again, that the axiom of extensionality proves that "1+4" and "5" must refer to the exact same thing in common applications. Further, although I am very interested in the fundamental incompatibility between the proposal of non-dimensional points, and the proposal of a continuous one-dimensional line, I believe no one until today has brought Weierstrass to my attention.

I tried dealing with 1. and 2. earlier, in mathematical analogues, but there was no interest. I could easily deal with 3. as well, but that takes the thread away from the spectacular leap from a first cause being something imaginable to an existential realm. — jgill

I dealt with the existential realm, but there was no interest in that either. So where does that leave us?

Suppose someone produces an axiom. Will it not be the case that that axiom will either be contradictory in relation to certain truths or consistent in relation to certain truths? Existence determines what is true and what is false. Whether any belief or axiom highlights truths or is contradictory to truth is determined by Existence/Truth. If not, there is no truth or semantics to work with to deduce further truths. — Philosopher19

I really don't think "truth" in this way is relevant. This is more of an issue of pragmatics, mathematics is a tool. You wouldn't say that one saw is more truly a saw than another saw, or on shovel is a more true shovel than another. So the axioms which are accepted, "which are bought", are the ones which mathematicians like to use. It may well be the case that existence determines truth, like you say, but that's not relevant to the selection of mathematical axioms.

I do believe we can bring "truth" into the picture in a different way though. Since mathematicians can choose to use whichever axioms they feel comfortable with, we can say that the axioms follow use. That means that they are a reflection of what mathematicians are doing. Therefore we can say that they are descriptive rather than prescriptive. The axioms do not give mathematicians rules for how to do things, because the mathematicians get to create and choose their own axioms. So the axioms simply provide a representation of what mathematicians are doing. Since they are descriptions, "truth" is to be found in how well the axioms represent what the mathematicians are actually doing. As an analogy, consider looking at a dictionary and judging how truthfully the definitions represent how people are actually using the words which are defined there.

Of course: nuclear reactions have emergent aspects by themselves, but you should distinguish these from emergent chemical features.
You can of course lump everything together and say that the universe, with everything in it, is emergent as a whole. But that means that the various properties of the universe are obscured. — Ypan1944

Well, I think if we're talking about emergency, we ought to consider the whole category, not just "emergent chemical features". If the reality of the situation is "that the various properties of the universe are obscured", then we need to respect that, rather than trying to obscure that fact by hiding it or denying it.

My belief is that we can't just produce axioms. We can only recognise truths about Existence such as 1 add 1 equals 2 or the angles in a triangle add up to 180 degrees or one cannot count to infinity. — Philosopher19

Axioms are simply produced, created. The ones which prove to be useful are put to use, and they persist by becoming conventional. "Truths about Existence" is irrelevant to the mathematicians who create axioms.

That is the heart of the issue. With logic, we might demonstrate the "validity" of the non-physical, but if the logic is sound, it would also demonstrate the truth or "reality" of the non-physical. The physicalist would argue that valid logic does nothing to prove the non-physical because so much logic proceeds from fictitious, fantasy, or imaginary premises, and such is the claim to a priori.

The task of metaphysicians then, is to ground the a priori in sound principles. Sound principles are derived from the way that we "experience" reality. Principles consistent with experience are considered to be sound. Now "experience" must be allowed to extend beyond simple sense observation (the trap that empiricism gets looked into), to include the inner most experiences of being, as phenomenology does for example. In this way the metaphysician brings the validity of the arguments for the non-physical into the position of being sound as well. We just need to escape the empiricist trap, which is a metaphysical belief that sound principles of "experience" can only be provided by sense observation.
And if the physicalist argues that all experience is simply a response to sense stimuli this is demonstrably false.

Nuclear reactions have nothing to do with the features of an atom or molecule. For reactions between atoms or molecules, only the "outside" of an atom (i.e. the outermost electrons of the atom) plays a role. The emergent feature of an atom or molecule depends only on its outermost electron configuration. — Ypan1944

So I assume you are restricting the concept of "emergence" so that the products of nuclear reactions are not included as forms of emergence. How would you classify this activity then? Surely it's not downward causation. What type of causation is it?

One is free to propose different axioms that prove differently. — TonesInDeepFreeze

Since there is a whole lot of difference between the different types of numbers you outline, I think it a very good idea for a mathematician to look for a whole new set of axioms to better deal with the problem of having different types of numbers. This could avoid the problem of needing principles to relate the different types of numbers to each other, in an attempt to reconcile the sometimes irreconcilable difference between them. Attempting to reconcile the incompatibility between them tends to create a new type of infinity. So every time a new type of number is produced to deal with a specific problem that has arisen, a new type of infinity is produced. One could get rid of a whole lot of unnecessary complexity with a more comprehensive set of axioms..

I don't agree with you. The features of an atom are totally dependent on the electron configuration of the atom, which you can describe with quantummechanics (harmonic oscillator etc.). The internal structure of the nucleus is irrelevant. You only need to know the electric charge of the nucleus. — Ypan1944

I believe 99.999 per cent of an atom's mass is in the nucleus. And you claim the internal structure is irrelevant to "emergent phenomenon". I think you've unnecessarily restricted your definition of "emergent phenomenon" to include only the activity of electrons.

Also in chemical reactions, only the electron configuration of the participating atoms or molecules is important. — Ypan1944

What about nuclear reactions though, as we find in the sun and other stars? On the scale of the universe as a whole, nuclear reactions are more significant than chemical reactions.

One might argue that the latter encompasses imagining that the count to infinity is complete, but one cannot imagine such a thing. — Philosopher19

The answer to your problem is quite simple. In mathematics things are done by axiom. If you want to count to infinity and beyond, simply produce an axiom which allows you to do that, and bingo the infinite is countable, and you're ready to go beyond. Look closely at the following:

The way set theory proves there exists a set with all and only the natural numbers is by an axiom from which we prove that there exists a set with all and only the natural numbers. — TonesInDeepFreeze

Then why do you think bijection requires counting? — Banno

Actually, the inverse is what is the case, counting is a form of bijection. But this does not necessarily imply that all bijections are a form of counting. And, some might still argue that there are forms of counting which wouldn't qualify as bijections. It all depends on how one might restrict these concepts though definition.

The reality of non-physical actuality is demonstrated by logic, and the logic proceeds from premises derived from physical activity. The aspects of physical activity which lead to the conclusion of the non-physical cannot be understood by "physics", so the non-physicalist concludes that these aspects of reality can be approached through other processes of understanding, metaphysical principle which allow for the reality of the non-physical.

The physicalist metaphysics however, renders these aspects of reality as fundamentally unintelligible. So for example, we have everything within the realm of science which gets designated as "random" (random mutations of genes and abiogenesis, random fluctuations of quantum fields and symmetry-breaking), being rendered as fundamentally unintelligible by physicalism, whereas the non-physicalist would argue that such things are actually intelligible, if approached through non-physicalist premises.

"Strong emergency" in the title is very eye-catching. I thought it was going to be a thread about the relationship between down-ward causation and global climate change, as an example of a strong emergency.

In my opinion, this addition by Bedau is superfluous: you do can actually describe the properties of an emergent phenomenon with "normal" physics, where its substructure is usually irrelevant, so a form of "coarse graining" will happen (f.i. with Bohr's atomic model the substructure of the atomic nucleus is irrelevant, only the electric charge of the nucleus plays a role; and you need a new theory (namely quantum mechanics) to describe that phenomenon) — Ypan1944

I don't think this is correct. The strong force of the atom's nucleus is not understood by physics. And since the negative charge of the electrons is balanced by the positive charge of the nucleus, separation of the electrons from the nucleus is not possible. There would no longer be electrons if separation occurs, but free energy as photons. So the electron's "electric charge" is dependent on the atom's nucleus for explanation, and the nucleus cannot be left irrelevant.

Supervenience is therefore completely different from "downward causation". — Ypan1944

I find you have not provided a good explanation of "downward causation", only giving a general outline, and stating distinctly what downward causation is not.

Problems occur if you consider the elements of a set to not be themselves sets. Set theory only talks about sets. It does not, for example, talk about individuals.

The lists only list other lists... — Banno

The question of what "a set" is, the definition of "set", becomes an issue when you consider the possibility of an empty set. If a "set" is taken to be the type, category, or definition which indicates the criteria of membership, then an empty set is possible. If "set" is supposed to refer to the group or collection of elements itself, then an empty set is impossible because that would be a non-existent collection of elements. The "non-existent collection" could not be understood by that definition, because that would mean "set" is understood by the category, not the elements in membership, So it is at this point, when we consider the possibility of an empty set, that we need to make a judgement about the relationship between individuals and sets.