These issues were not completely resolved in Plato's times and had to be worked out later. — Apollodorus

Actually, if you look really closely, you'll see that the issues haven't been resolved yet. I think you have something to work on Apollodorus. Get back to your studies!

As already stated, followers of Plato already referred to themselves as "Platonists" (Platonikoi) in antiquity and it would be absurd to claim that they were something else. Of course there were some variations according to different schools but that doesn't make the Platonism of one historical period a different system to the Platonism of other periods. — Apollodorus

Even by the time of Aristotle there was no such thing as one consistent "Platonism". Aristotle claimed to refute "some Platonists" with his cosmological argument. But Neo -Platonists took a position more consistent with Aristotle. So there must have been some fine tuning of "Platonism" and rejection of certain types of Platonism Therefore we can assume a reason for the name "Neo-Platonism" rather than simply "Platonism". And Aristotle himself is, in a way, a Platonist. The thing is, Plato had a lot of different teachings which could be interpreted in different ways.

Is there such a thing as "pure imagination" that does not arise ultimately from observations and experiences in the physical world? — jgill

That's a good question. But if all such principles can be said to have empirical causes, then how can you say that you have a mathematical concept which was produced purely for aesthetic beauty? If there are experiential concerns which enter into your conception then how can you say that your intention of aesthetic beauty is pure?

This is essentially the freewill vs determinism question. To make your conception pure, you'd need the capacity to mentally make a clean break between past and future. It is the assumed continuity between past and future, which forces real world concerns into our thinking. We cannot escape the reality of what we experience as what has just happened, and how this bears on what is about to happen. But if we make a break between past and future, then past experience has no necessary bearing on what we produce for the future because what has just happened will not influence our thinking about what is about to happen. Then your goal for the future, a creation of pure aesthetic beauty, could be completely free from the notion that past occurrences put a necessary constraint on your future production, and you could draw from your past experiences, in complete freedom. The creation of your aesthetic beauty could be done purely without any real world concerns, i.e. knowing that the past has no necessary causal relation with the future, allowing you complete freedom from real world pressure.

This is the issue of inherent order, which we've discussed for months now, in a nutshell. If there is order which inheres within a thing, then that order puts a necessary constraint on future possibilities of order, due to the continuity (causal relation) which we assume to exist between past and future. Logically, we want to start with the assumption of unlimited possibility, to give us the capacity to understand any possible ordering. So, we start with the premise fishfry suggested of "no inherent order". But this is not a real representation of the necessity imposed by inherent order. To remove the necessity of inherent order in a more realistic way, I think, requires that we make a clean break between past and future, annihilating the supposition of continuity, thus allowing that the order which inheres within a thing has no real bearing on the thing's future order. This would allow for the true possibility of any order, it doesn't start with the premise of no inherent order, but it rejects the order which is imposed by the supposition of continuity.

These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles. . . . — Wikipedia

I have a number of issues with this passage. Any concept which originates in real world concerns cannot be said to have been produced from an appeal to aesthetic beauty. And if we suppose that there are some of each, wouldn't it be the ones which deal with real world concerns which get accepted into the community. So, as much as I see the claim that "pure mathematicians" are motivated by aesthetic beauty, as opposed to real world concerns, I don't see that any such concepts as being produced purely for aesthetic beauty actually exist in mathematics.

The next is with the phrase " the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles". What we are discussing is the production of the basic principles themselves. If "pure mathematics" simply involves working out the logical consequences of already established principles, then it is not really relevant to what we were discussing, which is the derivation of those basic principles. The question is whether those principles ought to be derived from pure imagination, or ontology.

That is evidence that mathematics, and what fishfry calls "formal abstractions", are not separate, or independent from the world, as fishfry argues. They are not ideal perfections, separate Forms, but they must share in the imperfections of the material world, as only being useful in that world, if they are part of that world. The proposition that they could have independence from the material world is a false premise. So to create a formalism and present it as free from the negative influence of content, is to present a smoke and mirrors illusion, because the most one can do in this respect, is hide that negative influence.

I was just reading about the Frege-Hilbert dispute. As I understand it, Hilbert was saying that axioms are formal things and it doesn't matter what they stand for as long as we can talk about their logical relationships such as consistency. Frege thought that the axioms are supposed to represent real things. I'm not sure if I'm summarizing this correctly but this is the feeling I got when I was reading it. That I'm taking Hilbert's side, saying that the axioms don't mean anything at all; and you are with Frege, saying that the axioms must mean whatever they are intended to mean and nothing else.

https://plato.stanford.edu/entries/frege-hilbert/

Is this what you're getting at? — fishfry

Thanks for the reading material fishfry, I've read through the SEP article a couple times, and the other partially, and finally have time to get back to you.

I think Frege brings up similar issues to me. The main problem, relevant to what I'm arguing, mentioned in the referred SEP article, is the matter of content.

The difference of opinion over the success of Hilbert’s consistency and independence proofs is, as detailed below, the result of significant differences of opinion over such fundamental issues as: how to understand the content of a mathematical theory, what a successful axiomatization consists in, what the “truths” of a mathematical theory really are, and finally, what one is really asking when one asks about the consistency of a set of axioms or the independence of a given mathematical statement from others. — SEP

In critical analysis, we have the classical distinction of form and content. You can find very good examples of this usage in early Marx. Content is the various ideas themselves, which make up the piece, and form is the way the author relates the ideas to create an overall structured unity.

Hilbert appears to be claiming to remove content from logic, to create a formal structure without content. In my opinion, this is a misguided adventure, because it is actually not possible to pull it off, in reality. This is because of the nature of human thought, logic, and reality. Traditionally, content was the individual ideas, signified by words, which are brought together related to each other, through a formal structure. Under Hilbert's proposal, the only remaining idea is an ideal, the goal of a unified formal structure. So the "idea' has been moved from the bottom, as content, to the top, as goal, or end. This does not rid us of content though, as the content is now the relations between the words, and the form is now a final cause, as the ideal, the goal of a unified formal structure. The structure still has content, the described relations.

Following the Aristotelian principles of matter/form, content is a sort of matter, subject-matter, hence for Marx, ideas, as content, are the material aspect of any logical work. This underlies Marxist materialism

However, in the Aristotelian system, matter is fundamentally indeterminate, making it in some sense unintelligible, producing uncertainty. Matter is given the position of violating the LEM, by Aristotle, as potential is is what may or may not be. Some modern materialists, dialectical materialists, following Marx's interpretation of Hegel prefer a violation of the law of non-contradiction.

So the move toward formalism by Frege and Hilbert can be seen as an effort to deal with the uncertainty of content, Uncertainty is how the human being approaches content, as a sort of matter, there is a fundamental unintelligibility to it. Hilbert appears to be claiming to remove content from logic, to create a formal structure without content, thus improving certainty. In my opinion, what he has actually done is made content an inherent part of the formalized structure, thus bringing the indeterminacy and unintelligibility, which is fundamental to content, into the formal structure. The result is a formalism with inherent uncertainty.

I believe that this is the inevitable result of such an attempt. The reality is that there is a degree of uncertainty in any human expression. Traditionally, the effort was made to maintain a high degree of certainty within the formal aspects of logic, and relegate the uncertain aspects to a special category, as content.

Think of the classical distinction between the truth of premises, and the validity of the logic. We can know the validity of the logic with a high degree of certainty, that is the formal aspect. But the premises (or definitions, as argued by Frege) contain the content, the material element where indeterminateness, unintelligibility and incoherency may lurk underneath. We haven't got the same type of criteria to judge truth or falsity of premises, that we have to judge the validity of the logic. There is a much higher degree of uncertainty in our judgement of truth of premises, than there is of the validity of logic. So we separate the premises to be judged in a different way, a different system of criteria, knowing that uncertainty and unsoundness creeps into the logical procedures from this source.

Now, imagine that we remove this separation, between the truth of the premises, and the validity of the logic, because we want every part of the logical procedure to have the higher degree of certainty as valid logic has. However, the reality of the world is such that we cannot remove the uncertainty which lurks within human ideas, and thought. All we can do is create a formalism which lowers itself, to allow within it, the uncertainties which were formerly excluded, and relegated to content. Therefore we do not get rid of the uncertainty, we just incapacitate our ability to know where it lies, by allowing it to be scattered throughout the formal structure, hiding in various places, rather than being restricted to a particular aspect, the content.

I will not address directly, Hilbert's technique, described in the SEP article as his conceptualization of independence and consistency, unless I read primary sources from both Hilbert and Frege.

I take this to be a reference to the fact that geometers studied various models of geometry, such as Euclidean and non-Euclidean, and were no longer concerned with which was "true," but rather only that each model was individually consistent. And philosophers, who said math was supposed to be about truth, were not happy. — fishfry

Geometers, and mathematicians have taken a turn away from accepted philosophical principles. This I tried to describe to you in relation to the law of identity. So there is no doubt, that there is a division between the two. Take a look at the Wikipedia entry on "axiom" for example. Unlike mathematics, in philosophy an axiom is a self-evident truth. Principles in philosophy are grounded in ontology, but mathematics has turned away from this. One might try to argue that it's just a different ontology, but this is not true. There is simply a lack of ontology in mathematics, as evidenced by a lack of coherent and consistent ontological principles.

You might think that this is all good, that mathematics goes off in all sorts of different directions none of which is grounded in a solid ontology, but I don't see how that could be the case.

I'm saying that we can study aspects of the world by creating formal abstractions that, by design, have nothing much to do with the world; and that are studied formally, by manipulation of symbols. And that we use this process to then get insight about the world. — fishfry

I know you keep saying this, but you've provided no evidence, or proof. Suppose we want to say something insightful about the world. So we start with what you call a "formal abstraction", something produced from imagination, which has absolutely nothing to do with the world. Imagine the nature of such a statement, something which has nothing to do with the world. How do you propose that we can use this to say something about the world. It doesn't make any sense. Logic cannot proceed that way, there must be something which relates the abstraction to the world. But then we cannot say that the abstraction says nothing about the world. If the abstraction is in some way related to the world, it says something about the world. If it doesn't say anything about the world, then it's completely independent from any descriptions of the world, so how would we bring it into a system which is saying something about the world?

This is EXACTLY what you are lecturing me about! But as SEP notes, Hilbert stopped replying to Frege in 1900. Just as I ultimately had to stop replying to you. If you and Frege don't get the method of abstraction, Hilbert and I can only spend so much time listening to your complaints. I found a great sense of familiarity in reading about the Frege-Hilbert debate. — fishfry

Actually, I think that often when one stops replying to the other it is because they get an inkling that the other is right. So there's a matter of pride, where the person stops replying, and sticks to one's principles rather than going down the road of dismantling what one has already put a lot of work into, being too proud to face that prospect. You, it appears, do not suffer from this issue of pride so much, because you keep coming back, and looking further and further into the issues.

it is not the case that Frege and I do not get the method of abstraction. Being philosophers, we get abstraction very well, it is the subject matter of our discipline. You do not seem to have respect for this. This is the division of the upper realm of knowledge Plato described in The Republic. Mathematicians work with abstractions that is the lower part of the upper division, philosophers study and seek to understand the nature of abstractions, that is the upper half of the upper division. Really, it is people like you, who want to predicate to abstraction, some sort of idealized perfection, where it is free from the deprivations of the world in which us human beings, and our abstractions exist, who don't get abstraction.

i totally agree with that. i guess i just misunderstood your use of "see".

The Platonic Forms are "that which is seen" (eidos), they are not ideas or assumptions. — Apollodorus

I think Plato separates the intelligible objects (ideas and Forms) from the visible objecys.

The argument, laid out in the other thread, leads to the conclusion that there can only be opinion about the good itself. — Fooloso4

For Plato, opinion is still a type of knowledge.

The good is not something that is. — Fooloso4

What the good itself is in the intelligible realm, in relation to understanding and intelligible things, the sun is in the visible real, in relation to sight and visible things.
...
So what gives truth to the things known and the power to know to the knower is the form of the good. And though it is the cause of knowledge and truth, it is also an object of knowledge. Both knowledge and truth are beautiful things, but the good is other and more beautiful than they. In the visible realm, light and sight are rightly considered sunlike, but it is wrong to think that they are the sun, so here it is right to think of knowledge and truth as goodlike but wrong to think that either of them is the good - for the good is yet more prized. — 508, translation Grube

...as far as how Aristotle defined the good according to individual need...I would desire proof to believe it. The individual good, that is, the good for every man, is a product of the Enlightenment. — Leghorn

I believe the distinction between real good and apparent good (good as apprehended by the individual), was first presented in its primitive form by Aristotle. I do not have time now to look up a reference. However, it was definitely expounded on by Aquinas, who presented many thoughts on this distinction. So the idea was around long before The Enlightenment.

The problem as I understand it is with our inability to know the good itself. — Fooloso4

I guess that would depend on how you define "know". According to Plato In The Republic, the philosopher gets a glimpse of the good, enough to know of its existence. The good is what makes intelligible objects intelligible, just like the sun makes visible objects visible. In a way we see the sun, and in a way we know the good. So in the cave allegory, the philosopher sees that the things people look at, material objects, are just reflections, or shadows, of the immaterial objects, But the philosopher also sees the good, behind the real immaterial things, which is responsible for creating the shadows or reflections, the material things which the cave dwellers see as the real things.

From this, Aristotle proceeded to define the good, in the teleological terms of final cause, that for the sake of which. Plato gave priority to the immaterial ideas, in the cave allegory. But a principle is needed whereby material things might come into existence from the immaterial Forms (Timeaeus). This principle is "the good", which moves the will to act, in the case of human beings and artificial things, and also, the Divine Will in the case of natural things. Final cause (will) is an immaterial cause, which causes material things to come into existence from immaterial Forms.

Are all acts founded on reason? — jgill

No, not all acts are based in reason. Reason is a property of the conscious mind. But that they are not based in reason does not mean that there is not order to them, as there is order in the actions of the inanimate world. That's the difference I referred to. If all order is derived from consciousness, or "reason", we are lead toward panpsychism.

Is there an axiom in set theory that requires the display of elements of sets to be done in inherent order? — jgill

No I really do not believe there is such an axiom, as fishfry stated, a set has no inherent order. And regardless of the displayed order, one could express a statement like 'these elements without any order'. The issue is whether it is possible to conceive of something like that, 'elements without any order'. If not, it would be an incoherent concept. It is possible to state things which are contradictory, or something like that, making what is stated impossible to conceive.

But as simple symbols, rather than meaningful symbols, they may have no IO or a different IO. If I make up three random symbols from finite lines, say, would you then state the order in which I created them gives IO to the set? — jgill

You know it's an oxymoron to talk of a symbol without meaning. If it doesn't symbolize something we can't call it a symbol. Anyway, a physical symbol is not much different from any other physical thing, except it's produced with intention. Each symbol as a distinct entity has order inherent within as the relations of its parts, and as a group of three, there is order inherent in the relations between them, As symbols, with meaning, there is a reason for the order in which you write them. if they are supposed to be devoid of meaning, then why would you be making them in the first place? It's an intentional act, so there must be reasons, therefore order.

Yes, I think that's the distinction I've been describing to Luke, the difference between inherent order, and order as created by the human mind, what you call "ordering". The two are completely different, and how they relate is in some respects "unclear".

The issue I brought up with fishfry is the distinction between representation and imagination. Fishfry allows that "abstraction" might encompass both of these, such that imaginary ideas could be a useful part of a representative model. But this was not borne out by fishfry's map analogy. The purely imaginary in this case, is the thing with no inherent order, as a "set". Another example of the purely imaginary is infinity.

The point that I am making, is that the current order which a thing has, the inherent order, limits the possibility for future order. Therefore the possibilities for ordering (by the mind), in any true sense, must be limited by an assumed inherent order that a thing has. To remove the necessity of assuming an inherent order, for the sake of allowing infinite possibilities of ordering, is a fiction of the imagination, which has no place in a representative model.

So here's an example. Suppose there's a set which contains the "numbers" 3,4,5. The "numbers" are assumed to be abstract objects. As abstract objects, they necessarily have meaning. The meaning which the abstract objects have, necessarily gives them an order, an order which inheres within that group of objects, the set. You cannot have five without first having four, and you cannot have four without first having three. Or even if we define purely by quantity, there inheres within the meaning of "quantity", more and less. Therefore there is an order which inheres within that set of abstract objects, necessitated by the meaning of the objects. To assume to be able to remove that order, which is essential to the existence of those abstract objects, and therefore inheres with the set, for the sake of claiming that any ordering of the set is possible, is a mistaken adventure.

There is one thing about the relation between inherent order, and ordering, which is very clear. This is that to represent the possible orderings of any group of objects, one must have respect for the limitations imposed by the order that they already have. To deny that they have any inherent order is to produce a false representation based in imaginary fiction. And to give abstract objects a special status as fishfry does, to allow exemption from this rule, is to render them free from the restrictions of definition which enforces logical relations (order) between them, thus allowing the imagination to run free.

Anytime.

What do you think he believed? — frank

Socrates professed to know nothing, so his beliefs are scarce.

And could you explain the Euthyphro dilemma? What do you take away from it? — frank

I think that the Euthyphro dilemma was simply presented as an indication of the problems existing within the religious thinking of his day. It's not meant to be resolved, just like Zeno's paradoxes are not meant to be resolved, it's meant to show a problem in the habitual ways of thinking that are prevalent. It's a skeptic's tool, to show deficiency in the accepted ways of thinking.

If I remember correctly, the problem is in the relationship between God and good. I think the lesson is that it's wrong to even define "good" in relation to the gods, or even in relation to God, because it can't be logically done. And so with Plato and Aristotle there's a moving away from this, toward a pragmatic definition of "good", "that for the sake of which", in Aristotle, the end. This defines "good" in relation to what the individual person needs or wants. Then there's the further distinction of the real good, and the apparent good. The apparent good is what appears to the individual as what is wanted, and "real good" is an appeal to some reasoned or rational good, beyond the individual's wants. So the ethical human being seeks to produce consistency between the real good and the apparent good, such that what is wanted is the rational end.

You are assuming the existence of an inherent order that lies beyond conscious recognition. Is there another aspect of mind that might register this phenomena? Is the fact we can discuss IO due to this possibility? — jgill

I don't think I'd call it an aspect of mind, rather an aspect of life. It is evident, that at the most fundamental level, living beings make use of inherent order, by creating extremely complex molecules, etc.. I don't know how the inherent order is recognized at this level, but it must be, in some way, in order for this organizational activity to occur.

I wouldn't say it is an aspect of "mind" that recognizes it because then we get into panpsychism, or something like that, and I think that there is a clear separation, or difference between mind, along with consciousness, and what happens at this fundamental level. So I think it is good to keep them separate, and understand that this is not a part of mind though mind may be a part of this. But the part only performs its particular function, without understanding its relation to the whole.

To me, that means he did not uphold the conventional rules of what it means to be pious, in his society. I think it says very little about his beliefs, it says something about what he refused to believe. Do you acknowledge the difference between rejecting a belief, and replacing the believe with something else? Skepticism rejects beliefs, and many people would argue that skepticism is a matter of replacing those rejected beliefs with others, but I don't think that this is really a good description.

You would need to apprehend the inherent order in able to compare and judge the representation as good or bad. — Luke

This is not true, we use empirical evidence, and deductive logic. It's called science. The hypothesis is the representation, and it is judged according to the evidence and logic.

Hi Frank,
I wouldn't say that Socrates was atheist, I would say that his form of theism was non-conventional. Couldn't we say the same of Jesus, that his form of theism was non-conventional? But no one would say that Jesus was atheist. I think both were accused of blasphemy, but that doesn't make one atheist.

o you are saying that possibility has no regard for truth or falsity, i.e. no regard for the inherent order. I still have no idea what this means. — Luke

No, I am saying that the person who assigns possibility, in that situation, has no regard for truth or falsity, in that act. How could possibility be the type of thing which might have a regard for truth or falsity? Your interpretation is simply ridiculous, and I can't see any reason for such a ridiculous interpretation other than that you intentionally make an unreasonable interpretation in an attempt to make what I say appear to be unreasonable.

But what regard should the inherent order be given if it cannot be perceived or known? — Luke

You are ignoring the fact that I repeatedly said that we see the inherent order without apprehending it with the mind. You just can't seem to grasp this fact of reality, that we see things without understanding what is seen. To you, this is pure contradiction, but until you grasp it, you will never understand what I've been saying.

So, when the mind produces an order, which is supposed to be a representation of the inherent order, within the thing, the order which is being sensed must be regarded in order that the representation be a good one.

Yeah, that's why I asked. It's a bullshit assumption that can't be known. — Luke

If you are convinced that the assumption of an inherent order is a "bullshit assumption", then why didn't you just say this two months ago, and we could have avoided all of your nonsense bad interpretation, and out of context quotes, in your effort to make it look like what I am saying is contradictory?

But no, this you MO, to produce nonsense interpretations, and out of context quotes, with the intention of making it look like the author is inconsistent. So of course you couldn't have been up front with your difference of opinion. You had to carry on and on, in the pretense that you were trying to understand, but couldn't, to get more words, more phrases, sentences, and statements, as ammunition in your pointless attack, without any intention of trying to understand. And then you call me a troll!

Maybe it took you this long to figure it out, that the assumption of an inherent order is a bullshit assumption, but if this is what you believe, then there's no point in going any further with this discussion, because I have no desire to try to convince you otherwise. Because of this belief which you have, that inherent order is a bullshit assumption, there is no point in discussing the relation of inherent order to a set, because you do not believe there is any such thing as inherent order in the first place.

But isn't what is being measured simply the changes that take place in the relation objects have to one another? — Apollodorus

No, what is being measured is the passing of time, which is better described as the rate of such changes. The "rate" requires relations between changes themselves, not relations between objects, but relations between changes..

My question, again, what do you mean possibility has "no regard for truth or falsity"? — Luke

I replied to this, in the last post go back and read it. That is such an unreasonably bad interpretation of what I said, that I think any reasonable person could only have presented me with such a thing intentionally.

So tell me what is the order of the three coloured balls before they are drawn from the bag. — Luke

We've gone through this before. I can't tell you the inherent order.. Luke, you've got an extremely bad habit of getting me to spend endless time explaining something to you, then you start right back at square one, as if we've never talked about it before.

So, to assist you in understanding, I'll use different words than the last time. Then, you turn around to the last time, and say 'look, these words are different from the last time, therefore you contradict yourself'. And you say I'm a troll!

Can you tell us how the imperceptible, unapprehendable inherent order could be useful to anyone? — Luke

It's useful to recognize the reality of it, to understand the deficiencies of mathematics.

I think it might be useful for you to distinguish the two principal senses of "time", as Aristotle did.

First, "time" is primarily used to signify a measurement. So we say how long something takes by referring to a unit of measurement, hours, minutes, seconds, etc., In this sense, "time" refers to a measurement of duration, and from this usage we derive "velocity" and all sorts of concepts of physics.

In a secondary sense, "time" refers to the thing measured. This use of "time" is a little bit more difficult to grasp, but to give substance to our measurements of duration, we must assume that there is actually something being measured. We might call this the passing of time, or something like that.,

What's not true? You said: "(sets) can be assigned any possible order (in the sense of humanly created order), with absolutely no regard for truth or falsity." I asked what it means for the possibility (of the order) to have "absolutely no regard for truth or falsity". — Luke

I said, the assignment of possibility is done without regard for order.

Sorry Luke, your interpretation is so bad (no wonder you see contradiction in everything I say) , I have extreme difficulty communicating with you. I don't see how any reasonable mind could interpret the way you do, therefore I can only conclude that you make these unreasonable interpretations intentionally.

A set is not a bag of items, it's an abstraction, that's the point fishfry has been stressing. A bag of items has an inherent order, as I've spent months describing to you. If we want to represent that bag of items as a thing called "a set" we cannot truthfully predicate of that subject, the property of "no inherent order" because the thing being represented necessarily does have an inherent order. No inherent order is a false representation.

Fishfry claims that a pure abstraction is an imaginary fiction, so it doesn't matter that it's not a true representation, and claimed that the imaginary fiction is useful toward a "higher truth". However, fishfry insisted on using a map analogy for explaining abstractions, and a map is a representation, so there is inconsistency in what fishfry was presenting. Furthermore, fishfry could not explain how an imaginary fiction could be useful toward obtain a higher truth.

Has that been the basis of your argument from the start? Funny, since I've seen you argue against the eternalist block universe in other threads. You really are a troll. — Luke

Going through this thread, and taking statements out of context isn't enough for you, so now you have to refer to other threads. You bring "taking things out of context" to a whole new level. I really am a troll but you're just an ass hole.

Possibility has "no regard for truth or falsity"? What does that mean? — Luke

That's not true. Possibilities are limited by the actual state of the world. Anything claimed to be possible, which is not allowed for by the present state, is actually impossible.

You can't have impossibility without possibility. — Luke

And this is not even true. If determinism is the true description of reality, then true possibility is actually impossible, such that we would have the impossibility of changing the eternalist block universe, without any real possibility.

I've spent the last couple of posts saying that math is a lie, math is fiction, math is untruth in the service of higher truth, and you put words in my mouth. It's not fun and there's no point. — fishfry

I didn't see any mention of "higher truth". And I really don't think you've provided any explanation as to how lies, fictions, and untruths could be "in the service of higher truth". That sounds similar to Plato's "noble lie", where the rulers of the State lie to the common people for the sake of their own good. The proposal that such lies and deception are for the sake of a "higher truth" is extremely doubtful.

The problem we have here is that you seem to believe that mathematics give us idealized exactness, when really it fails at this. So this is a self-deception on your part. It all seems to stem from your idea that you can separate a "pure mathematics" as pure abstraction, away from applied mathematics, so that this pure mathematics deals with idealized exactness, while applied mathematics deals with the imprecision of real world measurement. You do not seem to understand that those who engage in so called "pure mathematics" are actually working on ways to solve the problems of imprecision in real world measurement, not trying to create pure abstractions. The problems of real world imprecision are not solved by axioms of idealized exactness, because of the fundamental incompatibility between the two.

Here's a proposal for you to consider. Imagine that human beings, when they first came out with mathematics many thousands of years ago, believed that mathematics provided them with ideal exactness. So we go way back, before Pythagoreans, Egyptians, or even Babylonians, and see that the people knew things from math, such as the example that they could derive a perfect right angle all the time, and they believed that mathematics gave to them idealized exactness. However, there were slight problems in applying numbers to spatial projections, such as the irrational nature of the square root of two, and pi.

Spatial projections are a creation based in real world appearances. I do not think you would deny this. So if we say that numbers are based in this idealized exactness, then when they are applied to real world problems, spatial projects, an incompatibility appears. The mathematician is faced with the problem of solving this incompatibility. But the mathematician is incapable of altering the reality of our spatial temporal existence, so there is no choice but to alter that lofty "idealized exactness" of the fundamental mathematical principles, numbers, if that incompatibility is to be resolved. The mathematician therefore is attempting to produce axioms which will bring numbers away from idealized exactness.

What I propose for you to consider then, is that mathematics may have been based in the idea of idealized exactness, many thousands of years ago, just like religion might have been based in the idea of an eternal immortal soul, but the progress which is made in mathematics, by mathematicians, is to bring us down from this idea, bring us away from it, to make mathematics something more compatible with the real world. For mathematicians to be working in some lofty world of ideal exactness, while this is completely incompatible with the real world, is pointless. So what the mathematicians are really doing is finding ways to bring the principles down out of that pointless realm of ideal exactness, compromise them in a way which makes them applicable, while maintaining as much exactness as possible.

You do need to understand the concept of the necessary approximateness of all physical measurement. I can't imagine why you would take a stance so fundamentally wrong. — fishfry

You see, my stance is directionally opposed to yours. Which one is fundamentally wrong is debatable. You seem to think that the way to "higher truth" is for the mathematician to work with "pure abstractions", of ideal exactness, which have no correlation to anything in the real world. I think the way to "higher truth" is to rid mathematics of such "pure abstractions", (because they are based in nothing but imagination), and to find "higher truth" we need to replace these principles with principles that correlate with the real world, even if this means to forfeit "ideal exactness". We both know that ideal exactness is impossible in the real world, therefore it cannot be a good principle upon which to judge mathematical principles which have the sole purpose of real world measurements.

You strongly imply that the inherent order is able to be apprehended in these quotes. We must be able to apprehend the inherent order if it is "describable" and we are able to see it. — Luke

You are not respecting the difference I described between what "order" refers to, and what "inherent order" refers to. The former we can apprehend, the latter we cannot, though they are both fundamentally intelligible. That's why I said, ultimately they can both be placed in the same category, I'll call it "intelligible".

I explained those differences already, and you are simply taking things out of context. The "inherent order" is fundamentally intelligible, but not by us, due to deficient capacity. Earlier I used the analogy of the way theologians speak of God. God is fundamentally intelligible, but not intelligible to use. Also, as Kant describes, noumena is fundamentally intelligible, but not by us.

That's right, and then you forced upon the conversation your idiosyncratic idea of "the inherent order" that is unrelated to sets or ordering in mathematics. Fishfry and Tones tried telling you this, but you weren't interested. — Luke

In this entire thread, no one but jgill has offered any sort of alternative explanation as to what "inherent order" could refer to. Clearly it refers to an order which inheres within something. Jgill proposed that it is the order which inheres within a biological organism. But I see no reason to restrict this term to living things, as inanimate things also display reason to believe they have an inherent order. Until you bring forth another idea for "inherent order", it appears like you have no reason to say that mine is idiosyncratic, it really seems to be the only coherent understanding of "inherent order" possible.

You seem to be missing the point of my argument. By removing "inherent order" from the things called sets, as fishfry did, with the assumption of "no inherent order", these things (sets) can be assigned any possible order (in the sense of humanly created order), with absolutely no regard for truth or falsity, because it is stipulated that the set has no inherent order. My point of contention is that there is no such thing as something with no inherent order, it is an impossibility as self-contradictory, a unity of parts without any order to those parts. So this concept of a set, as a thing with no inherent order, is fundamentally wrong, and ought to be rejected.

Thanks? I guess. But this does not answer the question of how your concept of "the inherent order" relates to "order" more generally. You could start with your own ideas of "order" and "the inherent order" and explain how these relate to each other. Why is "the inherent order" not a type of "order"? — Luke

What is referred to by "inherent order" is not a type of what is referred to as "order" because of the separation between them. This is described by Kant as noumenal and phenomenal. Inherent order is within the thing, as essential to the existence of the thing being the precise thing that it is. This is associated with the law of identity, which refers to the uniqueness of a thing. It has a unique inherent order, which makes it fundamentally intelligible as the unique thing which it is, and discernible from other unique things. However, the human mind does not grasp and understand the uniqueness of the thing, it grasps the thing relative to others, by similarities of universals, abstractions.

So the "order" understood by the human mind, created by the human mind as universal, is fundamentally different from the order inhering within the particular individual. If we said inherent order is a type of order (as conceive by humans), we'd dissolve the distinction between universal and particular, in a category mistake, making a particular into a universal. The law of identity prevents us from doing this.

How what is referred to as "inherent order" (within the particular) is related to what is referred to as "order" (a universal created by the human mind), is that they are both fundamentally intelligible. The former is not intelligible to the human being though. We could switch to the Aristotelian term "form" here. Aristotle distinguished two principal senses of the word "form", the form which inheres within the particular object, making it the unique object which it is, and the "form" which we assign, in describing the object, which involves universal abstractions. The latter, the form we assign to the object, neglects, or leaves out, the object's matter. "Matter" is assigned to the particular object to account for its accidents, the parts which are not grasped by the human mind, and this accounts for the object's uniqueness. Under this structure, the form of the particular, its inherent order, complete with accidents, is fundamentally intelligible, but not intelligible to the human intellect which understands through universal forms (orders), leaving the particular, inherent order, incomprehensible.

Have you considered that what you say might appear to be contradictory because it is contradictory, and that the problem is with your metaphysical edifice rather than with my understanding? — Luke

You haven't given me any real evidence that this might be the case, so no. I'll continue to wait for you to produce some substance, and indication that you understand, rather than demonstrating that you can search keywords throughout a lengthy thread, and take quotes out of context to produce the appearance of contradiction.

The Pythagorean theorem in the real world is literally false. It's close but no cigar. It's approximately true, that's the best you can say. But the point here is that you are on record claiming the Pythagorean theorem is "very true." So you are not in a position to deny saying that. — fishfry

"Idealized exactness" is not "truth". The Pythagorean theorem is very true in the real world. Where we disagree is on what constitutes being true. That has been obvious all along, you allow that fictions like "no inherent order", may be a part of your idealized exactness, thereby compromising your supposed truth with falsity.

Idealized mathematics (as opposed to say, numerical methods or engineering math, etc.) is perfectly exact. That's its supreme virtue. — fishfry

No fishfry, "infinity" is in no way perfectly exact. You did not address this, and there are a multitude of other examples of the imprecision of mathematics, such as the mathematician's ability to choose between incompatible axioms, and the various different roles which "zero" plays, as evidenced by imaginary numbers.

Now it seems to me that the starting point for an interesting discussion is to note that the Pythagorean theorem is literally false in the world, and perfectly exactly true in idealized math; and from there, to meditate on the nature of mathematical abstraction. How we can literally tell a lie about the world, that the theorem is true, and yet that lie is so valuable and comes to represent or model an idealized form or representation of the world. — fishfry

You keep insisting that the Pythagorean theorem is false in the world, but it is used many times every day, and every time it is used it proves itself. Where's your evidence that it's false? And if you produce competing mathematical principles as your evidence, you are only proving the inexactness of mathematics, not the falsity of that particular theorem..

But if you deny both these premises, one, that the P theorem is false in the world (close though it may be) and perfectly true in idealized math, then there is no conversation to be had. And for what it's worth, your opinions on these two statements are dramatically at odds with the overwhelming majority of informed opinion. — fishfry

Of course I deny those premises. They are both false. But we have different ideas of what constitute truth. I think truth is correspondence with reality, you think truth is some form of idealized exactness. But "idealized exactness" does not even describe mathematics in a true way. How can justify your claim that the Pythagorean theorem has idealized exactness when the square root of two is irrational? What defines an irrational ratio is that it is lacking in perfect exactitude. So both you premises are false. The Pythagorean theorem is not false in the real world, it gives us the right angle every single time, and there is no perfection to its truth in idealized math, because it gives us irrational ratios. See both your premises need to be rejected.

The fact that Moby Dick changed the name of the ship from the Essex to the Pequod, changed the names of the characters, and invented episodes and stories that never really happened, does not detract from the novel in the least. A representation or abstraction stands alone. — fishfry

You still do not seem to have any respect for the difference between a creative work of the imagination, and a representation. A "representation" cannot stand alone, because it necessarily represents something, that's why it's called a representation. If it stood alone it could not be called a representation. An imaginary fiction, like Moby Dick, stands alone as a creative piece of art, not meant to represent anything. It is not a representation.

This problem appears to deeply affect the way that you think about truth. You seem to think that a representation can be true without any rules of correspondence, simply by standing alone. Of course this is not true because it is the rules of correspondence which provide for the truth or falsity of any representation. Consider your map analogy, the key, or legend, tells you what the symbols stand for, allowing for the truth or falsity of the map.

Your map analogy fails because of your desire to extend it into the artistic world of fictitious creations which are not meant to represent anything, and therefore not similar to maps. Here, we have works of art, created by imaginative power, which are enjoyed for aesthetic beauty, This is where you place your "idealized exactness" striven for by mathematicians in their acts of imaginative creation, as a high form of beauty. If mathematicians could obtain to that highest level, ideal exactness, they might create the highest form of beauty, "truth". However, idealized exactness is not a part of the real world, just like "no inherent order" is not a part of the real world, nor is infinity part of the world, while mathematicians and mathematics are parts of the real world. So these beautiful works of art produced by the mathematicians, which have great aesthetic beauty, but do not represent anything, are simply beautiful works of art, which, as any other part of the world, contain imperfections.

Now, you present these works of art to me as "representations", and claim that there is truth within them, as "idealized exactness". However, they very clearly do not obtain to that level of "ideal exactness" so if ideal exactness is supposed to be truth for you, then these works of the mathematicians are obviously not true.

It obtains it every day of the week. It obtained idealized exactness in the time of Euclid. — fishfry

I think Euclid's parallel postulate is somewhat questionable in some modern geometry. You've just given me more proof that idealized exactness has not been obtained. If it had been obtained, there would be no need for new forms of geometry which cast doubt on the old. Geometry works in the field, in real world situations it gives us truth, but it clearly does not give us the ideal (absolutely perfect) exactness, which you seem to believe it does.

There are no dimensionless points, lines made up of points, and planes made up of lines in the world. — fishfry

Hmm, an infinity of dimensionless points could not produce a line with dimension, more evidence that ideal exactness has not been obtained.

The mathematical theory of infinity is a classic example of an abstraction that has nothing at all to do with the real world. And yet, without the mathematical theory of infinity we can't get calculus off the ground, and then there's no physics, no biology, no probability theory, no economics. So THAT is the start of an interesting philosophical conversation. How does such a massive fiction as transfinite set theory turn out to be so darn useful in the physical sciences? Where's Eugene Wigner now that we need him? — fishfry

I had no doubt that you'd have good things to say about infinity in mathematics, but you didn't address the point. The use of infinity in mathematics is clear evidence that mathematics does not not consist of idealized exactness. And now that you mention it, calculus itself is based in principles of allowing less than perfect exactness, with notions like infinitesimals.

Is this a reference to what you've been trying to talk to me about from time to time? Universals, and how they bear on mathematical abstraction? What does it mean, exactly? After all I frequently point out to you that mathematical ontology posits the existence of certain abstract entities, and this is exactly what you deny. If I understood this point about universals better (or at all, actually) I'd better understand where you're coming from. — fishfry

Yes, well maybe we'll continue this discussion for a few more years.

This was before you let anyone know that the inherent order was noumenal and invisible, which is right around the time I believe you changed your position. You started from this position: — Luke

I didn't change my position. We've been through this already. You misunderstood my use of words. I went back and explained how the position was consistent, but the choice of words was difficult.

That is, you started out telling us that the actual/inherent order can be perceived with the senses and apprehended, then you changed your position to say that the inherent order cannot be perceived with the senses or apprehended, and now you're saying that the inherent order is invisible but it can (again) be perceived with the senses. At least, that's your latest position. — Luke

This is false, I never said inherent order is apprehended. I've remained consistent and I've clarified this already.

If inherent order is not a type of order, then I don't understand what you have been arguing about regarding mathematical order. Why did you previously allow for other types of order, such as best-to-worst? — Luke

If you recall fishfry introduced "inherent order" by claiming that a set has no inherent order. I haven't been using "mathematical order" so I don't even know what you're talking about here. Near the beginning of the thread there was no consensus between the participants in the thread as to what "order" referred to. I developed the distinction between inherent order, and the order created by the mind as the thread moved on.

You previously spoke of "perceive" and "apprehend" as opposing concepts, but now you consider them synonymous? For a long stretch of the discussion, you repeated in various forms that we perceive with the senses, as distinct from apprehending with the mind: — Luke

OK, so I should have used "apprehend" then, and "perceive" was not a good option. As I said the choice of words is difficult, that is the nature of ontology. Just one little mistake after weeks or months of trying to explain the same thing to you over and over again, in as many different ways as possible, day after day, I think that's pretty good. You know, trying to explain the same thing in many different ways, so that a person who is having trouble understanding might have a better chance to understand, requires saying the same thing with different words. The appearance of contradiction is inevitable, to the person who refuses to look beyond the appearance, and try to understand what the other person is trying to say.

Your response to my last post makes it overwhelmingly clear that you are trying to see contradiction in my words, and not trying to understand. What a surprise!

Meta, I'm going to withdraw from this phase of our ongoing conversation. Perhaps we'll pick it up at some time in the future. If you can't agree that real world measurement is necessarily imprecise and that mathematical abstraction deals in idealized exactness, we are not using words the same way and there is no conversation to be had. I don't think you would be able to cite another thinker anywhere ever who would claim that physical measurement is exact. That's just factually wrong. — fishfry

I did not claim that physical measurement is exact. We agree that real world measurement is necessarily imprecise. Where I disagreed is with your claim that mathematics has obtained ideal exactness. That is what is factually wrong. Some mathematicians might strive for such perfection, and I would not deny that, but they have not obtained it, for the reasons I described.

Principally, mathematics has a relationship of dependency on physical world measurements which I described. This has ensured that the imprecision of physical world measurements has been accepted into the principles of mathematics. The lofty goal of idealized exactness has always been, and will continue to be, compromised by the need for principles to practise physical measurement, where idealized exactness is not a requirement. Therefore mathematics will never obtain idealized exactness. Look at the role of infinity in modern mathematics for a clear example of straying from that goal of idealized exactness.

Right. Sorry for assuming that we're talking about humans. Once we meet aliens or once we evolve to the point where we classify as a different species then yes, we may see more. — khaled

As I said, knowledge is a cumulative thing. Do you not agree that human beings have knowledge within themselves, instinctual knowledge, which was acquired by earlier life forms? If so, then you ought not define "what can and cannot be known" by the limitations of the human life form.

It is justified by definition. You have access to the reason in your mind. You don't have access to "the reason that orders the world". — khaled

This is completely untrue. Human beings communicate. Through communication we have access to what is in the minds of others. And we only have access to the minds of others through the medium which is the physical world. Therefore we must have access to the physical world. You can deny that this is "access", but what's the point to restricting the use of "access" in this way? You might as well say that we don't have access to anything and we know nothing. What good is such a claim?

Any theory about "the reason that orders the world" is just that, a theory. As long as it accounts for own reasoning and perceptions the only thing separating it from any other theory is Occam's razor. — khaled

So, what's wrong with having theories? Remember, you claimed that talking about "the reason that orders the world" is pointless. Are you now claiming that theories, in general, are useless? That's not true, theories are very useful.

If you do believe in evolution then you ought to believe that it is more likely than not that our reasoning and perceptions are incomplete. Not only is this supported by experimental evidence (Hoffman) but also we can easily find scenarios where there are things we cannot detect that affect us, such as UV light. — khaled

I'm sorry Khaled, but I cannot see your reasoning. You are claiming that because our reasoning and perceptions are incomplete, we ought not make any effort toward completion. How is such a defeatism ('because I don't have it I ought not try to get it') the approach of a rational being?

That you don't understand that all physical measurement is approximate, and that math deals in idealized exactness that does not correlate or hold true in the real world, is an issue I would have no patience to argue with you. You are simply wrong. Physical measurements are limited by the imprecision of our instruments. This is not up for debate. But I do see a relation between your misunderstanding of this point, and your general failure to comprehend mathematical equality. — fishfry

The fact that you believe that mathematics deals with "idealized exactness", is the real problem. Look at the role of things like irrational numbers and infinities in conventional mathematics, these are very clear evidence that the dream of "idealized exactness" for mathematics is just that, a dream, and not reality at all, it's an illusion only. Idealized exactness never has been there, and probably never will be there.

You seem to deny this brute fact concerning mathematics, to insist on a separation between real world measurement (deficient in exactness) and ideal mathematics (consisting of perfect exactitude). You hide behind this denial, to completely ignore the reality that the principles of mathematics have been created from the acts of, and for the purpose of, real world measurements.

These are two facts you need to recognize, 1) Mathematical principles have been derived from acts of measurement, and 2) Mathematical principles are created for the purpose of measurement. Since this separation which you espouse cannot be accomplished, due to the fact that the principles of mathematics have been derived from the practise of measurement (1), as I explained with the example of the Pythagorean theorem, you ought to dismiss that intent to separate, altogether. And, because measurement is the purpose of mathematics in its practise (2), it is itself an instrument of measurement. So your observations that physical measurements are limited by "the imprecision of our instruments" ought to inspire you to a recognition of the imprecision of our mathematics.

Inherent order is only one type of order (you also allow for other types such as best-to-worst). How is it that we do not perceive order with the senses in general, but that we do perceive inherent order with the senses specifically? — Luke

I can't answer the how, but I have answered the why. The two types of order are completely distinct and different.

It was not until recently that you began arguing that we do perceive inherent order with the senses and can "see" or otherwise "sense" invisible physical entities such as molecules, ultraviolet light, and the inherent order. — Luke

Go way back, to when I said "see" the inherent order in the dots on the plain in the diagram.

You will note I maintain the distinction here between order and inherent order. You must have been aware of this distinction in your own response when you contradicted your latest argument and affirmed that: "1) We do not perceive order with the senses". It is therefore a complete fabrication to attribute your own contradiction to my misunderstanding or lack of awareness of the distinction between order and inherent order. — Luke

Right, that's a good quote, showing context. I think I generally indicated inherent order with the word "inherent", or "inheres within", indicating order within the object itself, (noumenal if that helps). If I just said "order", I likely was referring to the type of order created within the mind.

You need to recognize the complete separation between what is referred to with "inherent order" and what is referred to with "order". Inherent order, as inhering within the object, is not a type of order, as created by the mind, like the description indicates, this is impossible. The complete separation is required by their contradictory natures. However, there may be similarities by which we could place both, order and inherent order, into one category, but we haven't approached that yet.

In the quoted passage you seem to be looking at what is referred to by "inherent order", as a type of what is referred to as "order". This would constitute a misunderstanding, they are completely distinct and one is not a type of the other.

In other words, you explicitly state here that we do not sense the inherent order specifically. — Luke

Sorry, that was a mistaken statement, instead of "sense" I should have used a better expression, like "perceive" or "apprehend". I was flustered by your ridiculous claim that I had earlier implied that sense was not involved at all. This is the complete context:

Of course sense perception is involved! Where have you been? We've been talking about seeing things and inferring an order. My point was that we do not sense the order which inheres within the thing, we produce an order in the mind. I never said anything ridiculous like we do not use the senses to see the thing, when we produce a representation of order for the thing. — Metaphysician Undercover

I should have said "my point was that we do not receive, from the senses into the mind (apprehend), the order which inheres within the thing, we produce an order within the mind". This would allow clearly that the inherent order is present to the senses (is that a better way to say it?), as I had been describing. The intent was to establish the complete separation between the order constructed, and the order inherent in the object, described above. To clarify, the inherent order is present to the senses, but not present to the mind, when the mind produces a representation. "Present to the senses" I have been arguing qualifies as being sensed, but in the quoted passage I mistakenly said that this is not a case of being sensed

Again, I apologize, that was a sloppy post. I was a little rushed. and extremely put off by your claim that I was saying sense was not at all involved in the act of showing, so my reply was a reflex, consisting of a poorly chosen word, rather than clearly thought out. If you understand what I have presented in this post, you'll see that the senses are the medium which separate the order produced in the mind, from the inherent order which exists within the object. And this is why the order produced by the mind is completely distinct from the order which inheres within the object, though it is very true that the senses, and sensation have a relation to both of these distinct things.

In this particular case, there is no direction towards the answer. Rather, every direction is as good as any other. — khaled

I see no reason to agree with you. And I did read your statements. You stated a personal opinion; "there is no use in talking about 'the reason that orders the world'". And you made a further statement about your personal resignation; "I don't have access to 'the reason that orders the world' so I don't care about it."

Nowhere have I seen the claim that a human being has no direct access to the independent ordering of the world justified. Plato argued that the philosopher does have access to it through the means of apprehending "the good". This is the point of the cave allegory. And, it is the described responsibility of the philosopher to turn around, and go back to the others to assist them in their enlightenment.

So the statement, "no human being can have access to the reason that orders the world" is absolutely unsupported, as far as I can tell, yet the statement "it is possible for a human being to access the reason that orders the world" is flimsily supported. Flimsy support out weighs no support by an infinitely large magnitude, so I choose the flimsy support for my opinion; while your opinion ought to be banished from the philosophical mind as that held by those who are satisfied to be trapped in the cave of illusion for all eternity.

And this is true of everyone. It's not about the limitations of the individual but the limitations of being human. — khaled

It appears like you do not believe in evolution then. If these limitations are truly the limitations of being human, as you believe, they are still not the limitations of being alive.

Only a fool would want to know something they know they can’t know. — khaled

Even if you know that you will never know the answer to a specific question, you can proceed in the direction toward finding the answer, and potentially help others, who are not so helpless as you, to find that answer. That the answer will not be found by you does not mean that it will not be found, so this ought not prevent you from working toward finding it. There's an interesting aspect of knowledge, it's cumulative, and not restricted by the limitations of the individual.

You have stated both that we do and do not perceive (i.e. see) order with the senses. This is not a failing on my part. — Luke

It suddenly occurred to me today, why you are having so much trouble understanding. It's not so much the ambiguous use of "see" which is throwing you off, but I now see that you are not respecting the distinction between the two completely different referents for "order", which I thoroughly explained to you.

In the case of "inherent order" the order is within the thing sensed. It is sensed (in the manner I described), but not apprehended by the mind due to the deficient capacity of the sensing being. I've also used "order" to refer to orders created by the mind, within the mind, sometimes intended to represent the inherent order, as a model does. This order is apprehended by the mind, being created within the mind, but it is in no way sensed, because it is created within the mind and is therefore not part of the thing sensed.

You can see that in one context the referent of the word "order" is sensed but not apprehended by the mind, while in the other context the referent order is apprehended by the mind, but not sensed. Without adhering to the particulars of the context, and maintaining the differentiation between the two very distinct things referred to with the word "order", it would appear like "order" is used in a contradictory way; both sensed and not sensed, apprehended by the mind and not apprehended by the mind This is what you have been doing, taking my statements concerning "order" out of their context, failing to respect the described difference between the two distinct types of order, and claiming that I have contradicted myself

This is an important point for you to recognize. It's not in the real world, (where truth and falsity is determined by correspondence), where the Pythagorean theorem is false, it's tried and tested in the real world, and very true. It's only in you imaginary world, of so-called pure abstraction, where the only test for truth is logical consistency, or coherency, that it appears to be false. All this indicates is that your imaginary world is not to be trusted, as it does not give us coherency between even the most simple mathematical principles. On the other hand the Pythagorean theorem alone, can be trusted, because it does give us the right angle. So the quest for logical consistency, or coherency, is not a quest for truth..

It's not complicated; you contradicted yourself. I think you see that now, which is why you have given up. — Luke

So, your failure to recognize the distinct ways that I used "see", which I explained over and over again, constitutes contradiction on my part. OK, I must have contradicted myself then, according to the way that you use "contradicted", therefore I give up.

As such, I don't care about "the reason that orders the world". Maybe it is the same as the reason in my mind, or maybe the reason in my mind is just an "evolutionary shortcut", a hack, a parody of the real thing optimized for survival. Either way, I don't have access to "the reason that orders the world" so I don't care about it. — khaled

The philosophical mind has the desire to know. So such statements are very unphilosophical.

That is the point that I was trying to make. I think it calls into question Kenosha Kid’s view that there is ‘one objective reality’ which all interpretations try to approximate or interpret. I agree that reality may be one, but that unity must necessarily transcend subject-object dualism, meaning that it’s out of scope for naturalism as such. — Wayfarer

What I find is the biggest problem with the materialist view is that it inevitably leads to determinism. The determinist perspective is "that there is 'one objective reality'", and this objective reality encompasses all of the past and future, in an eternalist sort of way.

This perspective completely ignores the very real, important and significant, difference between past and future, which we know very well through our experience. Ignoring this difference, and the fact that the undetermined nature of the future gives us the capacity for freely willed actions, while the fixed nature of the past renders us helpless in any desire to change what has already occurred, presents us with a very skewed conception of "one objective reality". The difficulty in understanding "objective reality" is the need to know how the undetermined becomes determined at the moment of the present.

Say, did you know that the Pythagorean theorem is false in the real world? — fishfry

No, you've got that wrong. The Pythagorean theorem is true in the real world, because it works well and has been proven. Where it is false is in your imaginary world. It works very well for me. I use it regularly. That you think my right angle is a wrong angle is a bit of a problem though. We know induction is not perfect, it just describes what is experienced or practised. (Am I spelling practise wrong?) That the Pythagorean theorem is false in your imaginary world which you call "abstraction", is just more evidence that what you call "abstraction" is not abstraction at all, but fiction.

Do you think that we can see infrared and ultraviolet light just because it exists in the world? — Luke

Yes, I think the eyes most likely do sense infrared and ultraviolet in some way: https://www.sciencedaily.com/releases/2014/12/141201161116.htm

I have not misunderstood. — Luke

Yes, you are very clearly misunderstanding, and I'm tired of trying to explain. You don't seem to have a mind which is inclined toward trying to understand complicated ontological problems, instead thinking that everything can be described simply by is or is not, because otherwise would be contradiction.

No middle 'e' in judgment. I can't take anyone seriously who can't spell. — fishfry

What kind of petty bullshit is this? Fuck you fishfry, I thought we were trying to be civil with one another. I see you've gone off the deep end already, and it's only Monday.

The point is that the process of abstraction necessarily, by its very nature, must omit many important aspects of the thing it's intended to model. You prefer not to engage with this point. — fishfry

I've engage with this point, explaining that I think it is wrong. If it's an important aspect, an essential feature, then if the abstraction processes "misses" it, the abstraction is wrong. If it is something which can be left out of the abstraction, it is in Aristotelian terms "accidental" or "an accident", and is not an important aspect. Abstraction separates the important from the unimportant, and if it omits important aspects it is faulty.

That's fantastic. The Pythagorean theorem is a beautiful, gorgeous, striking, brilliant, dazzling, elegant, sumptuous, and opulent example of an abstraction that is not based on ANYTHING in the real world and that has NO INDUCTIVE CORRELATE WHATSOEVER. I am thrilled that you brought up such an example that so thoroughly refutes your own point.

The Pythagorean theorem posits and contemplates a purely abstract, hypothetical, mathematical right triangle such that the sum of the squares on the legs is equal to the square on the hypotenuse. No such right triangle has ever, nor will ever, exist in the real world. — fishfry

That's amazingly wrong, to think that the Pythagorean theorem is not based in anything from the real world. It's based in the method used to produced parallel lines for marking out plots of land. Check into the history of "the right angle", and you will learn this. Clearly this is something in the real world.

Your own example falsifies this. I can never confirm the Pythagorean theorem in by observation of the world. I can only prove it deductively and never inductively. — fishfry

Huh? Construction workers prove the Pythagorean theorem in the real world, many times every day. Make a 3,4,5 triangle, tt never fails to produce the desired angle. How is this not proof? Try it yourself. Mark two points to produce a line. Use the Pythagorean theorem to make a right angle at each of the two points, and make two new points on those right angles, at equal distances from the original points. Measure the distance between the two new points, and you will see that it is the same as the distance between the two original points, and you have proven the Pythagorean theorem because you have used it to produce right angles, and have proven that the angles produced are in fact right angles by producing two more equivalent angles.

This is the complete opposite of induction. — fishfry

What you seem to not grasp, is that people were producing right angles long before the Pythagorean theorem was formalized. The Pythagorean theorem came into existence as a formalized description of what those people were doing. Therefore it is a generalization of what people were doing when they succeeded in producing the right angle, so it is an inductive conclusion. Try and see if you can apprehend pi as an inductive conclusion? It is a generalization, what all circles have in common, just like the Pythagorean theorem is a generalization, what all instances of "the right angle" have in common. If you produce an angle which is not consistent with what the Pythagorean theorem says, you have not produced the right angle.

But getting back to the larger point: A map is a formalization of certain aspects of reality that necessarily falsifies many other things. Just as a set is a formalization of the idea of a collection, that necessarily leaves out many other things. I have a bag of groceries. I formalize it as a set. The set doesn't have order, the grocery bag does. The set doesn't have milk, eggs, bowling balls, and rutabagas, the grocery bag does. — fishfry

As explained above, if an abstraction, or formalization, leaves out important aspects, then it is faulty. And if you insist on using the map analogy after I've explained why it is unacceptable, I will insist that if a map leaves out important things, then it is obviously a faulty map.

One reason why the map analogy is faulty, is because the map maker can decide, based on the purpose for which the map is being made, which aspects are important, and which are not. In the case of abstraction, formalizing, or generalizing, we have no choice but to adhere to the facts of reality, or else the formalizations will be incorrect.

I think what you are saying is that an abstraction faithfully represents some aspects of the thing, and leaves out others. — fishfry

An abstraction is a generalization. It does not represent "the thing" in any way, nor does it represent aspects of the thing. It represents a multitude of things, by creating a category or type, by which we can classify things. Again, another reason why the map analogy is misleading. It appears to make you think that an abstraction represents a thing, like a map does. That is incorrect, the abstraction is a generalization, a universal, which represents a multitude of things.

The essence of your argument is that "inherent order" is so tightly bound to the thing, that it can not be separated by an abstraction. I believe that's the core of our difference. Do I have that right? — fishfry

This is not really a good representation of my argument, because you don't seem to understand what abstraction is in anyway near to the way that I do. It's a good start anyway. But let me put it in another way. Let's suppose a category, or type called "thing". The abstraction, generalization, or formalization, would be a statement of definition, what it means to be a thing. This would be a statement as to what all things have in common, which makes it correct to call each of them a "thing". To be an acceptable definition, would be to be a good inductive conclusion. My argument is that the good inductive conclusion is that all things have inherent order therefore it would be a bad formalization, generalization, or abstraction, to posit a thing without inherent order because this is contrary to good inductive reasoning. Furthermore, I've argued that since such a principle, is not based in any inductive reasoning, it cannot truthfully be called an abstraction, generalization, or formalization, it is simply an imaginary fiction.

For purposes of the conversation, yes, I'll stipulate that. So what? It's how mathematical abstraction works. Just like the Pythagorean theorem does the same thing. — fishfry

As I've explained, it is false to call this an abstraction. To make up a purely imaginary, fictitious principle, is not abstraction. And, the Pythagorean theorem is not at all like this. Creating the Pythagorean theorem was a matter of taking what people had been doing on the ground, producing the right angle and parallel lines, and using inductive reasoning to determine what all these cases of producing the right angle had in common. Therefore it is not a purely imaginary and fictitious principle, it is a truthful inductive statement about what all instances of the producing the right angle have in common.

Well now you're just arguing about my semantics. If I don't use the word generalization (which by the way I have not -- please note); nor have I used the phrase inductive conclusion. I have not used those phrases, you are putting words in my mouth. I said mathematical models are formalized abstractions, or formal abstractions. Or formal representations. That's what they are. The need not and generally don't conform to the particulars of the thing they are intended to represent. Just as an idealized right triangle satisfies the Pythagorean theorem, and no actual triangle ever has or ever will. Oh what a great example, I wish I had thought of it. Thank you! — fishfry

Remember, you claimed a difference between a formalization, and an inductive conclusion. I did not accept such a difference, and asked you to validate this claim. You have not yet done so, but continue to speak as if your proposed distinction is a true distinction, while I have demonstrated that it is not. Therefore I suggest that you give up, as false, this claim to a difference.

You say that like it's a bad thing! Ok, imaginary fiction then. But a useful one! — fishfry

Yes, fictions are useful. The principal use of these is to mislead and deceive. A secondary use is entertainment, but this requires consent to the fact that what is presented is fiction.

So if you have a problem, it's your problem and not mine, and not math's. — fishfry

Of course, deception is a problem for the one being deceived, not the deceiver. Or maybe I'm just not entertained by your proposed entertainment. Again, still my problem, but perhaps you have made a poor presentation.

Well a set isn't actually a collection. A set is an undefined term whose meaning is derived from the way it behaves under a given axiom system. And there is more than one axiom system. So set is a very fuzzy term. Of course sets are inspired by collections, but sets are not collections. Only in high school math are sets collections. In actual math, sets are no longer collections, and it's not clear what sets are. Many mathematicians have expressed the idea that sets are not a coherent idea. It would be great to have a more sophisticated discussion of this point. I'm not even disagreeing with you about this. Sets are very murky. — fishfry

It was you who called a set a collection, and referred to some sort of mystical process of collecting, which allows for your proposed "no inherent order".

So therefore perhaps I should stop wasting my time trying to argue that math is inspired by reality, or that math is an abstraction of reality, and simply concede your point that it's utter fiction, and put the onus on you to deal with that. I could in fact argue the abstraction and inspired-by route, but you bog me down in semantics when I do that and it's tedious. — fishfry

I dealt with this. Most math is not fiction, as evident in the example of the Pythagorean theorem. I differentiated the types of mathematical principles which are imaginary fictions though, things like "no inherent order", and "infinity".

If I see 100 bowling balls fall down, "bowling balls always fall down" is an inductive conclusion. But F = ma and the law of Newtonian gravity are mathematical models from which you can derive the fact that bowling balls fall down. It's a physical law, meaning that if you assume it, you can explain (within the limits of observational technology) the thing you observe. — fishfry

The law of gravity is the more general statement, saying all things with mass will fall down. The statement that bowling balls will fall down is more specific. Inductive reasoning is to produce a general statement from empirical observations of particular instances. So, the law of gravity as a general statement, is an inductive conclusion. And, bowling balls may or may not have been observed in producing that inductive law, but the law extends to cover things not observed, due to the nature of inductive reasoning, and the generality of what is produced. This is why inductive reasoning gives us predictive capacity. That mathematics is used to enhance the predictive capacity of inductive reasoning is not relevant to this point.

But this is not an important point in the overall discussion. — fishfry

It is important, because induction, by its nature, requires observation of particular instances. And you seem to be arguing that there is a type of abstraction, pure abstraction, which does not require any inductive principles. So it is important that you understand exactly what induction is, and how it brings principles derived from observations of particular instances, into abstract formulae. Do you see that the Pythagorean theorem for example, as something produced from practice, is derived from induction?

Ok. I don't think the definition of induction versus a formal model is super important here. But "bowling balls always fall down" is simply a generalization of an inductive observation, whereas the law of gravity lets you derive the fact that bowling balls fall down; and that in fact on the Moon, they'd weigh less. The latter is not evident from "bowling balls always fall down," but it is evident from the equation for gravitational force. — fishfry

It is the inductive conclusion, which allows for the derivation, the prediction, which you refer to. As it is a general statement, it can be applied to things not yet observed. It is not the mathematics which provides the capacity for prediction. mathematics enhances the capacity

This point is not central to the main point, which is that models must necessarily omit key aspects of the thing being modeled. — fishfry

The central point is the difference between the inductive conclusion, which states something general, and the modeling of a "thing", which is a particular instance. At this point, we take the generalization, and apply it to the more specific. It must be determined how well the generalization is suited, or applicable to the situation. This requires a judgement of the thing, according to some criteria.

Ok fine, then order is physical and the mathematical theory of order is an abstraction or model that necessarily misses many important real-world aspects of order yet still allows us to get some insight. That's the point of abstraction, which I already beat to death in my last post. — fishfry

I think your description of abstraction as missing things, is a bit off the mark. What abstraction must do is derive what is essential (what is true in all cases of the named type), dismissing what is accidental (what may or may not be true of the thing). Now, if order is essential to being a thing, then we cannot abstract the order out of the thing, to have a thing without order, because it would no longer be a thing.

I'm explaining to you that whatever your concept of physical order is, mathematical order is an abstraction of it, which is necessarily a lie by virtue of being an abstraction or model, yet has value just as a map is not the territory yet lets us figure out how to get from here to there. — fishfry

This is not true in a number of ways. First, good abstractions, inductive conclusions, or generalizations, do not lie because they stipulate what is essential to the named type. They speak the truth because every instance of that named type will have the determined property.

Second, your proposed "mathematical order" is not an abstraction, inductive conclusion, or generalization. You started with the principle that there is a unity of things with no inherent order. So you have separated yourself from all abstraction, induction, or generalization, to produce a purely imaginary, and fictitious starting point. You cannot claim that the imaginary, and purely fictitious starting point, of "no inherent order" is a generalization, or an inductive conclusion, or in any way an abstraction of the physical order. You are removing what is essential to "order", by claiming "no order", therefore you have no justification in claiming that this is an abstraction of physical order.

Do you recognize the difference between abstracted and imaginary? Imagination has no stipulation for laws of intelligibility, while abstraction does.

I've conceded your point, now that I understand what you mean by inherent order. — fishfry

OK, now lets proceed to look at your imaginary "mathematical order". Do you concede as well, that by removing the necessity of order from your "set", we can no longer look at the set as any type of real thing. Nor is it a generalization, an inductive conclusion, or an abstraction of physical order. It is purely a product of the imagination, "no order", and as such it has no relationship with any real physical order, no bearing, therefore no modeling purposefulness. It ought to be disposed, dismissed, so that we can start with a new premise, a proper inductive conclusion which describes the necessity of order.

It's an abstraction that necessarily includes SOME aspects of the thing being modeled and excluces OTHER aspects. Just as a street map includes the orientation of the roads but ignores the traffic lights. — fishfry

The idea of something with "no inherent order" is not an abstraction, as explained above. It is a product of fantasy, imaginary fiction.

That's right. A map is correct about some aspects of the world and incorrect about others. It's an abstraction. — fishfry

A map is not an abstraction, it is a representation. I see that we need to distinguish between abstraction, which involves the process of induction, producing generalizations, and as different, the art of applying these generalizations toward making representations, models, or maps. Do you see, and accept the difference between these two? We cannot conflate these because they are fundamentally different. The process of abstraction, induction, seeks what is similar in all sorts of different thing, for the sake of producing generalizations. The art of making models, or maps, involves naming the differences between particulars. These are very distinct activities, one looking at similarities, the other at differences, and for this reason abstraction cannot be described as map making.

Maps are imaginary principles and don't formalize anything? Do you see why I think you're trolling? — fishfry

I was referring to the principles of "no inherent order", and "infinity", with the claim that these do not formalize anything. I wasn't talking about maps.

If you would engage with my examples of maps and globes, I would find that helpful. — fishfry

The map analogy is not very useful, for the reason explained above, it doesn't properly account for the nature of inductive principles, abstraction. Generalizations may be employed in map making, but they are not necessarily created for the purpose of making maps. Now the map maker takes the generalizations for granted, and proceeds from there, but must choose one's principles. In making a map, what do you think is better, to start with a true inductive abstraction like "all things have order", or start with a fictitious imaginary principle like "there is something without order"? Wouldn't the latter be extremely counterproductive to the art of map making, because it assumes something which cannot be mapped?

And sets represent aspects of collections, which exist in the world. And they omit "inherent order," which for sake of argument I'll agree collections in the world have. — fishfry

So, for the sake of argument, we can make the inductive conclusion, all collections which exist in the world have an inherent order. This is a valid abstraction, based in empirical observation, and it states that what is essential to, or what is a necessary property of, a collection, is that it has an inherent order. Do you agree then, that if we posit something without inherent order, this cannot be a collection? It doesn't have the essential property of a collection, i.e. order; therefore it is not a collection.

Well that has nothing to do with anything. Maps don't show things that aren't there. The question is, how do you feel when a map omits things that ARE there, like wet lakes and rivers, cars, and the size and scale of the actual territory being modeled. — fishfry

Each map maker, based on the needs of that map maker's intentions, chooses what to include in the map. Abstraction, inductive reasoning, is very distinct from this, because we are forced by the necessities of the world to make generalizations which are consistent with everything. That's what makes them generalizations

Then you don't see it. — Luke

Perhaps, but I disagree. It's a matter of opinion I suppose. You desire to put a restriction on the use of "see", such that we cannot be sensing things which we do not apprehend with the mind. I seem to apprehend a wider usage of "see" than you do, allowing that we sense things which are not apprehended. So in my mind, when one scans the horizon with the eyes, one "sees" all sorts of things which are not "forgotten" when the person looks away, because the person never acknowledged them in the first place, so they didn't even register in the memory to be forgotten, yet the person did see them.

And you claimed earlier that we could not possibly see it, in principle — Luke

No, I think you misunderstood. Perhaps it was the use of "perceive" which is like "apprehend". I said we could not apprehend it with the mind, the mind being deficient. This does not mean that we cannot sense, or "see" it at all. But your limiting of "see", to only that which is apprehended by the mind, instead of allowing (what in my opinion is the reality of the situation) that we are sensing things which are not being apprehended by the mind, not "perceived", is making you think that just because we cannot apprehend it with the mind, therefore we are not sensing it at all.

I know it's a difficult issue and it appears as incoherency, as ontological issues often do, because they are difficult to understand, but I think we need to establish a separation between what is sensed, and the apprehension of it, to account for the differences between how different people apprehend very similar sensations.

Have you rejected your claim that we can see the inherent order? — Luke

No, I think we see it in exactly the way that I explained.

2. How do you reconcile this with your statements that order is not visible? — Luke

I explained that. We see the object. The object exists as an instance of ordered parts, inherent order. Therefore we must be seeing the inherent order even though strictly speaking the order is not visible to the person who is seeing it. The "not visible" property is due to a deficiency in the capacity of the person who is seeing the order.

I used the molecule example. Molecules are not visible to the naked eye. But we see the object, and the object is composed of molecules, therefore we must be seeing the molecules. That the molecules are not visible to the person seeing them is due to a deficiency in that person's capacities.

It's the same principle as when someone is pointing something out to you, and you're looking right at it, so you're definitely seeing it, because it's right there in your field of vision, yet you don't see the particular thing that the person is pointing out. Have you ever looked at stars, and had someone try to point out specific constellations to you? You can be looking right at the stars, and see them all, therefore you are seeing the mentioned constellation, yet you still might not be able see that specific constellation.

See the different senses of "see", and how "visible" might be determined based on the capacity of the observer, or the capacity of the thing to be observed? The inherent order is not visible to us, due to our deficient capacities, yet we do see it, because it exists as what we are seeing. Go figure.

It is the content that is valuable, — Sir2u

So, who's the author?