It those truths precede in time our experience of reality then they cannot be dependent on experience. — Fooloso4

It those truths precede in time our experience of reality then they cannot be dependent on experience. Such is the case with non-Euclidean geometries. — Fooloso4

Truth is not a value, but a relation between mental judgements and reality. Since it depends on judgements, it can't be prior in time to them. Only being can be.

As another example consider infinitesimal calculus. There is no experience of infinitesimals. — Fooloso4

There are no actual infinitesimals in calculus. There are limits as quantities tend to zero. That is the whole point of the epsilons and deltas in the formal definitions of calculus.

Do you imagine that neither Kant nor those who followed him were aware of this? — Fooloso4

Having read Kant's reasoning, he seems to have been unaware of the errors he was making.

Instantiation is not abstraction. — Fooloso4

I did not say it was. I said that non-euclidean geometries could be abstracted from models instantiating them.

The historical fact of the matter is that they weren't abstracted. Non-Euclidean geometries were first developed as purely formal systems. — Fooloso4

If so, that would mean they had a hypothetical status until it was realized that they could be instantiated. The notion that one could be shown to be the true geometry of the universe was explicitly stated by János Bolyai.

What is at issue is your claim regarding the intelligibility of an object. Whether or not human knowing exhausts something's essence, if intelligibility inheres in the object then a sufficiently advanced intelligence should be able to know what a baseball is without knowing what the game is, or, perhaps, would know from the ball what the game is. But there is nothing in the ball that would provide this information. — Fooloso4

According to the Wikipedia article: "Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences."

I have answered all this previously. Knowing an object's intrinsic nature need not entail knowing its relationships.

By your logic the intelligibility of a car does not include the potential to know that it is a means of transportation. — Fooloso4

One might figure it out, but only if one knew there were beings that could use it so.

Truth is not a value, but a relation between mental judgements and reality. Since it depends on judgements, it can't be prior in time to them. — Dfpolis

There is no judgment of the truth of the deductions of non-Euclidean geometry that independent of reality, unless of course you maintain that there is a mathematical reality. They are formal logical truths. Whatever your theory of truth may be, non-Euclidean geometry works. They find their application in reality.

There are no actual infinitesimals in calculus. — Dfpolis

The point is that they are theoretical constructs. They are not abstracted from nature.

Having read Kant's reasoning, he seems to have been unaware of the errors he was making. — Dfpolis

Him and several generations of Kant scholars. When are you going to publish your findings in a peer reviewed journal?

I said that non-euclidean geometries could be abstracted from models instantiating them. — Dfpolis

But the fact that you are trying to dance around is that they didn't.

If so, that would mean they had a hypothetical status until it was realized that they could be instantiated. — Dfpolis

They did not have a hypothetical status because they were not hypotheses. They were formal logical systems that were not intended to relate to anything else.

According to the Wikipedia article: "Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences." — Dfpolis

This is besides the point. They were not constructed as models of the universe. The question is how is it that they do apply? Is it just coincidence?

I have answered all this previously. Knowing an object's intrinsic nature need not entail knowing its relationships. — Dfpolis

The problem is that a baseball being a baseball is not a relationship. It is intrinsic to what it is to be a baseball.

One might figure it out, but only if one knew there were beings that could use it so. — Dfpolis

You might claim that a car's being a mode of transportation is not intrinsic to it being a car, but that is only because you want to maintain your questionable claim about intelligibility. If not for that you would define it as everyone else does.

Truth is not a value, but a relation between mental judgements and reality. — Dfpolis

But there's a subtle recursion in this understanding, because it presumes we can attain a perspective where 'mental judgements' can be compared with reality.

'Truth, it is said, consists in the agreement of cognition with its object. In consequence of this mere nominal definition, my cognition, to count as true, is supposed to agree with its object. Now I can compare the object with my cognition, however, only by cognising it. Hence my cognition is supposed to confirm itself, which is far short of being sufficient for truth. For since the object is outside me, the cognition in me, all I can ever pass judgement on is whether my cognition of the object agrees with my cognition of the object”. (Kant, 1801. The Jasche Logic, in Lectures on Logic.)

math is not logic. That was Hilbert's view — Dfpolis

That was not Hilbert's view. It seems you are confusing Hilbert with Russell.

Godel's work shows more: it shows that there are truths that cannot be deduced from any knowable set of axioms. — Dfpolis

That is terribly incorrect. Godel's result is that, for any S that is a certain relevant kind of axiom system, there are true statements that cannot be deduced in S. However there are other systems, even of the relevant kind, in which the statement can be deduced. Then, in a followup:

In any lifetime, or finite number of lifetimes, we can only go through a finite number of axiom sets. So, there are true axioms we cannot deduce. — Dfpolis

No, again, that is terribly incorrect. There is no axiom such that there is no system in which the axiom can be deduced. And it is not needed to refer to Godel to point out that we can only look at finitely many systems.

David Hilbert's "program" (concept of math) was destroyed by Kurt Gödel. — Dfpolis

It is reasonable to argue that certain central aspects of Hilbert's program were shown by Godel to not be achievable. But that doesn't destroy Hilbert's concept entirely.

Godel's work means that we cannot prove the consistency of a set of axioms — Dfpolis

No, that is terribly incorrect. Godel's result is that for any S that is a certain kind of axiom system, the consistency of S cannot be deduced in S. But the consistency of S might be deducible by certain other systems.

We come to the notion of Aleph-1 (uncountable) infinity by proving that the numbers we assign to the points of continuous extents cannot be counted. — Dfpolis

'aleph_1' is not synonymous with 'uncountable'. aleph_1 is the least uncountable cardinal. And showing that there are uncountable sets does not rely on proving the uncountability of the continuum, but comes even more simply from proving that the power set of any set has more members than the set, so if there is an infiinite set then there is an uncountable set. And, just to be clear, Cantor didn't prove that the cardinality of the continuum is aleph_1. The proposition that the cardinality of the continuum is alelph_1 is the continuum hypothesis, famously not proven by Cantor.

Perhaps I'm wrong on C being unfalsifiable. Perhaps some consequent of C can be falsified. — Dfpolis

If a consequence of C is falsified, then C is falsified.

It has been claimed that formalists, such as David Hilbert (1862–1943), hold that mathematics is only a language and a series of games. — alcontali

Hilbert didn't say that mathematics is only a language game. He regarded certain aspects of mathematics as a kind of language game. But he explicitly said that certain parts of mathematics are meaningful, and even that the ideal mathematics that he regarded as literally meaningless is still instrumental and crucial for the mathematics of the sciences.

math is not logic. That was Hilbert's view — Dfpolis

That was not Hilbert's view. It seems you are confusing Hilbert with Russell. — GrandMinnow

Thank you. If you read the context, I was arguing against the position that math need only be logically self consistent, not Russell's more extreme position that math and logic were identical. In the SEP we read:

Hilbert believed that the proper way to develop any scientific subject rigorously required an axiomatic approach. In providing an axiomatic treatment, the theory would be developed independently of any need for intuition, and it would facilitate an analysis of the logical relationships between the basic concepts and the axioms. — Richard Zach

Godel's work shows more: it shows that there are truths that cannot be deduced from any knowable set of axioms. — Dfpolis

That is terribly incorrect. Godel's result is that, for any S that is a certain relevant kind of axiom system, there are true statements that cannot be deduced in S. However there are other systems, even of the relevant kind, in which the statement can be deduced. — GrandMinnow

One can always add a determinate and previously unprovable truth, or its equivalent (if one knows what it is and not merely that it is) to an axiom system and then "deduce" it. Still, the number of propositions we (all humans) can know is necessarily limited. So any knowable set of axioms is finite. No matter how large that finite set may be, there will be truths that cannot be deduced from it. Also, no computable procedure for generating new axioms will exhaust the possible axioms in a finite time. So, an exhaustive axiom set is unknowable. So there are truths we will never be able to deduce.

There is no axiom such that there is no system in which the axiom can be deduced. — GrandMinnow

That was not my claim. I do not deny that any particular truth is deducible from suitable axioms. Rather, I am saying we cannot generate actual axiomatic sets sufficient to deduce all truths in a finite time -- for any finite set of axioms will leave some truths undeducable.

'aleph_1' is not synonymous with 'uncountable' — GrandMinnow

Nor did I claim that it was. I was merely trying to provide a clue as to what was being discussed to those unfamiliar with aleph-1.

And showing that there are uncountable sets does not rely on proving the uncountability of the continuum — GrandMinnow

Did I say it was? I pointed to Cantor's 1874 proof as one way of knowing that the cardinality of the reals is not countable. The question asked was how can we come to concepts of countable and uncountable infinity from experience, not what are the principal findings of transfinite number theory.

comes even more simply from proving that the power set of any set has more members than the set, so if there is an infiinite set then there is an uncountable set. — GrandMinnow

And do you think that an explanation based on the concept of power sets is more comprehensible to a general philosophic audience than what I said?

And, just to be clear, Cantor didn't prove that the cardinality of the continuum is aleph_1. — GrandMinnow

I did not say that he did, but that he proved that the cardinality of the continuum was uncountable. You seem to think that I need to provide excruciating detail when that detail is not relevant to the point I'm making, namely that the foundations of mathematics have an adequate moderate realist interpretation.

The proposition that the cardinality of the continuum is alelph_1 is the continuum hypothesis, famously not proven by Cantor. — GrandMinnow

Again, I did not say that he did.

The cardinality of the set of real numbers (cardinality of the continuum) is 2^ℵo. It cannot be determined from ZFC (Zermelo–Fraenkel set theory with the axiom of choice) where this number fits exactly in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis, CH, is equivalent to the identity 2^ℵo = ℵ1. — Wikipedia

Perhaps I'm wrong on C being unfalsifiable. Perhaps some consequent of C can be falsified. — Dfpolis

If a consequence of C is falsified, then C is falsified. — GrandMinnow

Isn't that exactly what I said?

Hilbert didn't say that mathematics is only a language game. He regarded certain aspects of mathematics as a kind of language game. But he explicitly said that certain parts of mathematics are meaningful, and even that the ideal mathematics that he regarded as literally meaningless is still instrumental and crucial for the mathematics of the sciences. — GrandMinnow

If he was right, then the mathematical statements used by the natural science have to be instantiated in nature, and so are true in the sense of correspondence theory. That effectively vitiates formalism.

My question to you is, how do the details I have smoothed over serve to undermine my thesis? If they do not, then your criticisms are pedantic.

My question to you is, how do the details I have smoothed over serve to undermine my thesis? If they do not, then your criticisms are pedantic. — Dfpolis

Boom! :up:

The problem with Hilbert's "language game" is again connotational. John Nash's game theoritical contribution concerning equilibria in n-person strategy "games" was otherwise good for a Nobel prize.

There is nothing wrong with the technical term "game".

In common parlance it is considered something unserious but that merely reflects the notorious ignorance of the unwashed masses who often tend to be inspired by their fake morality.

There is no judgment of the truth of the deductions of non-Euclidean geometry that independent of reality, unless of course you maintain that there is a mathematical reality. They are formal logical truths. Whatever your theory of truth may be, non-Euclidean geometry works. They find their application in reality. — Fooloso4

This is a very confused statement. If a mathematical theory applies to reality accurately, it is instantiated in reality and the adequacy of the theory to that instantiation shows the truth of the theory with respect to that instantiation. Further, since we presumably know the instantiation, we can abstract the theory from it. So, one need not "maintain that there is a mathematical reality." only that empirical reality has a mathematical intelligibility.

There are no actual infinitesimals in calculus. — Dfpolis

The point is that they are theoretical constructs. They are not abstracted from nature. — Fooloso4

Since they do not exist, they are not constructs. The theory uses small quantities tending to zero, while always remaining finite.

Him and several generations of Kant scholars. When are you going to publish your findings in a peer reviewed journal? — Fooloso4

Do you think that I'm the first to notice that Kant's arguments are inadequate?

I said that non-euclidean geometries could be abstracted from models instantiating them. — Dfpolis

But the fact that you are trying to dance around is that they didn't. — Fooloso4

I have not read the original papers, so I don't know if they did or did not. I do know that the parallel postulate has been suspect since classical times precisely because it cannot be abstracted from experience -- which was my point.

They did not have a hypothetical status because they were not hypotheses. They were formal logical systems that were not intended to relate to anything else. — Fooloso4

That is you view. I already noted that Bolyai discussed which geometry described reality, which means that he saw geometry as potentially reflecting reality, and the status of the parallel axiom as a hypothesis to be studied by physics. I am not denying that math can be treated formally once we posit our axioms. I am discussing how we come to posit its axioms, and their epistemological status.

The problem is that a baseball being a baseball is not a relationship. It is intrinsic to what it is to be a baseball. — Fooloso4

Yes, still, the name is not intrinsic to it, but assigned in light of its relation to the game.

I do want to add that I was was unclear in discussing the relation of aleph-1 to the cardinality of the reals and that your point on that confusion was well-taken. Mea culpa.

Truth is not a value, but a relation between mental judgements and reality. — Dfpolis

But there's a subtle recursion in this understanding, because it presumes we can attain a perspective where 'mental judgements' can be compared with reality — Wayfarer

The statement presumes that experience gives us access to reality -- which is an independent, not a recursive, assumption. Books have been written on this assumption, but that is a topic for another thread. I would simply say that one can't deny this without twisting the meaning of "reality" as what is revealed by experience.

For since the object is outside me, the cognition in me, all I can ever pass judgement on is whether my cognition of the object agrees with my cognition of the object” — Wayfarer

This is why it is important to recognize that in both sensation and cognition we have an existential penetration of the subject by the object, Thus, Kant's claim that "the object is outside me" is only partly true. Every physical object is surrounded by a radiance of action, which is the indispensable means of our knowing it.

Kant's basic problem is that he wants knowing to be independent of knowers when it is actually a subject-object relation. Or, perhaps, he wants us to have divine omniscience of the noumena when we only have human knowledge -- knowledge, not of how reality is in se, but of how it relates to us. Yet, knowing how reality relates to us is exactly what humans need to know to be in reality.

How reality informs me, how I interact with its radiance of action, is immediately available to awareness -- not "outside me." So, Kant has misunderstood the issue.

This is a very confused statement. If a mathematical theory applies to reality accurately ... — Dfpolis

It is not a theory, it is a formal deductive system based on the negation of the parallel postulate.

... since we presumably know the instantiation, we can abstract the theory from it. So, one need not "maintain that there is a mathematical reality." — Dfpolis

Once again,it is a purely formal, logical system that was developed prior to and independently of any instantiation.

...empirical reality has a mathematical intelligibility. — Dfpolis

And in this case an intelligibility that was not empirically derived, suggesting that the physical world is structured mathematically, that the mathematics are fundamental, formative.

Since they do not exist, they are not constructs.The theory uses small quantities tending to zero, while always remaining finite. — Dfpolis

This is nonsense. With regard to Zeno, it is the divisibility that is infinite. With regard to infinitesimals the quantity is smaller than can be measured. In neither case is it something derived from experience. They are theoretical constructs. Whether reality is continuous or discrete remains an open question.

Do you think that I'm the first to notice that Kant's arguments are inadequate? — Dfpolis

Your claim was that Kant had no reason to claim that experience is constructed. This was followed in another post by:

Having read Kant's reasoning, he seems to have been unaware of the errors he was making. — Dfpolis

What do you provide in support of that? That you read Kant's reasoning. My response was sarcastic - The title of your paper: Kant's Reasoning Regarding Experience Faulty. The text of the paper: I read Kant's reasoning.

I do know that the parallel postulate has been suspect since classical times precisely because it cannot be abstracted from experience -- which was my point. — Dfpolis

Was it? Your claim is that mathematics is an abstraction from experience. But now you say that the parallel postulate cannot be abstracted from experience. That would make it a theoretical construct, but you have denied that there can be such a thing. You also say that:

non-euclidean geometries could be abstracted from models instantiating them. — Fooloso4

So, now a central part of Euclidean geometry cannot be abstracted from experience but non-Euclidean geometry can.

The fact is, though, once again, that non-Euclidean geometry was not abstracted from experience. All of this leaves your claim about mathematics being an abstraction muddled. But I take it that was not your point.

They did not have a hypothetical status because they were not hypotheses. They were formal logical systems that were not intended to relate to anything else.
— Fooloso4

That is you view. I already noted that Bolyai discussed which geometry described reality, which means that he saw geometry as potentially reflecting reality, and the status of the parallel axiom as a hypothesis to be studied by physics. — Dfpolis

This is what Bolyai is quoted saying in that article:

I have discovered such wonderful things that I was amazed...out of nothing I have created a strange new universe.

The article also states:

The discovery of a consistent alternative geometry that might correspond to the structure of the universe helped to free mathematicians to study abstract concepts irrespective of any possible connection with the physical world.

Clearly they were not hypothesis about the physical world, or, as your prefer, reality. They were neither abstracted from or hypothesis about the physical world.

I am discussing how we come to posit its axioms, and their epistemological status. — Dfpolis

And how do we come to posit the parallel postulate, if, according to you, it is not an abstraction from reality? Its negation is not an abstraction from reality either. Both, however, have their application in reality.

Yes, still, the name is not intrinsic to it, but assigned in light of its relation to the game. — Dfpolis

We have been through this. It is not a name assigned to a ball that came to exist independent of the game. It is the name of a ball specifically designed and made to be used to play the game of baseball. If not for baseball the ball would not exist.

I would simply say that one can't deny this without twisting the meaning of "reality" as what is revealed by experience. — Dfpolis

This is really a fundamental point. What you're arguing is British empiricism, per Locke and Hume. But does sensory apprehension qualify as 'revealed truth'? Certainly through scientific method, we can discover truth, but the assumption of the 'reality of the given' is precisely what is at issue in philosophy.

This is why it is important to recognize that in both sensation and cognition we have an existential penetration of the subject by the object, Thus, Kant's claim that "the object is outside me" is only partly true. Every physical object is surrounded by a radiance of action, which is the indispensable means of our knowing it. — Dfpolis

I'm sorry but I think this is mistaken. Again I'm no Aquinas scholar, but I think I grasp some of the rudiments of his hylomorphism, which says that

All cognition takes place through assimilation. But there is no assimilation possible between the mind and material things, because likeness depends on sameness of quality. However, the qualities of material things are bodily accidents which cannot exist in the mind. Therefore, the mind cannot know material things.

Aquinas, Thomas; Truth, Vol. II, Qs. 10, Article 4

And this is because, in the view of Christian philosophy, material things have no intrinsic reality; creatures are, as Aquinas' Dominican peer Meister Eckhardt said, 'mere nothings'. The locus of reality, as it were, is not the empirical domain, the realm of sensible objects, but the intelligible order inhering in the divine intellect of which the sensable domain is a product or creation.

Kant has misunderstood the issue. — Dfpolis

I think it's more likely that you're misunderstanding Kant.

Certainly through scientific method, we can discover truth — Wayfarer

Well, since the scientific method cannot possibly discover any truths about itself, how would it be able to discover the complete truth? While mathematics does have the self-knowledge that it is necessarily incomplete, the scientific method is simply not capable of that kind of self-inquiry.

Scientism is the Dunning-Kruger effect on steroids:

In the field of psychology, the Dunning–Kruger effect is a cognitive bias in which people mistakenly assess their cognitive ability as greater than it is. It is related to the cognitive bias of illusory superiority and comes from the inability of people to recognize their lack of ability. Without the self-awareness of metacognition, people cannot objectively evaluate their competence or incompetence.

In common parlance it is considered something unserious but that merely reflects the notorious ignorance of the unwashed masses who often tend to be inspired by their fake morality. — alcontali

LOL We all can’t be John Nash or Alan Turing types. We need guidance from the intellectual elites because we don’t know what’s good for us.

Well, since the scientific method cannot possibly discover any truths about itself, how would it be able to discover the complete truth? While mathematics does have the self-knowledge that it is necessarily incomplete, the scientific method is simply not capable of that kind of self-inquir — alcontali

I never would claim that science is omniscient and I myself am a critic of scientism. But it's implausible to deny the fact of scientific discoveries and principles. So we have to be able to grant science the considerable credit where it's due, without at the same time claiming that it is all-knowing, even in principle.

Kant never lost sight of the fact that while modern science is one of humanity's most impressive achievements, we are not just knowers: we are also agents who make choices and hold ourselves responsible for our actions. In addition, we have a peculiar capacity to be affected by beauty, and a strange inextinguishable sense of wonder about the world we find ourselves in. Feelings of awe, an appreciation of beauty, and an ability to make moral choices on the basis of rational deliberation do not constitute knowledge, but this doesn't mean they lack value. On the contrary. But a danger carried by the scientific understanding of the world is that its power and elegance may lead us to undervalue those things that don't count as science.

The Continuing Relevance of Immanuel Kant

This is a very confused statement. If a mathematical theory applies to reality accurately, it is instantiated in reality and the adequacy of the theory to that instantiation shows the truth of the theory with respect to that instantiation. Further, since we presumably know the instantiation, we can abstract the theory from it. So, one need not "maintain that there is a mathematical reality." only that empirical reality has a mathematical intelligibility. — Dfpolis

I had a friend, Mike Zielinski, who received his PhD in mathematics from the University of Wisconsin-Madison. He tried describing to me his dissertation and it was completely over my head. I asked him out of curiosity if the subject of his dissertation reflected anything in the physical world. He simply said, “I don’t know.” Sounds like a game to me, but what do I know with my fake morality and all.

It may describe some as of yet unknown physical process or it may just be a mathematical unicorn. Could these mathematical discoveries still be used in, say, cryptography?

This is why it is important to recognize that in both sensation and cognition we have an existential penetration of the subject by the object, Thus, Kant's claim that "the object is outside me" is only partly true. Every physical object is surrounded by a radiance of action, which is the indispensable means of our knowing it.

Kant's basic problem is that he wants knowing to be independent of knowers when it is actually a subject-object relation. Or, perhaps, he wants us to have divine omniscience of the noumena when we only have human knowledge -- knowledge, not of how reality is in se, but of how it relates to us. Yet, knowing how reality relates to us is exactly what humans need to know to be in reality.

How reality informs me, how I interact with its radiance of action, is immediately available to awareness -- not "outside me." So, Kant has misunderstood the issue. — Dfpolis

This is very profound. Who came up with this? Was it you? Also, could you flesh this out for me so I can understand it better: “Every physical object is surrounded by a radiance of action, which is the indispensable means of our knowing it.”

It’s totally up to your uncoerced will, of course.

Couldn’t it be the case that mathematics was first derived from empirical experience, and that newer maths were abstracted from these more fundamental maths? So doesn’t mathematics have as its ultimate foundation the physical world?

So doesn’t mathematics have as its ultimate foundation the physical world? — Noah Te Stroete

Or more accurately, our experience of the physical world. I remain a Kantian until I hear better arguments.

A mathematician or scientist does not necessarily a good philosopher make. I think it’s their pride that gets in the way. @Dfpolis, however, is a philosopher worth his salt. He takes criticism constructively, but the pedants don’t see the forest amongst the trees. The basic thesis is there and has not been shown to be refuted. A pedant does not make a good philosopher.

I never would claim that science is omniscient and I myself am a critic of scientism. But it's implausible to deny the fact of scientific discoveries and principles. So we have to be able to grant science the considerable credit where it's due, without at the same time claiming that it is all-knowing, even in principle. — Wayfarer

Agreed.

Scientism points to the fact that science badly lacks humility. That is a fundamental problem. The scientific method does not allow them to know when they do not know. The method simply does not allow for self-awareness. That is why they are so delusional to believe that they know everything; which is obviously ridiculous.

Seriously, this impossibility of self-inquiry is an enormous flaw in the scientific method. As a result, the false belief in its own delusional omnipotence has been snowballing for centuries now, and has even gone mainstream as to infect the unwashed masses with this dangerous disease. For heaven's sake, who is going to save the delusional populace from their delusions? I do not think that it can be done.

In comparison, mathematics is incredibly humble.

While reasoning from its arbitrary starting points, mathematics admits that it is unable to answer many otherwise applicable questions about these arbitrary starting points. Gödel's discovery of this theorem turned him into one of the most admired grandees in the field. Furthermore, the formalist philosophy admits that on the whole a good mathematical theory is meaningless (has nothing to do with the real world) and useless (no direct application possible).

Seriously, this impossibility of self-inquiry is an enormous flaw in the scientific method. — alcontali

You might appreciate this thread and the essay it points to. Actually, there's a book from a couple of years back on a similar theme, The Blind Spot: Science and the Crisis of Uncertainty by William Byers - he's an emeritus professor of maths, I think with your background and interest in math you might find it interesting.

who is going to save the delusional populace from their delusions? — alcontali

'The beasts are driven to the pasture by blows', said Heraclitus.

Could these mathematical discoveries still be used in, say, cryptography? — Noah Te Stroete

It is had to say without even knowing the area of research.

Who came up with this? Was it you? Also, could you flesh this out for me so I can understand it better: “Every physical object is surrounded by a radiance of action, which is the indispensable means of our knowing it.” — Noah Te Stroete

I came up with it reflecting on Aristotle and Aquinas. Aristotle classes action as an accident as something inhering in a substance. If we reflect on any object that we encounter, we see that it is inseparable from its environmental effects -- its radiance of action. This includes its gravitational field, the light that it radiates and scatters and the odors it emits -- all the means making it sensible, observable. The quantum description of matter also shows no hard boundaries -- its material fields extend becoming ever more tenuous. This action on us modifies our our neural state, and that modification of our neural state is identically our neural representation of the object. So, so that part of us is also the object's action.

We can and usually do abstract the object from its radiance of action, leaving us with the impression that it is no more than a core with well-defined boundaries. Still, if we remove the radiance of action from an actual object, it no longer acts as it does and no longer is what it is. Instead of being an integral part of reality, it becomes an isolated monad.

Couldn’t it be the case that mathematics was first derived from empirical experience, and that newer maths were abstracted from these more fundamental maths? — Noah Te Stroete

Yes, I think this is the case, for example with the structures studied in abstract algebra. Ultimately, however, the foundations can be traced to abstractions from reality or to hypotheses.

You made a number of unargued claims I will not respond to.

...empirical reality has a mathematical intelligibility. — Dfpolis

And in this case an intelligibility that was not empirically derived, suggesting that the physical world is structured mathematically, that the mathematics are fundamental, formative. — Fooloso4

Intelligibility is a potential that exists prior to being actually known. So, it is not "derived." It is in nature.

Since they do not exist, they are not constructs.The theory uses small quantities tending to zero, while always remaining finite. — Dfpolis

This is nonsense. — Fooloso4

I suggest you read a calculus book.

Having read Kant's reasoning, he seems to have been unaware of the errors he was making. — Dfpolis

What do you provide in support of that? — Fooloso4

I was challenging any Kantian to provide what they believed was an adequate argument. When an argument was provided I rebutted it.

Your claim is that mathematics is an abstraction from experience. But now you say that the parallel postulate cannot be abstracted from experience. — Fooloso4

Reread the OP.

I have discovered such wonderful things that I was amazed...out of nothing I have created a strange new universe.

One can be right about some things, and wrong about others. While I am happy to allow Bolyai his joy, his assessment is clearly inaccurate. Human creativity consists in imposing new form on old matter, not creation ex nihilo. Most of the axioms in non-Euclidean geometry are from Euclid. Concepts derive meaning from experience. So, his achievement was to impose new form on prior, empirically derived, content.

Clearly they were not hypothesis about the physical world, or, as your prefer, reality. They were neither abstracted from or hypothesis about the physical world. — Fooloso4

Yes, and no. I grant that most modern mathematicians are not thinking of the real world when they work. That does not mean that the content they work with is not derived from our experience of reality.

To be continued ...

Intelligibility is a potential that exists prior to being actually known. So, it is not "derived." It is in nature. — Dfpolis

This ignores the point. First, by derived I mean abstracted. Second, if the mathematical structure is in nature but that structure is knowable without being abstracted from nature then there is reason to think that structure might be independent of nature.

I suggest you read a calculus book. — Dfpolis

I suggest you read why I said it was nonsense and respond to that. Here it is once again:

With regard to Zeno, it is the divisibility that is infinite. With regard to infinitesimals the quantity is smaller than can be measured. In neither case is it something derived from experience. They are theoretical constructs. Whether reality is continuous or discrete remains an open question. — Fooloso4

First, Zeno's paradox is not something abstracted from nature. Second, both Newton and Leibniz used a concept of infinitesimals that was not abstracted from nature given that the infinitesimal is not measurable. Third, the question of whether reality is continuous or discrete is something that is dealt with in physics not mathematics.

Your claim is that mathematics is an abstraction from experience. But now you say that the parallel postulate cannot be abstracted from experience.
— Fooloso4

Reread the OP. — Dfpolis

If you are referring to 2a, an axiom or postulate is not a hypothesis.

One can be right about some things, and wrong about others. While I am happy to allow Bolyai his joy, his assessment is clearly inaccurate. Human creativity consists in imposing new form on old matter, not creation ex nihilo. — Dfpolis

Of course it is not creatio ex nihilo! He did not mean it literally. Nit picking does not address what is at issue. Once again, non-Euclidean geometries are not abstractions. The negation of the parallel postulate is not a hypothesis, it is an axiom. What is of interest is what follows from it, and what follows is completely independent of physical reality.

I grant that most modern mathematicians are not thinking of the real world when they work. That does not mean that the content they work with is not derived from our experience of reality. — Dfpolis

The negation of the parallel postulate is not derived from our experience of reality, nor is what follows from it.

And how do we come to posit the parallel postulate, if, according to you, it is not an abstraction from reality? — Fooloso4

You seem to have forgotten the OP, where I used it as an example of a hypothetical postulate. It is derived by assuming that our small-scale experience with parallel lines can be extended to infinity.

Its negation is not an abstraction from reality either. Both, however, have their application in reality. — Fooloso4

Whatever we know can be truly applied to reality can be abstracted from reality. We do not and cannot know that the parallel postulate is true because our experience is finite. We can only know if the space-time metric is approximately Euclidean.

We can abstract non-euclidean geometries from spherical and saddle shaped surfaces.

It is not a name assigned to a ball that came to exist independent of the game. It is the name of a ball specifically designed and made to be used to play the game of baseball. If not for baseball the ball would not exist. — Fooloso4

I agree with all of this, The point is that none of it, including the name, is intrinsic to the ball.

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First, by derived I mean abstracted. — Fooloso4

When intelligibility is abstracted it ceases being potential and commences being actually known. The whole point of intelligibility is that it is potential, not actual, knowledge.

Second, if the mathematical structure is in nature but that structure is knowable without being abstracted from nature then there is reason to think that structure might be independent of nature. — Fooloso4

I do not understand this at all. If it is in nature, there is no reason to think that it is not intrinsic to nature. Green leaves are in nature and intelligible. Does that mean they also have a Platonic existence independent of nature?

With regard to Zeno, it is the divisibility that is infinite. — Fooloso4

Yes, the potential to divide a continuous span is unlimited; however, any actual division is only finite. As we can only know what is actual, we cannot know anything infinitely divided. (Imagining an infinitesimal is not knowing it.) As I told you earlier, this is the reason for all of the epsilons and deltas in the definitions of calculus -- and it was to see those types of definitions that I referred you to a calculus book.

With regard to infinitesimals the quantity is smaller than can be measured. — Fooloso4

The question is not measurability, which is one for physics, but of being finite or not. Any actual quantity quantity greater then zero is finite. If we use '0' to define the concepts of calculus, they will be indeterminate. So, we us the limits of finite quantities tending to zero.

First, Zeno's paradox is not something abstracted from nature. — Fooloso4

I did not claim it was.

Second, both Newton and Leibniz used a concept of infinitesimals that was not abstracted from nature given that the infinitesimal is not measurable. — Fooloso4

I have not read their derivations. I know that they were defective and have been replaced by those now found in most calculus texts.

Third, the question of whether reality is continuous or discrete is something that is dealt with in physics not mathematics. — Fooloso4

Physics might well find limits to what is actually measurable, given the laws of nature. That is not deciding whether reality is continuous or not. The concept of continuity abstracts from the question of actual measurability.

Your claim is that mathematics is an abstraction from experience. ...
— Fooloso4

Reread the OP. — Dfpolis

If you are referring to 2a, an axiom or postulate is not a hypothesis. — Fooloso4

Regardless of whether I am right or wrong, I did not claim that all mathematics is an abstraction.

Of course it is not creatio ex nihilo! He did not mean it literally. — Fooloso4

If he did not mean it literally, does not support your position. If he would agree that he was imposing new form on old matter, then he might agree that the matter of math was abstracted from experience.

You keep repeating your dogmas, but you do not support them with arguments. You have not said why my analysis does not work beyond saying it does not agree with your belief system. I agree, my analysis is incompatible with your beliefs.

I would simply say that one can't deny this without twisting the meaning of "reality" as what is revealed by experience. — Dfpolis

This is really a fundamental point. What you're arguing is British empiricism, per Locke and Hume. — Wayfarer

No, I am not. I am arguing Aristotelian moderate realism.

But does sensory apprehension qualify as 'revealed truth'? Certainly through scientific method, we can discover truth, but the assumption of the 'reality of the given' is precisely what is at issue in philosophy. — Wayfarer

Experience is the data we have to work with. One can either work with experience, or one can simply cease thinking. The scientific method does not get one past this, as all it does is compare hypotheses to experience. Whatever you think reality is, experience is how we humans relate to it -- and we can only deal with it as we relate to it.

We do not and cannot have omniscience, so it is a trap to make omniscience the paradigm case of knowing. "Knowing" names a human activity. So as soon as you say "we do not know," you are abusing the foundations of language. "Reality" first means what we encounter in experience. So, if you say "we do not experience reality," you are again abusing language.

When you make "reality" mean more than, or something other than, what we encounter in experience, you are creating a mental construct. If you create that construct, and then claim that what you have constructed is inaccessible, you have said absolutely nothing about what we encounter in experience.

Doubt is an act of will. I can will to doubt anything, including my own consciousness, as eliminative materialists such as Dennett have chosen to do. What one cannot do is eliminate what we experience. We experience ourselves as subjects and everything else as objects. I know what I experience and no act of will, no doubt, can make me not know it.

Of course, I may misinterpret what I experience. I may think the elephant I experience is in nature rather than the result of intoxication. Still, if I did not have experiences I know to be veridical, I could not judge others to be errant.

the assumption of the 'reality of the given' is precisely what is at issue in philosophy. — Wayfarer

Only in post-Cartesian philosophy. The focus of pre-Cartesian philosophy was and continues to be being.

Again I'm no Aquinas scholar, but I think I grasp some of the rudiments of his hylomorphism, which says that — Wayfarer

What you quoted was a "difficulty" or objection Aquinas intends to resolve, not his position. His response is:

Although bodily qualities cannot exist in the mind, their representations can, and through these the mind is made like bodily things. — Aquinas De Veritate

And this is because, in the view of Christian philosophy, material things have no intrinsic reality; creatures are, as Aquinas' Dominican peer Meister Eckhardt said, 'mere nothings'. — Wayfarer

Eckhardt's is not Aquinas view. He sees material things as real and intrinsically good, as does Gen. 1, which sees God as judging each stage of creation as good.

Corporeal creatures according to their nature are good, though this good is not universal, but partial and limited, the consequence of which is a certain opposition of contrary qualities, though each quality is good in itself. To those, however, who estimate things, not by the nature thereof, but by the good they themselves can derive therefrom, everything which is harmful to themselves seems simply evil. For they do not reflect that what is in some way injurious to one person, to another is beneficial, and that even to themselves the same thing may be evil in some respects, but good in others. And this could not be, if bodies were essentially evil and harmful. — Aquinas ST I Q 65 Art 6 ad 6

I think it's more likely that you're misunderstanding Kant. — Wayfarer

If Kant is saying that we can know noumenal reality, but not exhaustively, I have indeed misunderstood him. I do not think he is saying that, do you?

Seriously, this impossibility of self-inquiry is an enormous flaw in the scientific method. — alcontali

I think the flaw is seeing the scientific method as the only acceptable means of inquiry. In its proper domain, the scientific method is fine.

You keep repeating your dogmas, but you do not support them with arguments. You have not said why my analysis does not work beyond saying it does not agree with your belief system. I agree, my analysis is incompatible with your beliefs. — Dfpolis

Here is the problem in a nutshell. You refer to your "analysis" as if it is not based on your own dogmas and beliefs. The fact that you indefatigably argue them demonstrates nothing more than your willingness do so.

Here is the problem in a nutshell. You refer to your "analysis" as if it is not based on your own dogmas and beliefs. The fact that you indefatigably argue them demonstrates nothing more than your willingness do so. — Fooloso4

I have explained how ere abstract concepts such as that of number from the realization that counting does not depend on what we count. You have not shown that this is an inadequate explanation of our natural number concept.