I support different projections of reality, but adhere to the thesis that because there is some general empirical data, re: experience and therefore knowledge, potentially common to all rational humans, reality in and of itself is most probably one iteration of all those various and sundry individual projections. — Mww

I am not sure I understand this. Are you saying, that as a projection, reality is a construct? That seems odd for an empiricist, for to be an empiricist, one must stand ready to be surprised by reality, and our own constructs have a hard time surprising us.

I would say that we try to model reality, and the more projections we incorporate into our model, the more adequate to reality it will be.

Calling it “instinct” or “innate knowledge” is splitting hairs in my view. — Noah Te Stroete

What are they aware of? Not some intellectual content, but a desire. In a sense it is knowledge, but not in the same sense that awareness of intelligibility is.

is there knowledge that can come from something other than sense data, or that doesn’t have as its foundation, sense experience? — Noah Te Stroete

After reading W. T. Stace, I started taking mystical experience seriously. I now think it is veridical, but not (usually) informative. It is veridical in that it is exactly what one would expect from an experience of God, but not informative because God is unlimited and information is the reduction of possibility. I think that a few mystics, such as John of the Cross, have grasped empirical reality via their awareness of God, but that this is extremely rare.

My other question is: in the case of JF Nash, he had insight into his illness. Someone else may not have this insight. Does someone who hallucinates and doesn’t recognize it not have useful knowledge of reality? — Noah Te Stroete

I don't know that "useful" is a relevant criterion for knowledge. I would say that if you don't know you are hallucinating, you could learn to recognize it, as Nash did, but in the meantime, you will be holding false views.

For example, how does a baby know how to suck on a bottle? Isn’t this an example of innate knowledge? — Noah Te Stroete

It is an example of instinctive behavior. If the child were old enough, it could know that had such instincts. I do not think that we should confuse behavioral propensities/desires with knowledge. For example adolescents have a sex drive, but not an innate knowledge of the mechanics of intercourse. It is rather that the things they want to do will get them there.

The "God's Helmet" experiments could not be replicated and are now considered debunked (by researchers in Holland, if memory serves -- I wrote about it in my book). The "results" were explained by suggestion.

But, supposing we had such a propensity, we would learn of it when we experienced its activation -- just as male transsexuals learn that they are "girls." Many people see religious behavior as a reflection of such a propensity.

My question is: is this empiricism, rationalism, or neither (such as in Kant’s view)? — Noah Te Stroete

The argument is mine. I'm a moderate, Aristotelian-Thomistic realist, who thinks that we can have different projections of reality, which is to say that we can represent the same reality using different conceptual spaces.

If there is no data outside our experience we are presented with two absurdities, 1.) we should know everything because all the experience we have is all the data there is, or 2.) data and experience are congruent which would force the impossibility of misunderstandings. — Mww

1. I have not assumed or implied that our experience exhausts being. I have only said that our concept of reality begins with what we experience. ("'Reality' first means what we encounter in experience.") We say "seeing is believing," not because seeing is exhaustive, or even inerrant, but because our concept of reality begins with things that can act on us in experience. I left out that we expand the extension of our concept of reality (being) far beyond this humble beginning because I was discussing the relation between experience and reality.

I think the problem is that we are not defining "data" in the same way. I am defining it as what is experienced, not things that could be experienced or known indirectly. You seem to be defining it as what we could know. I agree that\ much more is intelligible than we actually know, than our actual data.

2. I agree that being is not fully congruent with human understanding, but I think every experience is caused by an existent adequate to cause it. Our errors of understanding are due to misjudging/interpreting/classifying what we experience -- not to mis-experiencing. The adequate cause of my experience might be a neurological disorder -- and in time I might learn to recognize it for what it is. (As John Forbes Nash did.) Usually, the cause of my experience is just what I judge it to be.rly.

Be that as it may, I accept the gist of what you’re saying in the OP, so my little foray into the sublime can be disregarded without offense. — Mww

Even misunderstandings are opportunities for both parties to learn to communicate more clearly.

es, trees are often just present — Dfpolis

Sure. So going from that to "this is something I'm perceiving" etc. is theoretical, isn't it? That is, it's literally invoking a theory about what's going on. — Terrapin Station

It depends on how you define "theory." If you mean a hypothetical structure, then, no, it is not a theory. If you mean a way of organizing experience, then yes, noting that certain things (trees) are equally capable of evoking the concept <tree> does organize our experience.

So far we seem to share similar views, though I would not mix modes of neural response with modes of intentional response. (Not that I think that intentional response is independent of neural processing. I don't. Rather, they depend on different kinds of analysis, and if we are dealing with the question of realism, we have a lot of ground to cover before we can justify neuroscience.)

I am sorry that we have not been able to resolve our differences.

If one is positing that one has a body and is perceiving things via one's senses, etc., then one is already assuming realism, by the way. — Terrapin Station

I don't think we start by positing that. Rather we seek to organize our experience by classifying it, and in doing so we come to concepts that include our body. We come to understand things in terms of objects (ostensible unities) that persist and learn the conditions under which we can and cannot encounter them. It is out of such considerations that we develop notions of body, senses and sensible objects.

This is one reason the question of whether it's always the case of not just "tree" but "I'm conscious of a tree" (see my post above) is important. — Terrapin Station

You will need to expand on this, as I don't see your point. The center of our subjectivity is not given as material and so not as a body.

Excuse me for not quite seeing your point. I was not suggesting we really stop thinking, only that all the content we think about is experiential. So, if we were not thinking of content traceable to experience, we would have nothing to think about.

"Data" means what is given, and it is given exclusively in experience. I do not see any other way for there to be data. We even find out about our innate capabilities/propensities in the experience of using them. If, for example, I had an innate fear of 8-legged things, I would find out in experiencing my response to one -- and not as a priori content independent of experience.

I don't think that reflective thinking is the means of experience. I think that reflective thought is how we seek to integrate experience into a comprehensible whole.

Please forgive me if I missed your point.

If I make:
1) Reality synonymous with actuality,
2) Experience an awareness event, and
3) Awareness a perceptive and/or cognisant condition,

have we made similar assertions? — Galuchat

It depends on how you explicate your terms.
1) What does "actuality" mean to you? Is it accessible, or quarantined?
2) Are you thinking of "events" as disjoint, or simply points in a continuum we happen to be fixed upon? And, how do you conceive awareness?
3) By "Awareness" i mean what makes intelligibility known. So, it rises above sensory perception in that we can perceive and respond in complex ways without being aware in the sense required to know.

If I've over analyzed what you said, forgive me.

For example, it's never for you just that there's a tree, say. It's always that you have something like "I'm a conscious entity, aware of a tree" present? — Terrapin Station

For example, it's never for you just that there's a tree, say. It's always that you have something like "I'm a conscious entity, aware of a tree" present? — Terrapin Station

Yes, trees are often just present. Often they are not even identified. It is only when we fix on this or that aspect of experience that we distinguish trees and even ourselves as subjects. We are aware of the whole complex, but that does not mean that we have focused on any aspect of it so as to conceptualize/categorize it.

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.

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.

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?

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.

---

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.

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 ...

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.

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.

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.

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.

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.

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.

You said you were not a mathematical Platonist. — Dfpolis

I am not but your topic is an attack on mathematical platonism and if you are going to attack it you must accurately represent it. — Fooloso4

I wasn't representing it. I was telling you why abstract numbers do not occur in nature, which is what we were discussing.

If five is an abstraction from particular instances of five units or items then it is not actual except in that it is an actual abstraction. — Fooloso4

Exactly! At last we agree.

I think they might argue that the fact that mathematical truths are not dependent on experience is all the experience they need. — Fooloso4

People can argue whatever they like. There is no sound argument that "mathematical truths are not dependent on experience." How can we even know they are true unless they reflect our experience of reality?

... non-Euclidean geometries. They are not abstracted from experience. — Fooloso4

They can be. They are instantiated on spherical and saddle-shaped surfaces. If some axiom can't be, it's hypothetical.

They are not merely formally or internally consistent, they tell us something about the world without being dependent on it. — Fooloso4

Nothing can tell us something of the world without being instantiated in it -- and if it's instantiated in it, it can be abstracted from it.

to some extent (Kant would say completely) experience is itself constructed. — Fooloso4

Kant had no sound reason to claim that.

concepts that are constructed are not all "merely" constructed, the construct may be based on experience but cannot be reduced to experience. — Fooloso4

My claim is that what we know is based on our experience of reality, not that everything we can or do construct is reflected in reality. That is why some hypotheses are falsified.

The intelligibility of an object is the potential to know its essence. — Fooloso4

Perhaps, but as counting never exhausts the potential numbers, so human knowing never exhausts anything's essence. There is always more to learn.

The question is whether intelligibility inheres in the object. Whether or not our knowledge is partial is not at issue. — Fooloso4

Yes, that is the issue, but your argument is based on the fact that our knowledge is not exhaustive. That our knowledge is only partial does not show there is no potential to know more -- no greater intelligibility that that we have actualized.

Being a baseball is not incidental to it being a baseball. It is constructed according to specific rules for a specific purpose. — Fooloso4

Its purpose is in the minds of humans, not in the ball. We can use the ball for other purposes, such as to be a display or even a paperweight.

You said you were not a mathematical Platonist. — Dfpolis

I am not but your topic is an attack on mathematical platonism and if you are going to attack it you must accurately represent it. — Fooloso4

I wasn't representing it. I was telling you why abstract numbers do not occur in nature, which is what we were discussing.

If five is an abstraction from particular instances of five units or items then it is not actual except in that it is an actual abstraction. — Fooloso4

Exactly! At last we agree.

I think they might argue that the fact that mathematical truths are not dependent on experience is all the experience they need. — Fooloso4

People can argue whatever they like. There is no sound argument that "mathematical truths are not dependent on experience." How can we even know they are true unless they reflect our experience of reality?

... non-Euclidean geometries. They are not abstracted from experience. — Fooloso4

They can be. They are instantiated on spherical and saddle-shaped surfaces. If some axiom can't be, it's hypothetical.

They are not merely formally or internally consistent, they tell us something about the world without being dependent on it. — Fooloso4

Nothing can tell us something of the world without being instantiated in it -- and if it's instantiated in it, it can be abstracted from it.

to some extent (Kant would say completely) experience is itself constructed. — Fooloso4

Kant had no sound reason to claim that.

concepts that are constructed are not all "merely" constructed, the construct may be based on experience but cannot be reduced to experience. — Fooloso4

My claim is that what we know is based on our experience of reality, not that everything we can or do construct is reflected in reality. That is why some hypotheses are falsified.

The intelligibility of an object is the potential to know its essence. — Fooloso4

Perhaps, but as counting never exhausts the potential numbers, so human knowing never exhausts anything's essence. There is always more to learn.

The question is whether intelligibility inheres in the object. Whether or not our knowledge is partial is not at issue. — Fooloso4

Yes, that is the issue, but your argument is based on the fact that our knowledge is not exhaustive. That our knowledge is only partial does not show there is no potential to know more -- no greater intelligibility that that we have actualized.

Being a baseball is not incidental to it being a baseball. It is constructed according to specific rules for a specific purpose. — Fooloso4

Its purpose is in the minds of humans, not in the ball. We can use the ball for other purposes, such as to be a display or even a paperweight.

In the same way, there is no actual five in nature. — Dfpolis

The mathematical platonist does not claim that there is an actual five in nature. — Fooloso4

You said you were not a mathematical Platonist. I was explaining to you why the abstract five is not actual until abstracted.

What is not actual is abstract fiveness, i.e. the pure number. — Dfpolis

That is nothing more than an assertion. The platonist asserts that there is, but it is not in nature. — Fooloso4

No, it is not a mere assertion, but an appeal to experience. Platonists have no basis in experience for their position.

I agree with those who say we construct concepts rather than actualize them. — Fooloso4

If we merely constructed concepts, there would be no reason to think they apply to or are instantiated in, reality. It is only because our concepts actual prior intelligibility that what we have in mind relates to reality.

The intelligibility of an object is knowledge of its essence, that is, what it is to be the thing that it is. — Fooloso4

First, intelligibility is not knowledge. It is the potential to be known. Second, all human knowledge is partial, not exhaustive. We may, and usually do, know accidental traits rather than essences. Third, there is nothing intrinsic to a baseball that relates it to any particular game. The relation is a human convention, as games are human constructs.

What I said is that I actually have five fingers whether I count them or not. If I only get to three I still have five fingers. — Fooloso4

Yes. No universal exists abstractly in nature. There is no actual humanity in nature. There are men and women with the intelligibility to engender the concept <humanity>. What makes the universal concept actual is our awareness of this instantiated intelligibility. In the same way, there is no actual five in nature. There are these actual five fingers and those actual five toes, each with the intelligibility to engender the universal concept <five>. In other words, an instantiated concept is not a universal, abstract concept. Instatiated concepts like these five fingers come with additional notes of inteligibility (e.g. being fingers) that need to be separated/ignored by the mind in fixing on a universal such as five.

There is no actual count until they are counted, but there are actually five fingers, which is confirmed by the count. — Fooloso4

Yes, I was insufficiently clear earlier. I take responsiblity for the confusion. What is not actual is abstract fiveness, i.e. the pure number.

Knowledge is not passive reception of "intelligibility". Knowledge is conceptual. — Fooloso4

I did n't say knowledge was passive. We have to actively attend to intelligiblity to make it understood. That is why Aristotle calls awarenss "the agent intellect." Our act of attending/awareness actualizes intelligiblity, converting it into concepts.

And it follows from this that the intelligibility of a baseball is not something that inheres it the object. — Fooloso4

We have to distinguish inherrent intelligiblity from relational intelligiblity. All objects have both.

If she is not told, or as you would have it, learned what a number is, what she thinks a number is can vary. — Fooloso4

I have no problem with alternative conceptual spaces. There's nothing wrong with a concept of number that excludes 0 and 1. It just represents reality in a different way than a concept that includes them. Concepts aren't judgements and so they're neither true nor false.

The number is how many of whatever it is we are counting. If I count the number of fingers on one hand and I count correctly the number is 5. That is because I actually have 5 fingers on my hand. If one of my fingers was cut off I would count 4 and that is because I actually have 4 fingers on that hand. — Fooloso4

I am not denying that you have 5 fingers on your hand -- it is just that five fingers is not the abstract number 5 -- it is specific instance of five, not the universal five.

If we cannot determine the unit we cannot determine the count. — Fooloso4

If we cannot determine the unit, we can't count. The things we count are prior to our counting them.

No wonder you are confused! Counting something has nothing to do with determinism. — Fooloso4

It does not have to do with physical determinism, but with the fact that things can be predetermined without being actual. The count of your fingers was predetermined to be five before anyone counted them, but there was no actual count of five fingers.

I would say that the number is not determined until we count, but what we are counting, the items, as you said, are actual. It is because there is actually this item and this item that we can determine how many there are. We can call this determination the count. It we count six and we count correctly that is because there are actually six of the items to be counted. — Fooloso4

I agree. There are six items -- a specific instance of 6 -- not the abstract number 6.

It means that its intelligibility is actualized by someone's awareness. — Dfpolis

This is evasive. Intelligible in what way? Which is to say, as I asked, what does it mean to say the ball is known? — Fooloso4

I told you. The ball is intelligible as this kind of thing, with these specific properties, and someone has actualized part of its intelligibility by becoming aware of it. If it were not able to be known, no one could know it -- and if the knower were not able to be informed she could not be informed about the ball.

If you mean that it stands out (literally, exists) distinct from all else, that does not mean that intelligibility is a property of the object. — Fooloso4

This depends on how you define "property." What is intelligible is the whole, but we do not actually understand mall of it.

If intelligibility inheres in the object then someone would know what a baseball is even if they did not know what the game of baseball is. — Fooloso4

The ball is a baseball because of its relation to the game. Knowing the ball in itself will not tell us its relation to the game.

No, it would not necessarily be by abstracting. — Fooloso4

No, it would not necessarily be by abstracting. I gave several different things she might assume, stories she might tell herself. — Fooloso4

The assumptions are all after learning. You have provided no alternate account of learning the concept.

Both are dependent on us to determine, that is, to know or be informed of the number. In neither case is the number a potential number except with regard to our potential to know it. — Fooloso4

Let's try this a different way. Surely the number does not inhere in the objects we count, for they can be grouped and counted in different ways to give different numbers. So, if it is already actual, and we agree that it does not pre-exist in our minds, where is it?

I beg to differ. The items can be counted if and only if they are actual distinct items. — Dfpolis

I am not going to get into methods of counting bacteria. — Fooloso4

I am not confining my claim to bacteria, nor discussing methods that apply to them in particular. So, do you agree that items can be counted if and only if they are actual and distinct?

What we choose to count is up to us, how many there are of what we count is not — Fooloso4

Think of it this way. Classical physics is deterministic. So, given the initial conditions and the laws of nature, the system state at a later time is fully determined. That does not mean the later state is now actual. It is only potential. So it is with counting. The number is predetermined, but not actual until the count is complete.

What does it mean to say the ball is known? — Fooloso4

It means that its intelligibility is actualized by someone's awareness.

When someone identifies an object as a ball is the ball known? — Fooloso4

It has to be known as an object, as a tode ti (a this something) before it's classified.

If they cannot tell you whether the material is rubber or synthetic is the ball known? If they do not know the molecular or subatomic make-up is the ball known? — Fooloso4

Yes, but not exhaustively. We never know anything exhaustively.

If they know it is a baseball is being a baseball an intelligible property of the object? — Fooloso4

Being a baseball is intelligible, but it is the ball as a whole, not a property of the whole.

If some other ball is used to play baseball is being a baseball an intelligible property of the object? — Fooloso4

Not unless you change the definition of "baseball" to mean any ball you play baseball with. If you do, then the last response applies.

If the ball is used as a doorstop does someone who only knows it as it is used for this purpose know that it is a ball? A baseball? — Fooloso4

It is not necessary to know everything about a this something to know it in some way.

If they saw someone hitting it with a stick wouldn't they wonder why he was hitting the doorstop with a stick? Perhaps they might think that he does not know what a door stop is. — Fooloso4

Perhaps.

Now that I've answered your questions, can you explain their relevance?

She might be a platonist and assume that <four> must still exist even when the oranges are eaten and the pennies spent. — Fooloso4

That would not change how she came to the concept. It was by abstracting from her experience of counting real things -- not by mystic intuition.

The "experience" of abstract arithmetic concepts may only come as the result of being taught to think of numbers in a certain way. — Fooloso4

I am not saying that our conceptual space is independent of our cultural background. I am saying that whatever concepts we do have are abstracted from empirical experience.

For example, the fact that nothing can be and not be in one and the same way at one and the same time, contra if it were the case that something could be and not be in one and the same way at one and the same time. — Terrapin Station

I understand the contrast, but not its point.

Whether one is platonist or not, however, in such a case the number refers to the objects being counted. At any given moment that number is an actual number, even if we do not know what that number is. Here potential means we do not know what the actual number is. — Fooloso4

There are two potentials here. One is our potential to be informed, which belongs to us. The other is the set's potential to have its cardinality known, which belongs to what is countable, and is the basis in realty for the proper number to assign to the set.

The number of bacteria in the petri dish or fruit in the bowl or whatever it is that we are counting cannot be counted if that number is not an actual number of items. — Fooloso4

I beg to differ. The items can be counted if and only if they are actual distinct items. The number that results is one, abstract, way we can think of the set.

How many there are of whatever it is we choose to count is independent of us. — Fooloso4

This is self-contradictory. If the number is "How many there are of whatever it is we choose to count," it is not independent of us.

Rubber and spherical are properties of the object. Intelligibility is not a property. — Fooloso4

Necessarily, whatever is actually done can be done. If the ball is known, necessarily it can be known, and so is intelligible. As it can be known whether or not it is actually known, intelligibility inheres in objects. So, why do you say it is not a "property"?

The intelligible properties are those properties we understand, rubber and spherical. Intelligibility is not another property that is intelligible. — Fooloso4

Don't we understand that balls are knowable?

What depends on us is which notes of intelligibility we choose to fix upon. — Dfpolis

What depends on us is the ability to understand, to make the object intelligible to us. — Fooloso4

Rather, to make aspects of the object actually understood by us. Our understanding is not exhaustive and if we do choose not to look, we will not understand what we choose not to look at.

What we experience is not an assumption. It is data. — Dfpolis

We are talking about what a number is, the concept or ontology of numbers. That is not an experience or data. We do not experience numbers, we experience objects of a certain if indeterminate amount. — Fooloso4

And abstract arithmetic concepts from that experience. You let a child count four oranges, four pennies, etc., and she abstracts the concept <four>..

Okay but there is a limit in that being is some ways and not others. We've already gone over and agreed that it's some ways and not others. The ways it's not are the limits. — Terrapin Station

But, what being is not, is nothing.

The degrees of abstraction have real differences which our definitions are based on.

If "constituents" means preconditions, I have no objection to ideas having constituents.

We may have the potential to determine that number but that does not make it a "potential number" — Fooloso4

If numbers were objects in nature, you would be right, But they aren't objects in nature, they are the result of counting sets we chose to define. Why count only the fruit in this bowl instead of some other set we define? The objects in nature are fruit, bowls, and so on -- not integers. Integers are the counts of sets we arbitrarily define -- change your set definition, and the count changes. That makes the numbers partly dependent on us and partly dependent on the objects counted. So, numbers do not actually exist until we define what we're going to count and count it.

Universal ideas are not things. There is no "bigger than." There are pairs in which one is bigger than the other. In the same way there is no "seven." There are sets, some of which have seven elements, but that "seven=ness" ceases to be if we put those same elements in different sets.

The intelligibility of an object simply means that we are able to understand it in some way. That is not an aspect of the object. — Fooloso4

So, being rubber or spherical are not aspects of a rubber ball? Of course they are. Just because we can fix on the ball's matter or the form does not mean that the ball's intelligible properties depend on us (unless we're the ones defining the object). What depends on us is which notes of intelligibility we choose to fix upon.

If a state requires mental determination then that determination is not an aspect of the object but rather something we say or know or understand or have determined about the object. — Fooloso4

If it depends only on us, this is true, but knowing depends jointly on the properties of the object and what we choose to attend to. An object's properties do not force us to attend to them, nor does attending to an object typically create its properties.

No inquiry is free of assumptions. — Fooloso4

What we experience is not an assumption. It is data.

It lacks determinant reference, but it has a reference type. That type may be a numerical value or something else that can be represented by the formalism. — Dfpolis

Which means that it differs fundamentally from a number, which is always determine and, in addition, a variable may reference something that has no numerical value. — Fooloso4

Right. I never said that variables and determinate numbers were the same.

This unnamed authority was David Hilbert — alcontali

Thank you. Recall that David Hilbert's "program" (concept of math) was destroyed by Kurt Gödel.

Certainly the Circle of Vienna still happily amalgamated mathematics and science. — alcontali

The Vienna Circle hardly deserves to have its name attached to a movement started by Aristotle, and brought to fruition long before any of them were born.

These impossibilities give inescapable structure to nature. That is in my impression the core of the esoteric link between nature and mathematics. — alcontali

If so, we can certainly know that structure, and abstract it to form the axiomatic basis of mathematics -- making Platonism unnecessary and formalism inadequate.

2+2=4 is not a "Platonic relationship". That 2+2=4 is true, according to mathematical platonism is due to the nature of numbers. The relationship is made possible by their nature. The relationship itself is not another platonic object. — Fooloso4

Yes, the content of the Platonic realm is usually supposed to be prototypes of universal concepts, such as number and equality. Excuse my shorthand description. I don't think it impacts my point that the relation between the Platonic realm and empirical reality is fuzzy at best.

The number of pieces of fruit in the bowl is undetermined until counted. This does not mean that the number of pieces is a potential number. It is an actual number that before we count we might say it could be six or seven or eight. There are actually seven pieces whether we count them or miscount them. They do not become seven by counting them. We are able to count seven because there are actually seven pieces of fruit in the bowl. — Fooloso4

Yes, the cardinality of the fruit in the bowl is seven whether we count or not. That does not mean that the concept <seven> can exist outside of the mind. While the set we have chosen to define has a determinate cardinality, the fact is that we choose to define the set. So, the concept seven is not determined solely by the experienced situation. If we count only oranges we might get three. If we count objects, we may include the bowl and get eight, or the bowl and table, and get nine. We might count pits and seeds, and get twenty of thirty. While each of these counts exists in our experience potentially, the actual count/number will depend on how we choose to conceptualize our experience. So, actual numbers depend both on objective reality and how the subject thinks of that reality by defining the sets whose cardinality we seek to know.

So, an aspect of something known is that it is knowable. Aside from being tautological and trivially true it raises questions that go beyond the current topic and so I will leave it there. — Fooloso4

It is not trivial that the intelligibility of an object does not constitute an actual concept. A state's potential for a seven count does not exclude is simultaneous potential for other counts when conceived in other ways. So, it is not trivial that states require further (mental) determination to be assigned actual numbers.

Of course it is interpretative! What is at issue is the concept of number. That is an interpretive question. — Fooloso4

Exactly, and so one that requires justification. It seems to me there is inadequate justification for both Platonism and pure formalism. Saying that mathematicians have such beliefs is not justification. One needs to look at how we learn and apply mathematics to have a theory that is coherent with the rest of our knowledge.

It does not have any reference until it is assigned one. — Fooloso4

It lacks determinant reference, but it has a reference type. That type may be a numerical value or something else that can be represented by the formalism. A variable might, for example, be assigned any real number, or perhaps, a complex tensor of rank 12, depending on its type. So it has a determinant (well-defined) potential reference -- just as does any universal term.

Again, we see the importance of distinguishing what is actual from what is merely potential.

The nature of being and God IS being. — Dfpolis

So that's the same thing as saying "the nature of God" no? — Terrapin Station

Yes, as long as you do not take "nature" to entail limiting determinations.

there are approximations and generalizations etc. that simply don't make sciences as rigorously logical as mathematics. For starters, every measurement is an approximation. — ssu

I do agree that physicists tend to think more eclectically and in a less structured way than mathematicians. Still, I think logic is logic and the validity of consequences depend only on the claims made in the premises, not on the accuracy of those claims.

Perhaps now I understand your point. (I'm btw happy with pragmatism: usefulness is far more important than we typically think.) — ssu

I interpret Aquinas's veritas est adaequatio rei et intellectus in a way that spans from correspondence to pragmatism. Adaequatio means "approach to equality," not correspondence per se. The question is how close do we need to approach reality for our understanding to be true? My answer is that the approach has to be adequate to our needs in context. In metaphysics this is very close to correspondence. In science, it is very close to pragmatism.

when you talk about 'unscientific' math that is "merely a game, no different in principle than any other game with well-defined rules" is that it's actually not applicable and/or the axioms simply aren't in line with reality. — ssu

Yes.

So the foundations aren't so narrow that everything starts from simple arithmetic. — ssu

Agreed. I also agree that there is always more to learn.

If God willed "something" other than being, God would will no-thing. — Dfpolis

What makes this the case, God or something else? — Terrapin Station

The nature of being and God IS being.

First, thank you for posting Frege's argument.

ii. It is possible for simple statements with singular terms as components to be true only if the objects to which those singular terms refer exist.
....
v. the natural numbers are existent abstract objects that are independent of all rational activities, that is, arithmetic-object platonism is true.

Your example of counting fruit is a straw man. — Fooloso4

My comment is directly on point, and does not attack a straw man, but premise ii. It misstates the conditions required for qa statement to be true, by taking the correspondence theory of truth too literally. If is not necessary that the predicates of true simple statements with singular terms as components to exist actually, but only potentially, That was Aristotle's insight in his definition of quantity in Metaphysics Delta. Quantity in nature is countable or measurable -- potential not actual numbers. "There are seven pieces of fruit in the bowl" is true, if on counting the pieces of fruit, we come to seven and no more.

This being adequate account of the numerical claim shows that we need make no appeal to an actual seven existing independently of a counting operation. In other words, "true only if the objects to which those singular terms refer exist" is false if we tale "exist" to mean "actually exist," but true it we take it to mean "potentially exist" or "exist as intelligible".

And, yes, abstraction does not create content, it actualizes intelligibility already present in reality. — Dfpolis

This strikes me as a form of Platonism, as if intelligibility is something somehow present in but other than the objects of inquiry. — Fooloso4

It is a form of realism -- Aristotelian

Do you mean different concepts that were in prior use? — Fooloso4

moderate realism, not Platonic extreme realism. Moderate realism sees content as deriving from objects (their intelligibility), and awareness of content as deriving from knowing subjects. So, I ask, does not data derive from what we are studying? And, is unexamined data thought?

I'm not saying "intelligibility is something somehow present in but other than the objects of inquiry." I'm saying that every note of intelligibility is an aspect of the object known. It is not the whole object, but an aspect (rubber is not all there is to being a rubber ball). I say "aspect" instead of "part" because parts can be separated in nature, but aspects may be separable only in the mind (by abstraction). E.g. we can separate rubber from the ball, but we can think of it in abstraction.

Do you mean different concepts that were in prior use? — Fooloso4

No, I mean that concepts don't change. New concepts are necessarily different concepts. The may replace an old concept, but they are not the old concept transformed.

in modern math a number, '4' for example, is itself an object. With the move to symbols, 'x' does not signify anything but itself. — Fooloso4

This is an interpretive, not a mathematical, claim. If you're a Platonist, "4" is an object, if you're more reflective, you see that it's only an object of thought. No, "x" does not mean the letter "x." It has reference beyond itself. It may mean an unknown we seek to determine, a variable we can instantiate as we will, or possibly other things, but it never signifies itself, which is always a particular image -- because text images are not what math deals with.

I suppose you could mean that 'x' is just an object that can be formally manipulated according to a set of rules. That it is only that is also an interpretive claim, formalism. Nothing in the view I am proposing prevents rote, formal manipulation according to rules. My view just says "x" is usually more than that, but we can abstract away from its meaning in formal manipulation.

Clearly, mathematical symbols are not invariably free of meaning. Godel uses arithmetic forms to represent axiom sets, and his major theorems are restricted to systems representable in arithmetic.

Thank for the book review reference, It may take me a while to get to it.

I am speaking here specifically about the concept of number, that is, what a number is. — Fooloso4

OK.

It is an intellible whole that becomes increasingly actualized (actually known) over time. — Dfpolis

Either you think that each of these ways are retained in the development of the intelligibility of the whole or some are modified and rejected. — Fooloso4

Hypothetical understandings are modified and/or rejected over time. Abstractive understanding is partial and grows over time without need of replacement. Still, parts of it can be forgotten or fall out of vogue.