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.

In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. — alcontali

Which is why a consequent of formalism is that math, as a meaningless game, is of no intrinsic value. This view is incompatible both with our experience of learning math by reflecting on examples, and with the fact that mathematical propositions are treated as truths in scientific thought.

According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other contensive subject matter — in fact, they gi't "about" anything at all. — alcontali

The Aristotelian-Thomistic view also rejects actual numbers, sets, and triangles in extramental reality, but sees an alternative other that empty formalism. Reflecting on the role of examples in learning math and on its applicability in science, it sees that numbers, sets, and triangles are intelligible (potential, able to be understood) in reality. The act of abstraction, which is one function of awareness (the agent intellect), makes what was merely intelligible in nature actually understood. This provides a middle ground between Platonism and formalism.

There may be an esoteric link between the abstract, Platonic world of mathematics and the real, physical world, but this hypothetical link cannot be used for any practical purpose. — alcontali

As there is no Platonic world, there is no possibility of a link to it, There is, however, a natural world with well-known links to mathematical thought.

This formula game enables us to express the entire thought-content of the science of mathematics in a uniform manner ... — alcontali

Your unnamed authority agrees that math is a science. Games are not sciences. Being a science (an organized body of knowledge) means that math is an understanding of reality.

Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. — alcontali

Clearly, this is nonsense, We know that there is no intrinsic necessity to the parallel postulate in the context of geometry, or to the axiom of choice in the context of Zermelo–Fraenkel set theory. If the rules are arbitrary, so are the results.

These rules form a closed system that can be discovered and definitively stated. — alcontali

No one can discover what does not pre-exist. Yet, the pre-existence of mathematical axioms is the exact premise formalists reject. So, again your authority is inconsistent.

What "truth" or objectivity can be ascribed to this theoretic construction of the world, which presses far beyond the given, is a profound philosophical problem. — alcontali

This is the problem of universals, solved by the moderate realist insight that ideogenesis involves the actualization of intelligibility in nature via abstraction by the agent intellect.

It is closely connected with the further question: what impels us to take as a basis precisely the particular axiom system developed by Hilbert? — alcontali

The answer to this is, nothing. Abstraction fixes on certain notes of intelligibility and certain intelligible relations to the exclusion of others. Thinkers may have different conceptual spaces in light of their individual experiences and needs. So, the same reality can be understood in different, partial. ways -- depending on the perspective we take and the conceptual space into which we project our experience.

Mathematics is consistent by design while the real, physical world is consistent by assumption. — alcontali

No, the physical world is consistent in virtue of its existence. The nature of being is such that it cannot instantiate a contradiction. That does not mean that hypothetical theories, in physics or in math, need be consistent.

Therefore, it is sometimes possible to construct consistency isomorphisms between both, that will be uncannily effective in mirroring some sector of reality inside an abstract, Platonic model. — alcontali

This view makes the applicability of math to nature entirely accidental. If you think about it, you'll see that you can't construct such an isomorphism unless the relevant mathematical relations are already instantiated in nature -- and we can understand that they are. But, if they are already instantiated and intelligible, both Platonism and formalism are wrong. We can construct the relevant math on the basis of our understanding of those intelligible relations.

Yes, in the Metaphysics where he discusses the principle of contradiction.

A secondary source is not a citation from Aristotle. I've read his analysis of axiomatic foundations. While he says we cannot deduce everything, he is convinced that we can justify axioms non-deductively and does so in a number of instances.

You are avoiding the question. Science does not simply "assume its principles". It determines them through observation, hypothesis, testing, theory, modeling, and so on. — Fooloso4

This is just a verbal difference. Scientists certainly do, and that it my point: axioms need justification. The verbal difference is in how to define a science like math. Some would say that no science justifies its own assumptions, others look at what people who call themselves scientists or mathematicians actually do. I don't care how you define a science such as math. We seem to be agreeing on what is and needs to be done.

First, someone has to do the abstracting. Second, the properties of say a triangle are not determined by abstraction. — Fooloso4

Yes, we need knowing/observing subjects. And, yes, abstraction does not create content, it actualizes intelligibility already present in reality.

2+2=4 does not exist only in the "Platonic realm", does not need to "apply to reality", and it is meaningless to call it a Platonic relationship. It does not apply to reality because it is counting something real. — Fooloso4

The context was that of showing the consequence of the questionable claim that 2 + 2 = 4 exists in a Platonic realm. It was not me stating my own position.

So, in a topic entitled The Foundations of Mathematics, the actual foundations of mathematics is not your present interest. — Fooloso4

My interest is how the foundations are justified. The actual foundations are of interest only as examples of claims made and needing justification. I have used a number of actual postulates in that way. Enumerating all the postulates in all branches of mathematics would not help us understand the justification processes. It would only be a distraction.

It is not simply adding new concepts, it is a matter of different concepts. — Fooloso4

I am not sure how you distinguish different concepts that were not in prior use from new concepts. Perhaps examples would help.

This does not vitiate old concepts in the sense that they are wrong, but that mathematics no longer operates according to the older concepts. — Fooloso4

I think we are using "concept" in different senses. I am thinking of <number>, <line>, <irrational number> and so on when I say "concept." You seem to be thinking rules of procedure. Yes, there are new procedures, but Euclidean geometry is still being pretty much as he wrote it in the Elements. So, the old procedures are not abandoned.

Thank you for the link/reference.

As I pointed out, there is no number 0 or 1 in Greek mathematics. You might dismiss this as simply wrong, but in doing so what you miss is the ability to understand a way of looking at the world that is not our own. — Fooloso4

No, I don't dismiss different conceptual spaces as wrong -- they are just different ways of thinking about the same reality.

Thank you. I will read it when I have time.

So in the way as history is a science? Some in the natural sciences would shudder at the idea, but I'm totally OK with it. — ssu

We seem to be converging. I see good history as the result of rigorous method, but not as explaining events from first principles.

The way I see here math to be logical that simply every mathematical truth has to be logical. It doesn't state AT ALL that everything in math has to begin from a small finite set of axioms. What Hilbert was looking for was something else, especially with things like his Entscheidungsproblem. — ssu

I have no problem with this. My point was that logic is necessary in all sciences. Of course, the amount of empirical data and the role of hypotheses varies widely. The point of my classical mechanics example was that it is a closed, axiomatic structure, within which one may deduce theorems in the same way that one deduces them in math. Still it is not math, and it is not true in any absolute sense.

What you are making is a hugely reductionist argument that everything has to be deduced from the same axioms. — ssu

Not at all. I believe in open philosophy -- the idea that we should be open to all sources of truth and not restrict our inquiries with a priori assumptions or conceptual spaces. I do, however, see each mathematical theory as defined by its axioms.

If something doesn't fit to be the universal foundation, in your terms it has to be false and whole fields have to be false. — ssu

That is precisely the notion I reject.

I am saying that axioms are no different than any other claims. They are either justifiable, or not. Either adequate to reality (true) or not. Mathematics cannot be exempted from epistemological scrutiny just because it has a canonical, axiomatic form.

There being quantum mechanics or geometries of spheres etc. simply don't refute one another and make the other untrue or false. What is only wrong is the reductionist idea that everything can be deduced from one system or the other. — ssu

Of course the fact that classical mechanics fails to predict phenomena at quantum or relativistic scales, means that it is inadequate to these kinds of realities and so false in an absolute sense.

Geometry is a little different, as it lacks operational definitions of basic concepts such as <straight line>. If we define "straight" to be the path taken by a light ray, then Euclidean geometry is inadequate on a cosmic scale and so false.

I am not a reductionist. For example, biology cannot be reduced to physics because some of the contextualizing data that is abstracted away in physics is the data on which biology is built. So, we must continually return to reality, to the experience of being, to correct our conceits.

Math isn't like this. Mathematics has for example incommensurability, which is totally logical. — ssu

I have no idea what you mean by "totally local." Are you claiming that the concept <incommensurability> came to be independently of any experience of reality? History would seem to rebut this.

It seems to me that you are looking at final canonical forms and forgetting the mental processes that got us there.

Yes, well, crocodiles and dragonflies have some degree of awareness, but zero intellect! — Wayfarer

They have a degree of responsiveness that seems fully explainable neurophysiologically. We have no data implying such animals can actualize intelligibility as opposed to sensibility. I disambiguated my use of "awareness" by saying that it was the same as Aristotle's agent intellect -- whose function is the actualization of intelligibility.

the problem boils down to the fact that consciousness is intrinsically first-person, something of which one is subjectively and immediately aware, or rather, 'that which is aware', and as such is never an object of experience (except for by abstraction). The precise reason why Daniel Dennett refuses to accept that it's real, is because it's not an object of experience. — Wayfarer

Yes, and no. I agree with most of what you say, but we would not know we were aware if we did not experience our own awareness. The problem, then, is not lack of experience, but lack of third person experience. Dennett rejects consciousness because he is unwilling to credit first person experience.

Logical positivists used to say consciousness/awareness was not intersubjectively available. It is. What is unavailable is multiple subjects observing the same token of consciousness. Science is not concerned with token availability, but with type availability. Natural science experiments are intersubjectively available even if only one person at a time observes a result -- as, for example, in the Rutherford scattering experiment. So, the number of observers of result token is irrelevant. What is relevant is type replicability -- and that is intersubjectively available.

Thank you for very lucid explanation. — Wayfarer

You're welcome.

I have the idea that numbers and other intelligible objects are not existent (as are sensory objects), but that they are real. Numbers do not come into or go out of existence, and when we know them, we know them purely intelligibly, i.e. they are only discernible to a rational intellect (which is the thrust of the passage in Augustine). — Wayfarer

Yes, Augustine was inclined to Neoplatonism and that's a powerful intuition. I just don't see evidence to support it. It seems to me that we should rule it out on grounds of parsimony -- we don't need it to explain how we know mathematical truths.

But they're not really objects, they're constituents of thought - so the word 'object of thought' is in some sense a metaphor. (I regard 'objects' as exactly that - things that you have a subject-object relationship with, i.e. everything around you.) — Wayfarer

I agree with most of this, but "constituents of thought" bothers me. While we often reify ideas, it seems to me that the idea <apple> is simply the act of thinking of apples, not thing that can have constituent parts. Still, ideas can have multiple notes of intelligibility that can be separated in thought.

I tried Googling "Augustine intelligible objects" but only found secondary sources so far.

If you justify the axioms, then the justifications will become the new axioms. — alcontali

Only if the justification is axiomatic. It is not. See my new thread on the foundations of math.

As Aristotle wrote: If nothing is assumed, nothing can be concluded. — alcontali

Citation? We need not assume what we know by experience.

I subscribe to mathematical Platonism. However, for practical reasons, I do not make use of the possible link between the real, physical world and the abstract, Platonic world of mathematics. I rather leave this link unspecified. In fact, so does everybody else. — alcontali

Which is why this irrational belief continues to find adherents in mathematics.

First, sciences do not establish their own principles — Dfpolis

Where do you imagine these principles come from? — Fooloso4

Each field of math assumes its principles (its postulates and axioms), but that does not mean that the principles can't be investigated and justified by nonmathematically. I have said how the principles of math derive -- mostly via abstraction, some hypothetically. Investigating and justifying these means is outside of the scope of mathematics and the axiomatic method.

After a full paragraph on Platonism you said:

Platonic relationship 2 + 2 = 4 — Fooloso4

Please read sentences in context.

I said most of the foundations are the result of abstraction. — Dfpolis

To say what they are the result of is not to say what they are — Fooloso4

What they are is not my present interest.

The most basic concepts of of Western mathematics underwent a fundamental change with the origin of algebra, that is when numbers were replaced by symbols. Which leads to the question of whose mathematics? — Fooloso4

The concepts that existed before the addition of unknowns, variables, functions and distributions continue in use today. Adding new concepts does not vitiate old concepts.

Which leads to the question of whose mathematics? — Fooloso4

Mathematics is not personal property. It is an intellible whole that becomes increasingly actualized (actually known) over time. At any time some people know more than others. That does not make them owners, but knowers.

Are you disagreeing with his reading of Aquinas? If so, where do the mistakes lie? Or is it that you are disagreeing with Aquinas? — Fooloso4

As I have thought about the topic, but not read Maurer's paper yet, I am not in a position to say where the errors originate. If they are in Aquinas, they would not be my first disagreement with the Angelic Doctor.

God willed the being, Humans, not God, are the direct efficient causes of sentences by reflecting on the intelligibility of being. God is not the direct efficient cause of the sentences, and so did not will them, but their causes. i explained this in my last reply by saying God was the remote cause by willing both the known object and the knowing subject.

But then God could have willed being so that it's other than it is, right? — Terrapin Station

If God willed "something" other than being, God would will no-thing. That is the whole point of the ontological principle of excluded middle. Being is not the kind of thing that admits of intrinsic change.

"Being" is not a definable term, as it is intrinsically indeterminate. The only way to "change" it would be to make it what it is not, viz. either not-being or something determinate -- so replacing being simpliciter with a determinate kind of being.

Once again, the title of your topic is "The Foundations of Mathematics". Those foundations are not in modern mathematical theory or methodology. Greek mathematics is part of that foundation. Greek mathematics is not "Platonism". — Fooloso4

First, sciences do not establish their own principles, so it would be very surprising if math did. So, we agree on the first part.

Second, I did not claim that Greek math was Platonism. So, I have no idea what you aim to show by the last sentence.

To say:

... our mathematical concepts have a foundation in reality. — Dfpolis

Is like saying a building has a foundation in the ground. It says nothing about that foundation. — Fooloso4

If that were all I said, your criticism would be justified. it is not all I said. I said most of the foundations are the result of abstraction. In response to questions, I went on to explain how that worked. I also said that the rest were hypothetical -- and some of those were falsifiable and the rest not.

Instead, it would be more useful to direct the reader to Maurer's "Thomists and Thomas Aquinas On the Foundation of Mathematics", available free online — Fooloso4

Thank you for the reference, I did not know it.

I disagree with much of the quote you gave from Maurer. Every degree of abstraction is grounded in intelligible reality. It is true that there are no prefect triangles, etc. in empirical realty, but abstraction can leave behind the intelligibility of defects. I don't think anyone has ever come to the idea of a triangle without experiencing an imperfect instance in reality. The same is true of the other examples Maurer gives.

So re those three statements, are they the case because God willed it so, or are they prior to God so that God has no choice in them, either? — Terrapin Station

Neither.

Nothing is prior to God, because if something were, God would be dependent on it, and so not self-explaining, Further, statements are the expression of discursive (time-sequenced) thinking, and as God is unchanging, God does not engage in discursive thinking.

The statements are the result of finite, discursive minds expressing a partial understanding of being. As all being is willed by God, His will makes the statements possible in two ways: (1) by creating intelligible being to be understood and (2) by creating discursive minds to understand it.

In sum, God wills being, and the statements are the result of humans grasping part of the intelligibility of being.

You have completely ignored the foundation of Greek mathematics which makes your pseudo-problem of counting disappear. 2 + 2 = 4 is not a "Platonic relationship", at least not for Plato or the foundation of Greek mathematics. — Fooloso4

And that relates to my OP how?

First of all, with 'scientific' we describe that we are using the scientific method, an empirical way to make objective observations, experiments, tests or measurements, about reality, the physical world as you mention, to solve if our hypothesis are correct or not. Mathematics is logical system. — ssu

That is certainly the modern usage, but not the only one. Traditionally, scientia meant an organized body of knowledge -- organized in terms of explanations reducible to first principles. So, I would say that mathematics is a science in the sense of being an organized body of knowledge -- and that knowledge is an understanding of reality.

Second, "logical system" needs more explanation. All sciences proceed logically. I suspect that you mean a "closed system," i.e. one that simply elaborates an axiom set. Clearly, mathematics does more than that. I agree that mathematicians seek to put their science into a canonical form which is axiomatic, but they also have a history of examining and adding to their axiom sets. For example, we have the questioning of the parallel postulate and the development of non-euclidean geometries, the development of set theory, and discussion of the axiom of choice.

I'd suggest that the finished form of a science is a poor starting point for examining the nature of that science. Isn't it better to reflect on the process leading to the canonical form?

Also, "logical system" is inadequate if there are truths within the scope of the science that cannot be deduced axiomatically. The very existence of such truths implies that there are means of knowing truth more fundamental than the system's logical/axiomatic foundations.

Applicability of mathematics to the physical world isn't the logic that glues mathematics into a rigorous system, but logic itself. — ssu

No, math is not logic. That was Hilbert's view and Godel killed it. Math provides physicists with a set of assuredly true premises, which, combined with other, more empirical, premises will provide a sound conclusion if the empirical premises are true. What is special about mathematical premises is their reliability compared to less reliable Hume-Mill inductions and the hypothesis under consideration. Thus, when an experiment falsifies a conclusion, we can rule out the falsity of the mathematical premises as the cause, and fix our attention, first on our hypothesis, and second on our empirical inductions.

Typically a physicist uses a argument of the form:
Major (mathematical) premise: All systems instantiating mathematical property p are such that they instantiate mathematical property q.
Hypothesis: System A instantiates mathematical property p.
Conclusion: System A instantiates mathematical property q,
where we hope that the conclusion can be experimentally tested. The only "logic" here is a valid syllogistic form (Barbara), not mathematics. Our mathematical knowledge enters as a premise on the same footing as any other premise. For the argument to work, the mathematical premise must be adequate to reality (true).

Above all, something that we have thought to be a mathematical axiom isn't shown to be false from physical reality, but with mathematical logic. — ssu

This is inconsistent with your earlier claim that mathematics as a logical system. In such a system the truth of axioms is unquestioned. An axiom set may prove to be inconsistent, but if it is that only shows that some member of the set is false, not that a specific axiom is false -- unless one has metamathematical reasons for suspecting a particular axiom. But, if one does, then we are justified in examining the truth of axioms, and not merely accepting them as given a priori.

Here you seem to have the idea that if the axiom of choice is independent of ZF, it is somehow 'unscientific' as if other axioms would be the 'scientifically' approved. — ssu

No, not at all. If you go back to the OP, you will see that I divide axioms into three groups: (1) justified by abstraction, (2) falsifiable, and (3) unjustified by abstraction and unfalsifiable, and so unscientific. The question is not whether we are doing math, but how we justify truths in general. It is special pleading to say that all scientific (in the older sense) truths need be justified, but mathematical premises get a pass.

Perhaps the problem here is that some mathematicians see themselves as enlightened by mystic insight into the realm of Platonic truth. I do not.

So, the problem is not that C is independent of ZF, it is that C is unjustified and seems to be unjustifiable. Perhaps I'm wrong on C being unfalsifiable. Perhaps some consequent of C can be falsified. Whether my example holds, however, is a contingent matter,and irrelevant to the need for justification.

To say in this case that all of the math in all of those various fields of mathematics are unscientific is, should I say, out of whack. — ssu

I do not see this. After Laplace published his Celestial Mechanics and dispensed with Newton's ill-conceived "hypothesis of God," physics was seen as axiomatic and given canonical form by mathematical minds such as Hamilton and Lagrange. That did not make it true in any absolute sense. The classical mechanics they developed is still useful and taught today, but it is not true. Unjustified foundations necessarily give unjustified conclusions.

To make my argument short, scientific/unscientific is a poor definition in math, far better would be to speak of logical and illogical. We have had and can indeed still have illogical presumptions (or axioms) of the nature of math, just like some Greeks thought that all numbers had to be rational and were truly disappointed when finding out that there indeed were irrational numbers. — ssu

First, I have problems with the theory of types as a solution to Russell's paradox, but that is for another day.

What validly follows from axioms is necessarily "logical," but if the axioms are unjustified it has little claim on being true. If we understand science as organized knowledge, and what is unjustified does not count as known, then any consequent of an unjustified axiom is unscientific.

What I'm questioning is the notion that an account can be given of intelligible objects (such as number) in purely mentalistic terms. I think that Platonic realism posits that numbers are real for anyone who can count. So they are only knowable to a mind, but they are not the product of an individual's mind. — Wayfarer

Providing a purely mentalistic account is exactly what I am not doing. I am saying that our mathematical concepts have a foundation in reality. If there were no countable beings and no measurable beings, we could have no experience from which to abstract mathematical concepts.

While we may combine concepts in ways not found in reality, ultimately our concepts are traceable to the actualization of intelligibility found in nature. Objects act on our neural system is our senses, presenting intelligibility to awareness, Still, until we turn our awareness to these encoded contents, they are not actual concepts. It is our being aware of contents that makes what was merely intelligible actually known.

Further, engendered concepts are dynamically united to a corresponding intelligibility in nature, to the engendering object. First, the object's modification of my neural state is identically my neural representation of the object (encoding contents I can become aware of). Second, a single act of awareness simultaneously actualizes both the object's intelligibility (making it actually known) and the subject's capacity to learn (making it actually informed). So, there is no isolated mental construct here, but a (partial) ontological penetration of the intelligible object into the knowing subject.

This analysis does not, therefore, make numbers a mental product. It makes them the actualization of a prior intelligibility in nature -- eliminating the need for numbers to actually exist prior to being known -- in other words, the need for a Platonic realm.

think it's reason that naturalism has no account of. — Wayfarer

To me, "awareness" means agent intellect, without which there is no reason or conscious experience.

The link to Augustine is not available. I would like to see it if you have it (message it?). I am a great admirer of Augustine's insight, and think his account of the soul coming to know intelligibility is largely compatible with my understanding of Aristotle and Aquinas.

The page in Britannica is good starting point to answer your objections:

Axiomatic method, in logic, a procedure by which an entire system (e.g., a science) is generated in accordance with specified rules by logical deduction from certain basic propositions (axioms or postulates), which in turn are constructed from a few terms taken as primitive. These terms and axioms may either be arbitrarily defined and constructed or else be conceived according to a model in which some intuitive warrant for their truth is felt to exist. — alcontali

Defining a method is not an argument justifying the application of the method. Clearly, many mathematicians are concerned the justifying their axioms. I am also concerned about this issue. You seem not to be. So, we do not share a common interest.

You continue to make unargued claims. You are attacking Platonism, which I do not hold. You are attacking my claims, not by offering a substantive critique, but by the ad hominem that I am unqualified to comment, despite having studied advanced math and its history. So there is no point in our continuing to dialogue on this topic.

Yes. (1) Whatever is, is, and whatever is not, is not. (2) Something must either be or not be. And, (3) nothing can be and not be in one and the same way at one and the same time. These are reflected in the kinds of assertions that can be true.

Still, these are not constraints, because they exclude no-thing.

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

Does the Incompleteness Theorems say really this? Correct me if I'm wrong, but doesn't (the first Gödel Incompleteness Theorem) say that for any 'set of axioms' or consistent formal system there exists specific true but unprovable statements. That's a bit different — ssu

It does not seem different enough to vitiate my point. 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. Or, am I missing your point?

I think maybe "unscientific" in this context is wrong usage. In any case, mathematics is not an experimental science. — tim wood

I agree here with tim wood, talking about scientific/unscientific here with foundations mathematics is totally out of whack. — ssu

Is it? If we cannot justify certain axioms, how can we rely on the conclusions? As a physicist, I want my mathematics to be not merely consistent, but applicable to the physical world. We know that, however reasonable, the parallel postulate is not so applicable if we define straight lines as geodesics. How are we to know that the consequences of ZFC fair any better?

So, you will have to explain why my criticism is "totally out of whack."

What is the axiom of choice? — Noah Te Stroete

The Wikipedia provides a good discussion. It says "Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if the collection is infinite."

The basic problem with it is the same as that with the parallel postulate -- we have no experience of the infinite per se, from which to abstract it.

Personally, it seems reasonable, but then so did the parallel postulate when I studied geometry. My criticism is based on purely its lack of epistemological justification.

Thank you for your comments. I have no problem with the neoplatonic One Identified as God.

So do you think the law of the excluded middle, or the Pythagorean theorem, only came into existence with h. sapiens; or that such principles are eternal, and are discovered by any intelligence sufficiently rational to discern them? — Wayfarer

I am not positing that we're the only rational animals. I am saying that the principle of excluded middle reflects the nature of being and so is intelligible to anything capable of abstracting being as being. I am not saying that the Pythagorean theorem is true in general, only in flat spaces (where the parallel postulate holds).

I think the error you're making with the 'Platonic world' is to try and conceive of it as a literal domain. But what of the 'domain of natural numbers'? — Wayfarer

If someone wishes to define a "Platonic world" in non-literal way, I will be happy to comment on their effort. If it's undefined, it is irrational to appeal to it as an explanation.

The domain of rational numbers is countable objects. It is unlimited because we can partition unities into countable parts indefinitely.

Surely that is something real, as real numbers are included in it, and irrational numbers are not. — Wayfarer

Irrational numbers are not based on countable objects. That does not mean that they are not based on other aspects of nature, i.e. measurable quantity. In neither case do actual numbers exist in nature. What exists in nature is the potential to be counted or measured. Actual numbers exist only in minds actually thinking them.

And how does your account differ from run-of-the-mill evolutionary naturalism, in which there is nothing corresponding to what Aquinas deem the soul, which is 'capable of existence apart from the body at death'? Your account most resembles that of John Stuart Mill, whom I'm sure would not be the least inclined to agree with Aquinas. — Wayfarer

I distinguished abstraction from the Hume-Mill model of induction in my previous response (to Tim Wood). I wouldn't mind if my account were based on the aspects of reality fixed upon by naturalists, but, in point of fact, it does not. It requires the operation of an intellect in act (Aristotle's agent intellect) to make what is merely intelligible actually known.

Naturalism has no problem describing neurophysically encoded contents, but it has no rational account of awareness. Daniel Dennett showed in Consciousness Explained that naturalism cannot account for the experience of consciousness, and David Chalmers has pointed out the difference between the kind of progress made by neuroscience and the "hard problem of consciousness." In a previous thread I discussed the difference between intentional and material reality.

So, any account that hinges on the actualization of notes of intelligibility by awareness (as mind does) is beyond the scope of evolutionary naturalism.

Do you know that Godel considered himself a mathematical Platonist? — Wayfarer

No, I did not, but as many mathematicians are, it's not surprising. Being a good mathematician doesn't make one a good philosopher.

and surely we cannot abstract universal truths that are not instantiated. — Dfpolis

Are you sure of that? Are not all instantiated truths particular truths, and are not all universal truths abstract? You have left out how to get from particular to "universal." And how do you instantiate the truths of transfinite arithmetic? — tim wood

Yes, I am sure.

Yes, to be instantiated is to be particular. We move from the particular to the universal by removing particularizing notes of comprehension. For example, we ignore that we are counting apples, that apples are fruit, etc. and fix on the noted that matter to counting, i.e. that we are dealing with an instance of a unit and that unities are countable.

Contrast this with the Hume-Mill model of induction. In it you see 100 black crows, no white crows and posit "All crows are black." Whereas abstraction is a subtractive process, Hume-Mill induction is an additive process. We add the assumption that the cases we have not seen are like the cases we have seen. In abstraction, we add nothing. We merely remove notes of comprehension that don't interest us.

Re transfinite numbers: We come to the notion of Aleph-0 (countable) infinity by noting that the counting process has no intrinsic limit. 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. This was done by Georg Cantor in his 1874 uncountability proof.

Further, it may be that we can trace to a ground in nature, but is that all? — tim wood

I am not saying that nature is all that is, only that we have no reason to posit a Platonic realm of ideas.

And while the idea of concepts needing minds to have and hold them informally and intuitively seems right, is it altogether right? — tim wood

It is, unless you have a different definition of "concept," but then, you're not talking about what I'm talking about.

the content of thinking - the that that is thought about, is pretty much always already there prior to a mind thinking it. — tim wood

Yes, the content may well be "there" as intelligible, as something capable of being known, but not yet known -- and so as not yet a concept. That's what it is for a concept to be grounded in reality.

This means there is an accounting/definition problem. — tim wood

Specifically?

I think maybe "unscientific" in this context is wrong usage. In any case, mathematics is not an experimental science. — tim wood

Well, that is not how the game is played now, but that doesn't mean the game is played rationally, does it? The value of Popper's falsifiability criterion is that it restricts hypotheses to ones we can gain intellectual traction on. If you allow hypotheses that cannot be tested, then any guess, no matter how irrational, can be posited.

Of course, the test might not be experimental. While Godel's work means that we cannot prove the consistency of a set of axioms, it doesn't prevent us from proving their inconsistency. So, we could have mathematicians deducing hypothetical consequences in the hope of finding an inconsistency. If they do, they'd have proven an axiom was false. But, if they don't, they won't know that any of their hard-won conclusions are true.

As you note, we can't prove that the axiom of choice is true or false in the context of ZF. That leaves it unfalsifiable.

As for "games," what would you call playing by rules that are either false, or ungrounded in reality?

the pursuit of the consequences of which, which may be "no different in principle," are in practice and in fact altogether and entirely different from a mere game. — tim wood

What would the difference be? It's a game many mathematicians enjoy?

Being is some way(s) rather than other ways, no? — Terrapin Station

I'm not sure what you mean. Surely anything that can act in any way must be, and nothing can be unless it can act in some way -- for it it could not, it would be indistinguishable from no-thing.