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. — Dfpolis

I do not wish to defend mathematical Platonism, but I think you misrepresent the position. The problem stems, at least in part, from jumping from Aristotle's criticism of Plato's Forms to mathematical platonism.

From the IEP article on Mathematical Platonism:

Formulated succinctly, Frege’s argument for arithmetic-object platonism proceeds as follows:

i. Singular terms referring to natural numbers appear in true simple statements.

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.

Therefore,

iii. the natural numbers exist.

iv. If the natural numbers exist, they are abstract objects that are independent of all rational activities.

Therefore,

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.

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.

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

Do you mean different concepts that were in prior use? In the briefest terms, the arithmos is always a definite number of definite things,a collection of countable units, whereas in modern math a number, '4' for example, is itself an object. With the move to symbols, 'x' does not signify anything but itself.

This is a wide-ranging topic that goes far beyond the concept of number. The second part of this book review that addresses Klein will give a better sense of what is at issue as it relates to modern philosophy and science: https://ndpr.nd.edu/news/the-origin-of-the-logic-of-symbolic-mathematics-edmund-husserl-and-jacob-klein/

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. — Dfpolis

No, I am speaking here specifically about the concept of number, that is, what a number is.

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

What you said was:

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.

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. — Dfpolis

I agree, yet when modeling reality, it's apparent that there are approximations and generalizations etc. that simply don't make sciences as rigorously logical as mathematics. For starters, every measurement is an approximation. Logic is of course necessary. I studied myself economics and economic history and noticed that a lot of variables are rudimentary models of very complex phenomena, like 'inflation', 'GDP' or 'aggregate demand', and that one shouldn't forget it when calculating math formulas with them.

That is precisely the notion I reject. — Dfpolis

Ok, then I think I've misunderstood your point.

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. — Dfpolis

Perhaps now I understand your point. (I'm btw happy with pragmatism: usefulness is far more important than we typically think.) So if I understood you correctly, 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. Like astrophysics using a helical model of the universe simply might not be useful...especially if the universe simply isn't optimally modeled using a helix.

The standard mathematicians answer would be "Well, it could be useful someday". Modelling the universe using a helix might have those not yet known nice 'mathematical properties' that future physicists make better models and can avoid today's problems. And some mathematicians are totally happy with the "math-is-just-various-kinds-of-rules" approach and declare every kind of math as worthy as long as it's logically correct.

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. — Dfpolis

Logical (not local). No, I'm not saying that. What I'm saying that a field that has developed from the need to count and calculate to solve real world problems doesn't have it's axiomatic foundations solely on arithmetic as it has also incommensurability and uncomputability. So the foundations aren't so narrow that everything starts from simple arithmetic.

Some comments about your classification. You define in the first class to be math that has axioms rooted in our experience and reality.

Most axioms are abstracted from our experience of nature as countable and measurable. — Dfpolis

Yet we can have logical problems with those too: Zeno's paradoxes and the huge debate over infinitesimals have shown that we stumble to the problems of infinity from quite normal experiences. (And those who think limits have solved all the questions, well, how about the Continuum Hypothesis then?)

There's still a lot that we don't know.

I'm btw happy with pragmatism: usefulness is far more important than we typically think. — ssu

The term "usefulness" is quite controversial in mathematics. I tend to agree with Hardy on the matter:

I have never done anything "useful". No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.

We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not.

As I see it, and in line with what Hardy said, while the low-hanging fruit is almost immediately useful, but only moderately so, the real game changers may take even centuries to find their way into applications. That is why it is necessary to abstract away "usefulness" when exploring the abstract, Platonic world of mathematics for new discoveries.

I have never done anything "useful". No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. (Hardy) — alcontali

Hardy was a number theorist. At the time he wrote those words, number theory was regarded as beautiful but useless. Today it's the mathematical foundation of public key cryptography, underlying all Internet security and cryptocurrencies. I wonder what he would say if he came back and discovered that his belovedly useless number theory was intensely studied by the spies at the NSA.

For a discussion of the pragmatics of the axioms of set theory, see Penelope Maddy's Believing the Axioms parts I and II.

https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf

https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms2.pdf

All these bloody pure mathematicians trying to stop us from occupying their lawn. They forget the rest of us squatters were here first.

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.

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.

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.

My comment is directly on point, and does not attack a straw man, but premise ii. — Dfpolis

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.

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. — Dfpolis

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.

I'm saying that every note of intelligibility is an aspect of the object known. — Dfpolis

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.

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

No, I mean that concepts don't change. — Dfpolis

The question was about your wording. Whether the 'not' in "not in prior use" was a typo.

This is an interpretive, not a mathematical, claim. — Dfpolis

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

No, "x" does not mean the letter "x." It has reference beyond itself. — Dfpolis

It does not have any reference until it is assigned one. That is the point. It is a variable that can stand for any unknown. In this sense it is different from both "4" as how many or "4" as an object.

[Added trivia note: I read somewhere that Descartes' publisher used x because he was low on letters an x was not frequently used in French. Whether that is true or not I did not verify.]

It may mean an unknown we seek to determine, a variable we can instantiate as we will, or possibly other things ... — Dfpolis

Right.

but it never signifies itself — Dfpolis

It is because it is indeterminate that it does not signify something other than itself, which is to say, unlike a number it has no signification until or unless assigned one. It could stand for any number or no number at all.

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. — Dfpolis

This unnamed authority was David Hilbert:

It has been claimed that formalists, such as David Hilbert (1862–1943), hold that mathematics is only a language and a series of games. Indeed, he used the words "formula game" in his 1927 response to L. E. J. Brouwer's criticisms: "And to what extent has the formula game thus made possible been successful? This formula game enables us to express the entire thought-content of the science of mathematics in a uniform manner and develop it..."

This letter predates Karl Popper's "Science as Falsification" by almost half a century. The dust hadn't settled yet on the impossibility of verificationism. Certainly the Circle of Vienna still happily amalgamated mathematics and science.

The other objections to David Hilbert's view came from Hermann Weyl: What "truth" or objectivity can be ascribed to this theoretic construction of the world ...

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. — Dfpolis

Consistency is indeed assumed to be already instantiated in nature. The existence of consistency makes particular things impossible. These impossibilities give inescapable structure to nature. That is in my impression the core of the esoteric link between nature and mathematics. The structure visible in the Platonic world of math will therefore tend to be also visible in the real, physical world.

I personally refute neither Platonism nor formalism (Hilbert). They are a dual view on the abstract, Platonic objects versus the structures that constrain them in math.

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.

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.

Yes, the cardinality of the fruit in the bowl is seven whether we count or not. — Dfpolis

My issue is with what you call "potential numbers". The number of pieces of fruit in the bowl or the number of seeds in the pieces of fruit in the bowl in never a potential but an actual number. We may have the potential to determine that number but that does not make it a "potential number".

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. — Dfpolis

This is really convoluted and seems to be contradictory. 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. The way in which something is understood is not an aspect of the object but rather of our ability to see it or understand it in different ways. 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.

Exactly, and so one that requires justification. — Dfpolis

No inquiry is free of assumptions. The ontology of mathematical objects is an open question. It is not that different theories of mathematical objects are without justification it is that there is no universal agreement regarding their justification.

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 has a determine value.

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.

, Thomas Aquinas distinguishes three degrees of abstraction as fundamental to the difference between physical science, mathematics and metaphysics. — Dfpolis

By definition none the less.

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. — Dfpolis

Thinking of apples...

What are the requirements, the necessary pre-requisites, the sufficient pre-conditions...

What must also be the case in order for that to be?

What is the act of thinking existentially dependent upon?

That's a step in the direction of necessary elemental constituents.

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? — Dfpolis

We might say, for example, that the number of bacteria in a petri dish is potentially thousands or tens of thousands. 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.

What you said was:

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. — Dfpolis

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.

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. — Dfpolis

What is dependent on us is what we choose to count. How many there are of whatever it is we choose to count is independent of us. Here we are not talking about the concept of number but how many of something.

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? — Dfpolis

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

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). — Dfpolis

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

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.

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.

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

You were responding to the following:

In the briefest terms, the arithmos is always a definite number of definite things,a collection of countable units, whereas 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

The point was the one you now acknowledge. Klein's insight is into the radical shift in mathematics from numbers to symbols. Although we treat them as interchangeable when we assign value to the variable, numbers and symbols are not interchangeable. We do not assign values to numbers, we must assign value to variables. Numbers are determinate. Symbols are indeterminate. 3+2=5 is true. a+b=5 may be true or false. 3+2=5 is not dependent on us. a+b=5 is dependent on the values we assign to a and b.

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.

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>..

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. — Dfpolis

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.

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.

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. — Dfpolis

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

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"? — Dfpolis

You ignore a great number of questions. What does it mean to say the ball is known? When someone identifies an object as a ball is the ball known? 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? If they know it is a baseball is being a baseball an intelligible property of the object? If some other ball is used to play baseball is being a baseball an intelligible property of the object? 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? 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.

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

She might be a platonist and assume that <four> must still exist even when the oranges are eaten and the pennies spent. Or she might assume that <four> vanishes with the oranges and pennies. She might assume that there is only <four> when there are this many objects, even if they are not oranges and pennies. The "experience" of abstract arithmetic concepts may only come as the result of being taught to think of numbers in a certain way.

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.

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? — Dfpolis

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.

So, do you agree that items can be counted if and only if they are actual and distinct? — Dfpolis

As I said from the beginning, the count depends on the unit. If we cannot determine the unit we cannot determine the count. It the items to be counted are actual then their total number is also actual.

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. — Dfpolis

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

So it is with counting. The number is predetermined, but not actual until the count is complete. — Dfpolis

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.

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?

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

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. To be is not a property of what is. To be is a necessary condition for having properties. "This" is not a property of this something. To be intelligible a thing must be distinguishable as separate from other things but to be intelligible there must also be some subject to which it is intelligible. Without subjects there is no intelligibility.

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

What you said was:

intelligibility inheres in objects — Dfpolis

If someone from a tribe that knows nothing about baseball were to find a baseball what it it about it that would make it intelligible to the tribesman that it is a baseball? Its intelligibility as a baseball is not something that inhere in the ball. To be intelligible as a baseball one must know what baseball is.

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

The relevance can be seen in the what I just said. 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.

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. — Dfpolis

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

I am saying that whatever concepts we do have are abstracted from empirical experience. — Dfpolis

I would say that since none of us are without experience we cannot say what if any concepts we would form, but that concepts are not always abstractions, the can be something we add to rather than something we take away from experience.

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.

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. — Dfpolis

What you seemed to be claiming is that the number, whether it is fingers or fruit, is not actual but potential until it is counted. One iteration of what you said is:

The number is predetermined, but not actual until the count is complete. — Dfpolis

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.

As I said early on, I do not intend to defend platonic mathematics. For one, I am not well versed in the arguments. For another, I am agnostic on the matter.

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

Agreed. I have said this from the beginning in my discussion of Greek mathematics.

The count of your fingers was predetermined to be five before anyone counted them, but there was no actual count of five fingers. — Dfpolis

Here we go again. There is no actual count until they are counted, but there are actually five fingers, which is confirmed by the count.

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. — Dfpolis

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

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. — Dfpolis

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

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

The child has learned to count the objects. If she is not told, or as you would have it, learned what a number is, what she thinks a number is can vary. Is she is taught by a mathematical platonist what she learns the concept is is not what she learns if you tell her what you think it is.

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.

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.

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.

Our act of attending/awareness actualizes intelligiblity, converting it into concepts. — Dfpolis

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

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

The intelligibility of an object is knowledge of its essence, that is, what it is to be the thing that it is. What it is to be a baseball is not something that inheres in the object. It is to that extent not intelligible unless we know it as a baseball. To know it in its role in the game is not relational, it is essential to what it is.

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.

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.

I was explaining to you why the abstract five is not actual until abstracted. — Dfpolis

Your talk of potential and actual is misleading. 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.

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

I think they might argue that the fact that mathematical truths are not dependent on experience is all the experience they need. Consider, for example, non-Euclidean geometries. They are not abstracted from experience. They were initially seen an useless, mere curiosities. But with the discovery of the curvature of space, they found their application. They work. They are not merely formally or internally consistent, they tell us something about the world without being dependent on it.

If we merely constructed concepts, there would be no reason to think they apply to or are instantiated in, reality. — Dfpolis

First, see above regarding non-Euclidean geometries. Second, to some extent (Kant would say completely) experience is itself constructed. Third, concepts that are constructed are not all "merely" constructed, the construct may be based on experience but cannot be reduced to experience.

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. — Dfpolis

Okay. Let me rephrase it: The intelligibility of an object is the potential to know its essence. This changes nothing about what I said that follows from this. To use your favored language, knowledge of an object's essence is the actualization of its intelligibility.

Second, all human knowledge is partial, not exhaustive. We may, and usually do, know accidental traits rather than essences. — Dfpolis

The question is whether intelligibility inheres in the object. Whether or not our knowledge is partial is not at issue. The question is whether from the baseball alone it can be known that it is a baseball. An intelligence far greater than ours would not know this unless it also knows what the game is.

Third, there is nothing intrinsic to a baseball that relates it to any particular game. — Dfpolis

Of course there is! Being a baseball is not incidental to it being a baseball. It is constructed according to specific rules for a specific purpose. 'Baseball' is not simply a name attached to it. But if there is nothing intrinsic to a baseball that relates it to any particular game then your argument fails. We could not tell from it that it is a ball designed, manufactured, and used for one specific purpose.

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.

I was telling you why abstract numbers do not occur in nature, which is what we were discussing. — Dfpolis

Of course abstract numbers do not occur in nature, nothing abstracted from nature exists in nature.

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. — Dfpolis

At last? I have never said anything to the contrary. What was at issue was your denial five of something is actually rather than potentially five of something. You recently corrected yourself on that matter.

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? — Dfpolis

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. As another example consider infinitesimal calculus. There is no experience of infinitesimals. Not only are they not found in experience, they confound experience, as Zeno's paradoxes show. They are not abstracted from nature, they are theoretical constructs applied to it. In addition, the experience of motion or change does not yield the mathematics that adequately describes it.

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. — Dfpolis

Instantiation is not abstraction.

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. — Dfpolis

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

Kant had no sound reason to claim that. — Dfpolis

I won't bother getting into this. Do you imagine that neither Kant nor those who followed him were aware of this?

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

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.

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. — Dfpolis

No, my argument has nothing to do with the limits of human intelligence. It has to do with what is knowable from the object itself. Not knowable within the limits of human intelligence but from an intelligence without our limits.

Its purpose is in the minds of humans, not in the ball. — Dfpolis

That is right and that is why you cannot tell from the ball what its purpose is. To the extent the ball is intelligible its purpose is not part of that intelligibility. By your logic the intelligibility of a car does not include the potential to know that it is a means of transportation.

We can use the ball for other purposes, such as to be a display or even a paperweight. — Dfpolis

Yes, we have been through this already.