I think this is a very interesting subject. I had a similar discussion not too long ago.

When you say 'exist in a platonic sense', what exactly do you mean?

I am inclined to argue that maths do not 'exist' in any objective sense.

Math is a product of the human mind, and a very useful for modeling reality for human purposes. It's a way of describing ratios and relations between things. The actual objective nature of such relations seems inaccessible to humans though.

When you say 'exist in a platonic sense', what exactly do you mean? — Tzeentch

Platonism in the Philosophy of Mathematics

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented.

Why do you think realism is the prevailing view in Phil of math? Why is it found to be a valuable perspective in spite of its drawbacks?

Why do you think realism is the prevailing view in Phil of math? Why is it found to be a valuable perspective in spite of its drawbacks? — frank

I don't know, I'm not a psychologist.

As far as my understanding of platonism goes, it argues that ultimate reality can be accessed (with great difficulty) via mystical experiences which go beyond the intellect, and are thus unintelligible?

So platonic mathematics implies someone had a mystical experience and discovered math still exists 'beyond the veil'?

I don't know, I'm not a psychologist. — Michael

Dude. Really?

As far as my understanding of platonism goes, it argues that ultimate reality can be accessed (with great difficulty) via mystical experiences which go beyond the intellect, and are thus unintelligible?

So any platonic mathematics implies someone had a mystical experience and discovered math still exists 'beyond the veil'? — Tzeentch

It's certainly unclear, and is precisely what gives rise to the epistemological argument against platonism:

1. Human beings exist entirely within spacetime.
2. If there exist any abstract mathematical objects, then they do not exist in spacetime. Therefore, it seems very plausible that:
3. If there exist any abstract mathematical objects, then human beings could not attain knowledge of them. Therefore,
4. If mathematical platonism is correct, then human beings could not attain mathematical knowledge.
5. Human beings have mathematical knowledge. Therefore,
6. Mathematical platonism is not correct.

Isn't it easier then to accept that mathematics does not exist objectively, and is simply a very useful tool conceived by the human mind?

Tying it back to the OP, who cares if infinitesimals exist objectively, as long as they are useful in creating more accurate models of reality?

Isn't it easier then to accept that mathematics does not exist objectively, and is simply a very useful tool? — Tzeentch

I certainly believe so. Given my thoughts in the OP and Occam's razor, I think that mathematical platonism ought be rejected.

:up:

Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented.

I know this isn't your definition, but I would suggest a modification to just:

"Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is not dependent on us and our language, thought, and practices."

"Independent" might suggest that the two don't interact, but it seems obvious that they must for platonism to be an interesting thesis. The whole second part is problematic in that it seems to assume that "statements" are also independent of us (and true or false independent of us), and I am not sure if all mathematical platonists would like to be committed to those implied premises. It seems to require being a platonist about "statements" in order to be a platonist about any mathematical objects. But, at least for me, "threeness exists without humans around" seems a lot more plausible than "sentences exist without humans around."

I am inclined to argue that maths do not 'exist' in any objective sense.

Math is a product of the human mind, and a very useful for modeling reality for human purposes. It's a way of describing ratios and relations between things. The actual objective nature of such relations seems inaccessible to humans though.

Well, my turn to ask for a definition: what does "objective" mean here? I've noticed it tends to get used in extremely diverse ways. I assume this is not "objective" in the same sense that news is said to be "more or less objective?"

As a follow-up, I would tend to think that the game of chess does not exist independently from the human mind. Chess depends on us; we created it. However, are the rules of chess thus not objective? Are there no objective facts about what constitutes a valid move in chess?

I suppose this gets at the need for a definition.

Isn't it easier then to accept that mathematics does not exist objectively, and is simply a very useful tool conceived by the human mind?

But isn't the follow up question: "why is it useful?" Not all of our inventions end up being useful. In virtue of what is mathematics so useful? Depending on our answer, the platonist might be able to appeal to Occam's razor too. A (relatively) straight-forward explanation for "why is math useful?" is "because mathematical objects are real and instantiated in the world."

This also helps to explain mathematics from a naturalist perspective vis-a-vis its causes. What caused us the create math? Being surrounded by mathematical objects. Why do we have the cognitive skills required to do math? Because math is all around the organism, making the ability to do mathematics adaptive.

1. Human beings exist entirely within spacetime.
2. If there exist any abstract mathematical objects, then they do not exist in spacetime. Therefore, it seems very plausible that:
3. If there exist any abstract mathematical objects, then human beings could not attain knowledge of them. Therefore,
4. If mathematical platonism is correct, then human beings could not attain mathematical knowledge.
5. Human beings have mathematical knowledge. Therefore,
6. Mathematical platonism is not correct.

I think the platonist response would be that premise 2 is false. Mathematical objects exist in spacetime. There is twoness everywhere there are two of something (e.g. in binary solar systems). Premise two seems to imply that any transcendent, Platonic form is absent from what it transcends. Yet this is not how Plato saw things. The Good, for instance, is involved in everything that ever even appears to be good. Plus, my understanding is that many mathematical platonists (lower case p) are immanent realists, along the lines of Aristotle. So, numbers exist precisely where they are instantiated (in space-time). A Hegelian theory would similarly still allow that numbers exist "in history."

I think the platonist response would be that premise 2 is false. — Count Timothy von Icarus

At least according to the SEP article here, (2) is platonism:

Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental.

Mathematical objects exist in spacetime. There is twoness everywhere there are two of something (e.g. in binary solar systems). — Count Timothy von Icarus

This is moderate/immanent realism:

Immanent Realism: Advocates of this view agree with platonists that there do exist such things as mathematical objects — or universals, or whatever category of alleged abstract objects we're talking about — and that these things are independent of us and our thinking; but immanent realists differ from platonists in holding that these objects exist in the physical world.

Well, my turn to ask for a definition: what does "objective" mean here? — Count Timothy von Icarus

Objective in the platonic sense refers to the reality that underlies our 'reality' of sense experience.

We infer its existence, because we are able to consistently predict outcomes accurately enough for human endeavors. Mathematics and science help us do so.

As a follow-up, I would tend to think that the game of chess does not exist independently from the human mind. Chess depends on us; we created it. However, are the rules of chess thus not objective? Are there no objective facts about what constitutes a valid move in chess? — Count Timothy von Icarus

Hmm.. I'm inclined to say that there are indeed no objective facts related to chess. Chess tells us nothing about this underlying reality.

But isn't the follow up question: "why is it useful?" Not all of our inventions end up being useful. In virtue of what is mathematics so useful? Depending on our answer, the platonist might be able to appeal to Occam's razor too. A (relatively) straight-forward explanation for "why is math useful?" is "because mathematical objects are real and instantiated in the world." — Count Timothy von Icarus

Math is a very useful way of describing relations and ratios between things.

Claiming things are real runs into all sorts of prickly problems, though. Have you peeked beyond the veil and seen it was so?

Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental. — Michael

I'm actually kind of curious what passages of Plato this refers to.

Although I'm not a mathematical platonist, I do have some questions about the epistemological argument, specifically with premise (3):

3. If there exist any abstract mathematical objects, then human beings could not attain knowledge of them.

If knowledge is justified true belief then this can be rephrased as:

3. If there exist any abstract mathematical objects, then human beings could not attain justified true beliefs of them.

Is this saying that if mathematical objects are abstract then we cannot conceive of an equation, that we cannot believe that the equation is true, or that we cannot be justified in believing that the equation is true?

Or is there some other sense of knowledge, distinct from justified true belief, at play here?

You would not know whether equations describe true things. Maybe the universe does not work according to such rules, but we can make equations accurate enough to 'do the job' for our human purposes?

I think knowledge here refers to absolute certainty, or objective knowledge, and the platonists were highly skeptical of that.

Do infinitesimals exist (in the platonic sense)?

1. If they don't exist then any number system that includes them is "wrong" — Michael

An infinitesimal is not a real number, so it doesn't exist in the set of real numbers, but that's in the sense of existential quantification. I don't see what the purpose of platonic existence is. 3 and 5 seem to add up to 8 whether or not 3 and 5 exist in the platonic sense. Lack of that does not prevent the usage of the number system. You seem to say something along these lines in the OP.

2. If they do exist then any number system that excludes them is "incomplete" (not to be confused with incompleteness in the sense of Gödel).

There is an 'extended real numbers' that includes infinity. I'm sure we can name a set that includes infinitesimals as well. Still not complete since I think octonians is necessary for that, extended octonians at that.

I did not follow the bit about fictional numbers

The definition you gave in 2nd post about numbers essentially being real, well, the definition seems to identify them being objective: Independently gleaned by isolated groups. This is a good argument against any specific god since isolated groups might all claim divine communication, but none of them come up with the same story. With mathematics, this is not the case.

Still, the sum of 3 and 5 being 8, is that a property of this universe, or does it work anywhere? Is it truly an objective fact? I don't equate objectivity with being real, but the definition you gave seems to equate the two.

I am inclined to argue that maths do not 'exist' in any objective sense. — Tzeentch

Cool. An opposing viewpoint. What's the alternaitve?

This distinction seems more Kantian than Platonic to me. I think "noumenal" might be a better tern here, i.e. "a thing that exists independently of human senses." At least, Plato himself would reject such a cleavage in reality, as well as existence without any edios (quiddity, intelligibility, form).

Claiming things are real runs into all sorts of prickly problems, though. Have you peeked beyond the veil and seen it was so?

Have you looked on both sides to see if the veil itself is real? I am not sure if you can have a "reality versus appearances" dichotomy if there is only appearances. If there are just appearances, then appearances are reality. But then how do we justify the claim that there is a reality that is completely isolated from appearances?

On the other hand, if we can "infer" the "'reality' behind the veil," then why can't we likewise infer that this reality includes numbers?

This is, BTW, Hegel's critique of Kant. Kant himself is dogmatic. He doesn't justify the assumption that perceptions are of something, that they are in some sense "caused" by noumena (although of course, "cause" itself is phenomenological and so suspect). He just presupposes it and goes from there (and look, he just happens to deduce Aristotle's categories, convenient!). The Logics are pretty much Hegel's attempt to start over without this assumption.

Math is a very useful way of describing relations and ratios between things.

But then wouldn't these objective/noumenal things need to be the sort of things that have ratios? If they don't have ratios, why is it useful to describe them so? If they do, then numbers (multitude and magnitude) seem to apply to the noumena.

Hmm.. I'm inclined to say that there are indeed no objective facts related to chess. Chess tells us nothing about this underlying reality.

But presumably it tells us something about the reality of chess. This is why I don't know about making "objective" and "noumenal" synonyms. For one, it seems likely to me that many people will find a use for the former while rejecting the assumptions that make the latter meaningful. Second, we wouldn't want to have to be committed to the idea that facts about chess, or the game itself, are illusory.

Sort of besides the point though.

I'm actually kind of curious what passages of Plato this refers to.

Platonism in many areas is lower case "p" platonism, which tends to be only ancillary related to Platonism. For instance, :

At least according to the SEP article here, (2) is platonism:

Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental.

This could apply to Plato in some sense, but you'd really need a lot of caveats. Plato's metaphysics works on the idea of "vertical" levels to reality. Forms are "more real" they aren't located in some sort of space out of spacetime. But at the same time, even for Plato and the ancient and medieval Platonists, the Forms aren't absent from the realm of appearances. The reason medieval talk of God can be so sensuous without giving offense is because they thought all good, even the good of what merely appears to be good, is still a participation in/possession of the Good.

So, the world of the senses and spacetime would be deeply related to the Forms, not isolated from them. However, I am not super familiar with platonism in contemporary philosophy of mathematics.

But isn't the follow up question: "why is it useful?" Not all of our inventions end up being useful. In virtue of what is mathematics so useful? Depending on our answer, the platonist might be able to appeal to Occam's razor too. A (relatively) straight-forward explanation for "why is math useful?" is "because mathematical objects are real and instantiated in the world."

This also helps to explain mathematics from a naturalist perspective vis-a-vis its causes. What caused us the create math? Being surrounded by mathematical objects. Why do we have the cognitive skills required to do math? Because math is all around the organism, making the ability to do mathematics adaptive. — Count Timothy von Icarus

One can always answer the question of why something is useful by attaching it to a sovereign ground. This works equally well for the true , the good and the numerical. But it may be more illuminating to ask the question of how something is useful, that is, what are the consequences of resorting to a sovereign ground rather than a pragmatic explanation based on subjectively and intersubjectively constructed norms. For instance, the consequence of asserting that mathematical objects are real things in the world is the risk of skepticism and arbitrariness. Why should there be unanimous agreement about the meaning of enumeration when there is disagreement out every other fact of nature?

By contrast, if we were to argue that the concept of number is a conceptual abstraction derived from the selective noticing of individual elements of a collective multiplicity, wherein one deliberately abstracts away everything about those individual elements other than the idea of ‘same thing different time’. ‘Same thing different time’ is not found anywhere in the world, it is an invention which, when applied to real objects, flattens differences in kind in order to accomplish certain useful goals. One consequence of understanding the usefulness of number as a pragmatic tool rather than as a sovereign fact of nature is to bypass the risk of skepticism and arbitrariness. Numeration arises out of the ground of practical human need for relating and keeping track of disparate objects. Its meaning is universal precisely because it is a pure, because empty, idealization and therefore not subject to the intersubjective tribunal that objective facts of nature must undergo.

Interesting discussion of this topic was published in The Smithsonian Institute magazine, from which:

Some scholars feel very strongly that mathematical truths are “out there,” waiting to be discovered—a position known as Platonism. It takes its name from the ancient Greek thinker Plato, who imagined that mathematical truths inhabit a world of their own—not a physical world, but rather a non-physical realm of unchanging perfection; a realm that exists outside of space and time. Roger Penrose, the renowned British mathematical physicist, is a staunch Platonist. In The Emperor’s New Mind, he wrote that there appears “to be some profound reality about these mathematical concepts, going quite beyond the mental deliberations of any particular mathematician. It is as though human thought is, instead, being guided towards some external truth—a truth which has a reality of its own...”

Many mathematicians seem to support this view. The things they’ve discovered over the centuries—that there is no highest prime number; that the square root of two is an irrational number; that the number pi, when expressed as a decimal, goes on forever—seem to be eternal truths, independent of the minds that found them. If we were to one day encounter intelligent aliens from another galaxy, they would not share our language or culture, but, the Platonist would argue, they might very well have made these same mathematical discoveries.

“I believe that the only way to make sense of mathematics is to believe that there are objective mathematical facts, and that they are discovered by mathematicians,” says James Robert Brown, a philosopher of science recently retired from the University of Toronto. “Working mathematicians overwhelmingly are Platonists. They don't always call themselves Platonists, but if you ask them relevant questions, it’s always the Platonistic answer that they give you.”

Other scholars—especially those working in other branches of science—view Platonism with skepticism. Scientists tend to be empiricists; they imagine the universe to be made up of things we can touch and taste and so on; things we can learn about through observation and experiment. The idea of something existing “outside of space and time” makes empiricists nervous: It sounds embarrassingly like the way religious believers talk about God, and God was banished from respectable scientific discourse a long time ago.

Platonism, as mathematician Brian Davies has put it, “has more in common with mystical religions than it does with modern science.” The fear is that if mathematicians give Plato an inch, he’ll take a mile. If the truth of mathematical statements can be confirmed just by thinking about them, then why not ethical problems, or even religious questions? Why bother with empiricism at all?

Massimo Pigliucci, a philosopher at the City University of New York, was initially attracted to Platonism—but has since come to see it as problematic. If something doesn’t have a physical existence, he asks, then what kind of existence could it possibly have? “If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.” (If the proof of the Pythagorean theorem exists outside of space and time, why not the “golden rule,” or even the divinity of Jesus Christ?) — What is Math?

Why not, indeed? But I think that extended passage brings out the underlying animus against mathematical Platonism, which is mainly that it undermines empiricism. And empiricism is deeply entrenched in our worldview.

Mathematical platonism has considerable philosophical significance. If the view is true, it will put great pressure on the physicalist idea that reality is exhausted by the physical. For platonism entails that reality extends far beyond the physical world and includes objects that aren’t part of the causal and spatiotemporal order studied by the physical sciences.[1] Mathematical platonism, if true, will also put great pressure on many naturalistic theories of knowledge. For there is little doubt that we possess mathematical knowledge. The truth of mathematical platonism would therefore establish that we have knowledge of abstract (and thus causally inefficacious) objects. This would be an important discovery, which many naturalistic theories of knowledge would struggle to accommodate. — SEP, Platonism in the Philosophy of Mathematics

I resolve the conundrum by saying that numbers (etc) are real but not existent in a phenomenal sense. They are intelligible or noumenal objects (in a Platonic rather than a Kantian sense) and as such are indispensable elements of rational judgement.

Why not, indeed? But I think that extended passage brings out the underlying animus against mathematical Platonism, which is mainly that it undermines empiricism. And empiricism is deeply entrenched in our worldview.

Mathematical platonism has considerable philosophical significance. If the view is true, it will put great pressure on the physicalist idea that reality is exhausted by the physical. For platonism entails that reality extends far beyond the physical world and includes objects that aren’t part of the causal and spatiotemporal order studied by the physical sciences.[1] Mathematical platonism, if true, will also put great pressure on many naturalistic theories of knowledge. For there is little doubt that we possess mathematical knowledge. The truth of mathematical platonism would therefore establish that we have knowledge of abstract (and thus causally inefficacious) objects. This would be an important discovery, which many naturalistic theories of knowledge would struggle to accommodate.
— SEP, Platonism in the Philosophy of Mathematics — Wayfarer

If Platonism seems to ‘undercut’ empiricism, it does so only by occupying the opposing pole of the binary implicating both physicalism and platonism within the same tired dualistic subject-object metaphysics. Why not undercut both empiricism and platonism in one fell swoop, and see both numbers and physical things as pragmatic constructions, neither strictly ideal nor empirical, subjective nor objective, inner nor outer, but real nonetheless?

If Platonism seems to ‘undercut’ empiricism, it does so only by occupying the opposing pole of the binary implicating both physicalism and platonism within the same tired dualistic subject-object metaphysics. — Joshs

I don't know if I agree with your diagnosis that the opposition to Platonism arises from 'subject-object metaphysics'. I think it goes back to the decline of Aristotelian realism and the ascendancy of nominalism in late medieval Europe. From which comes the oxymoronic notion of mind-independence of the empirical domain, when whatever we know of the empirical domain is dependent on sensory perception and judgement (per Kant). Hence those objections in that passage I quoted, 'The idea of something existing “outside of space and time” makes empiricists nervous'. Anything real has to be 'out there somewhere' - otherwise it's 'in the mind'. That is the origin of subject-object metaphysics.

see both numbers and physical things as pragmatic constructions, neither strictly ideal nor empirical, subjective nor objective, inner nor outer, but real nonetheless? — Joshs

But there are imaginary numbers, and also imaginary objects, even imaginary worlds. There are degrees of reality, and there is a such a thing as delusion, and delusions can be very deep indeed, in today's panoptical culture. Agree with the constructivist attitude overall, but still want to honour the epistemology of the Divided Line.

Yeah, why answer a difficult question when we can just engage in question begging? And if everyone just assumes the asserted conclusion is right, this will prevent any skepticism or charges arbitrariness!.

Have you looked on both sides to see if the veil itself is real? — Count Timothy von Icarus

It is inferred that there exists our world of sense experience, and a reality underlies it. Science has gone a long way in confirming this, showing how our senses mislead us, and only show us the tip of the iceberg.

At least, Plato himself would reject such a cleavage in reality, — Count Timothy von Icarus

It is pretty much the central theme of Plato. It's not that reality is cleaved, but that we do not experience reality - only a reflection of it. That's the cave.

But presumably it tells us something about the reality of chess. — Count Timothy von Icarus

I think the word 'reality' is a misnomer here. Chess is something we made up. Would you accept it if people were arguing for the reality of the flying spaghetti monster?

Abstract mathematical objects such as numbers, functions, operational symbols are descriptive language for the existence in the real world. They are not the existence themselves. In that regard, I agree that Platonic math objects don't exist.

Do infinitesimals exist (in the platonic sense)? — Michael

They are convenient and useful descriptive tools to denote and express the small objects and motions in the real world such as the information or movements of particles and atoms.

It is inferred that there exists our world of sense experience, and a reality underlies it. Science has gone a long way in confirming this, showing how our senses mislead us, and only show us the tip of the iceberg.

Ok, but you didn't answer how this "reality" can be inferred "by science," but numbers absolutely cannot be. It seems to me that the empirical sciences only ever deal with phenomena.

Further, can we do physics, cognitive science, or biology without mathematics? More importantly, you haven't given any answer for why or how ratios are useful if they don't *really* apply to or exist in your noumenon. If the noumenal isn't the sort of thing that can be accurately described by number and ratio (we have many things like this in the world of phenomena) then why is it "useful" to describe them that way anyhow? Shouldn't the usefulness of mathematics in science lead us to "infer" that it says something about reality?

It just seems strange to me to appeal to all the ways in which science shows our senses can be misled, when those same sciences often rely on mathematics to point out these illusions, but then to turn around and say that the math you used to discover the illusions (and so to infer their "reality") is itself illusory. And of course any corrections to perception made by "by empirical science" are also discovered through the senses, so if the senses and intellect "mislead us," they're also responsible for correcting this.

It seems that at best you're arguing for nescience: "we can never know if numbers are *really* in the noumena with total certainty." But your positions seems to require actually demonstrating that there is no good reason to infer that ratios/mathematics apply to "things-in-themselves." Having some avenue for skepticism is not enough, people can also doubt that there is any reality that is distinct from appearances (e.g. solipsism, subjective idealism, etc.), but clearly you don't think this is good grounds for accepting that reality is just appearances yourself.

(Note: both noumena [plural] and things-in-themselves imply plurality, number—this is why people who want to go along with Kant's distinction normally speak of simply "a noumenon.")

It is pretty much the central theme of Plato. It's not that reality is cleaved, but that we do not experience reality - only a reflection of it. That's the cave.

Plato makes a distinction between reality and appearances. He does not make a distinction between appearances as "subjectivity," and reality as the "objective/noumenal"—i.e., reality as "things-in-themselves" as set over and against appearances. This Kantian division makes no sense given Plato's philosophy of appearances and images as participation. Kant's view requires the presuppositions of modern representationalism, i.e., that "what we experience" are our own "mental representations of ideas" and that such representations are "what we know" instead of "how we know." The later Platonists allowed that "everything is received in the manner of the receiver," but not that things' appearances are disconnected from what they are (i.e "act follows on being" and "appearing" is an act of the subject of predication).

For instance, Plato's Good is absolute. The absolute is not reality as separated off from appearances. It must encompass all of reality and appearances to be truly absolutely. Thing's appearances are really how they appear. Likewise, the transcendent Good isn't absent from the very finitude it is supposed to transcend. This would make it less than truly transcedent.

It seems like a lot of people, when it comes to philosophy, think "objective" is a synonym for "noumenal." But this is certainly not how the term is employed by many philosophers, and this leads to all sorts of confusions, like the idea that an "objective" goodness or beauty is somehow one that is wholly absent and disconnected from experiences (Sam Harris has this misreading of the Platonic Good in The Moral Landscape for instance). In which case, no wonder such ideas seen farcical. On this misunderstanding they are incoherent, the objective Good must be, by definition, "good for precisely no one." But Platonic eidos (forms), as the term's usual connotations in ancient Greek suggest ("shape," "something seen") are not unrelated to, or absent from, appearances—a reality as set apart from appearance.

I think the word 'reality' is a misnomer here. Chess is something we made up. Would you accept it if people were arguing for the reality of the flying spaghetti monster?

Presumably, the latter is an intentional fiction created to critique religion. It is one thing to claim that Homer's Achilles is a "fictional character." It is another to claim that the Iliad doesn't "really exist" because Homer wrote it. Do airplanes also not exist because they are the invention of man? States? World history? Chess?

I think a view that commits us to claims like: "there are no objective facts about what constitutes a valid move in chess," or "the proposition 'Kasparov is a better chess player than the average preschooler' is one with no truth value because it refers to the "subjective" game of chess," has serious deficits. Does "the Declaration of Independence was signed in 1776" also become subjective because our calendar system is the creation of man? But then temperature would also have to be subjective because it involves both man-made scales and measurement from particular perspectives.

Shouldn't the usefulness of mathematics in science lead us to "infer" that it says something about reality? — Count Timothy von Icarus

They're both tools for modeling an inferred underlying reality. But they themselves are human creations, accurate enough for our human purposes.

They're useful because they're accurate enough. But it would be a mistake to believe they convey the objective nature of reality.

He does not make a distinction between appearances as "subjectivity," and reality as the "objective/noumenal" — Count Timothy von Icarus

Neither am I, as far as I am aware.

Presumably, the latter is an intentional fiction created to critique religion. It is one thing to claim that Homer's Achilles is a "fictional character." It is another to claim that the Iliad doesn't "really exist" because Homer wrote it. Do airplanes also not exist because they are the invention of man? States? World history? Chess? — Count Timothy von Icarus

If someone were to create a gigantic effigy of a flying spaghetti monster, would that suddenly make the flying spaghetti monster real?

I'd argue all of those things you named are human creations, and therefore not 'real' in the sense that we are talking about right now.

Obviously, we can make all sorts of practical concessions in what we colloquially refer to as 'real'.

I don't know if I agree with your diagnosis that the opposition to Platonism arises from 'subject-object metaphysics'. I think it goes back to the decline of Aristotelian realism and the ascendancy of nominalism in late medieval Europe. From which comes the oxymoronic notion of mind-independence of the empirical domain, when whatever we know of the empirical domain is dependent on sensory perception and judgement (per Kant). Hence those objections in that passage I quoted, 'The idea of something existing “outside of space and time” makes empiricists nervous'. Anything real has to be 'out there somewhere' - otherwise it's 'in the mind'. That is the origin of subject-object metaphysics.

:up: :100:

Yes, the philosophy of Plato does not seem to be commensurate with modern subject-object dualism. It seems even less applicable to later Platonists, such as Plotinus, St. Augustine, or St. Bonaventure.

Nominalism seems to me to be the larger issue and I think it has generally been nominalism that has motivated to errection of subject-object dualism, rather than the other way around (although obviously the influence is bi-directional).

They're both tools for modeling an inferred underlying reality. But they themselves are human creations, accurate enough for our human purposes.

Yes, you seem to be asserting this as a premise and then arguing from there. But this is to assume as true the very thing you're setting out to prove, that platonism is false.

What's the argument for mathematics being a sui generis human creation unaffected by the reality of multitude or magnitude? What caused us to create it? If it's useful, why?



To say that these questions are unanswerable suggests nescience, not one answer re platonism being supported over the other.

Neither am I, as far as I am aware

You certainly seem to be. Your claim is that, for something to be properly "real" it must exist wholly outside appearances. How is this not defining reality in terms of the noumenal? For all those following Parmenides, Plato included, there is no reality as totally divorced from appearances and intelligibility. Thought and being are two sides of the same non-composite whole.

If someone were to create a gigantic effigy of a flying spaghetti monster, would that suddenly make the flying spaghetti monster real?

Do you think making a statue of a fictional character makes them real? I don't. Yet is chess fictional? Is world history fiction? Temperature? Dates?

Scientific theories and paradigms are human creations. Yet if these are thereby fictions, then your appeal to "inferring reality from science" would amount to "inferring what is real from fiction."

You certainly seem to be. Your claim is that, for something to be properly "real" it must exist wholly outside appearances. — Count Timothy von Icarus

That's not what I'm asserting, because how would I know?

The core of what I'm saying is that, as Plato argued, it is very difficult to even access the reality that underlies our world of sense experience, let alone make statements about this reality.

So rather I am expressing skepticism towards those who would claim mathematics is 'objectively real', and also pointing out the contradiction in the term 'mathematical platonism'.

Does that make sense?

Do you think making a statue of a fictional character makes them real? I don't. Yet is chess fictional? Is world history fiction? Temperature? Dates?

Scientific theories and paradigms are human creations. Yet if these are thereby fictions, then your appeal to "inferring reality from science" would amount to "inferring what is real from fiction." — Count Timothy von Icarus

In the context of a philosophical debate, I would argue all of those things are indeed human 'fictions', that serve a purpose for our human needs.

Note that I am not saying that science shows us what is real, rather it seems to heavily suggest the existence of an underlying reality because it is able to make models of how that reality works to a degree that is at least accurate enough for our human endeavors.