Philosophies of Mathematics

[skepticalbip]

The notion "if something exists, it exists somewhere" is faulty.

I had to think about this for a while because the notion as you stated it is a tautology (if something exists, it exists) so necessarily true.

Could it be that you meant "'if something can exist (meaning not prohibited by physical laws), it exist somewhere' is faulty"? This would seem to be arguable - what are the chances of something with an infinitely small chance in an infinite universe?

No, the way you worded it, it's a tautology.

I've left open room for something to exist without location.

??? Was this then an argument for or against the notion of a god or gods???

 

[fast]

No, the way you worded it, it's a tautology.

I've left open room for something to exist without location.

??? Was this then an argument for or against the notion of a god or gods???

Suppose there exists a planet with three moons tightly clustered together. On the opposite side of the planet, there are twice as many tightly clustered moons. That is a fact or state of worldly affairs that is independent of the human mind.
As to numbers, sure, we invented numerals, but they refer to numbers, like the invented word, "Sol" refers to something 'out there' that was discovered, not invented.

The form of the planets is of a physical form while the relationship which we say exists cannot itself be physically isolated--like, we can have a ball that is rolling or not rolling, but never can we have a rolling. It's important not to confuse objects with (oh say) movement. The objects are physical, but movement exists, but it itself is not physical--only the objects that are physical.

The relationship that exists between objects are just as real as the objects themselves, but while the relationship is a physical relationship, the relationship that exists does so independent of our awareness. If it didn't, we would never discover the relationships that exist in nature. I think it's important to compartmentalize the two.

 

[fast]

Left room for something to exist without location.

I like it. But surely we need to invoke the word "virtually" at this point. Or maybe "undetectable".

The brain is a physical thing, and as far as physical things go, they exist in some place. The brain, for instance, is located in the cranium.

The activity of the brain (the actual physical processes) or the physical objects in motion too are within the brain and thus in the cranium.

But, you cannot hand me a mind. You cannot put activity in your hand. You cannot hold up processes for anyone to see. We speak as if (as if, I say) the mind is in the brain, but never can a surgeon isolate it and put it in a jar. People are too quick to deny the immaterial and insist that if something exists that it therefore must exist SOMEWHERE. It's probably out of fear of being led down the road to accepting things as existing that don't.

Unicorns, if they did exist, would have a physical existence. We deny they exist (as we should) when we grasp that they have no physical existence.

Numbers have no physical existence, but that is not the form they are supposed to be, but they also exist independent of the mind, but if they are neither physical nor mental, people would want to deny they exist, but that is an error, as they exist just as they always have, not as a physical object and not as a mental abstraction. They are abstract (without location).

Also, I have a broader scope of "exist." Numbers have properties, and so, they exist.

 

[Lion IRC]

 

[skepticalbip]

No, the way you worded it, it's a tautology.

I've left open room for something to exist without location.

??? Was this then an argument for or against the notion of a god or gods???

Suppose there exists a planet with three moons tightly clustered together. On the opposite side of the planet, there are twice as many tightly clustered moons. That is a fact or state of worldly affairs that is independent of the human mind.
As to numbers, sure, we invented numerals, but they refer to numbers, like the invented word, "Sol" refers to something 'out there' that was discovered, not invented.

The form of the planets is of a physical form while the relationship which we say exists cannot itself be physically isolated--like, we can have a ball that is rolling or not rolling, but never can we have a rolling. It's important not to confuse objects with (oh say) movement. The objects are physical, but movement exists, but it itself is not physical--only the objects that are physical.

The relationship that exists between objects are just as real as the objects themselves, but while the relationship is a physical relationship, the relationship that exists does so independent of our awareness. If it didn't, we would never discover the relationships that exist in nature. I think it's important to compartmentalize the two.

Ah yes. So you are talking about some action or some process rather than some thing. Indeed verbs have different properties than nouns. If you had said in the original post that the notion that if some process exists, then it exists in some physical location is false, the post would have been clearer. But then does anyone ever have such a notion?

 

[steve_bank]

Left room for something to exist without location.

I like it. But surely we need to invoke the word "virtually" at this point. Or maybe "undetectable".

The brain is a physical thing, and as far as physical things go, they exist in some place. The brain, for instance, is located in the cranium.

The activity of the brain (the actual physical processes) or the physical objects in motion too are within the brain and thus in the cranium.

But, you cannot hand me a mind. You cannot put activity in your hand. You cannot hold up processes for anyone to see. We speak as if (as if, I say) the mind is in the brain, but never can a surgeon isolate it and put it in a jar. People are too quick to deny the immaterial and insist that if something exists that it therefore must exist SOMEWHERE. It's probably out of fear of being led down the road to accepting things as existing that don't.

Unicorns, if they did exist, would have a physical existence. We deny they exist (as we should) when we grasp that they have no physical existence.

Numbers have no physical existence, but that is not the form they are supposed to be, but they also exist independent of the mind, but if they are neither physical nor mental, people would want to deny they exist, but that is an error, as they exist just as they always have, not as a physical object and not as a mental abstraction. They are abstract (without location).

Also, I have a broader scope of "exist." Numbers have properties, and so, they exist.

There is a rolling ball, but not a rolling.

OK. Grammar school syntax diagrams. Adverbs and adjectives are not nouns, subjects, or objects. The point?

BTW 'a rolling' can be a noun in some areas of engineering. It can describe the result of a mechanical manufacturing process. The inescapable reality of syntax and context for words.

As to numbers, your are adding a mystical separate existence of human thought forms which are the result our physical brain.

 

[steve_bank]

Philosophy bakes no bread, math got us to the moon.

Is anyone suggesting one neds a pjilosohy to learn, apply, and research math?

If so them does a carpenter need a philosohy? The differnece between the skills is a matter of degree, both skills require knolwdge coupled with creativity.

In Calculus Integration By Parts is taught. It does not have a lot of utility. It is an apprtroch that has no specic rules and does not always work. One learns it by woring problems and trial and error. Is there a philosophy to that?

A great little book is How To Read And Do Proofs. Again, no specific rules that always ensures success. It is learned by stuying what has been done and through experience. The creative intutive part is part experince and part our brains. If you want insight even without a math bbackground I reccomend ithe book. It is short. Far more entertaining than video games.

Haven't played a video game since around 1983. I'd rather explore math problems.

Philosophy is simply ”use you rationality to be wise”. As soon as a subject gets so much worked on and gets a a crowd it is given a specific name, as maths, economics, etc. all these subjects are philosophy. Its just that what is called philosophy in US today is the remaining thought-stuff that hasnt yet formed into subjects of their own.

Philosophy bakes no bread .. QED

A story from one of my phil profs. Hegel was on his death bed. His closest student said 'Don't worry, I will makes sure your work is finished'. With great effort Hegel pulled himself up and said 'I only had one student who understood, and even he did not understand'.

As the prof put it in a university the top academics walked the halls with close students trailing behind and a canopy over the professor. People like Kant lived in an insulated ivory tower in an imaginary world of metaphysics.

Philosophy as it was and as a pursuit was filled with mysticism and power politics going back to the Greeks.

 

[fast]

Left room for something to exist without location.

I like it. But surely we need to invoke the word "virtually" at this point. Or maybe "undetectable".

The brain is a physical thing, and as far as physical things go, they exist in some place. The brain, for instance, is located in the cranium.

The activity of the brain (the actual physical processes) or the physical objects in motion too are within the brain and thus in the cranium.

But, you cannot hand me a mind. You cannot put activity in your hand. You cannot hold up processes for anyone to see. We speak as if (as if, I say) the mind is in the brain, but never can a surgeon isolate it and put it in a jar. People are too quick to deny the immaterial and insist that if something exists that it therefore must exist SOMEWHERE. It's probably out of fear of being led down the road to accepting things as existing that don't.

Unicorns, if they did exist, would have a physical existence. We deny they exist (as we should) when we grasp that they have no physical existence.

Numbers have no physical existence, but that is not the form they are supposed to be, but they also exist independent of the mind, but if they are neither physical nor mental, people would want to deny they exist, but that is an error, as they exist just as they always have, not as a physical object and not as a mental abstraction. They are abstract (without location).

Also, I have a broader scope of "exist." Numbers have properties, and so, they exist.

There is a rolling ball, but not a rolling.

OK. Grammar school syntax diagrams. Adverbs and adjectives are not nouns, subjects, or objects. The point?

BTW 'a rolling' can be a noun in some areas of engineering. It can describe the result of a mechanical manufacturing process. The inescapable reality of syntax and context for words.

As to numbers, your are adding a mystical separate existence of human thought forms which are the result our physical brain.

The class of groupings is not physical, but it's neither mystical nor mental.

A piece of paper that contains a written numeral 3 is physical.

The idea of the numeral 3 and the idea of the number 3 is mental. (Ideas are Abstract)

The material substances necessary for idea formation are physical. The ideas themselves are not physical.

The idea of a dog is not physical. A dog is physical.
The idea of the number 3 is not physical, but the idea of the number 3 is no more like the actual number 3 anymore than the idea of a dog is like an actual dog.

People always want to insist that if something exists yet not physical that it's mental--except those that want to insist that even mental objects are physical because their existence is dependent of physical substance. Either way, it's all wrong.

To say of something that it exists is to say of something that it has properties. The number three has properties (e.g. It's an odd number, the next whole number after two, divisible by other numbers, etc.). The number three exists.

The form, however, is neither physical nor mental. It's not concrete. It's an abstract object. That's not to be confused with a mental abstraction, like either the idea of, concept of, or thought of the number three.

The term, "abstract object" is inherently confusing, as it's a complex term. It's no more an actual kind of object than is an imaginary object. An imaginary object, for instance, is the denial that there's an object. It's not like anyone (sensible) is asserting that there is an object and the kind of object it is is imaginary. No, it's not an assertion but rather a denial. A similar confusion surrounds the notion of abstract objects--and explains historical notions of them being spooky--or mystical.

Grasping the relationship between objects (actual physical objects) is a human endeavor, but what we are grasping (regardless of the terms we provide explanations) are independent of us. The relationship between objects are mind-independent. The relationship between objects (not the objects but the relationship) is not physical (and certainly not mental). We can talk of physical relationships, but what's the number three?

It's a class of triples. We cannot place our hands upon a class. They are not physical. They are not mental.

 

[steve_bank]

Last month on CSPAN I listened to a woman talk about her book. She contrasted Analytic vs Continental philosophy and argued that the differences are the basis for fundamental differences on views and approaches between Europe and the USA.After looking it up it seems like the science-philosophy debates I get into around here reduce to Analytic vs Continental philosophy.

Which dominates, sconce and math as the source of understanding reality vs more pure conceptual metaphysics as the best approach.

I started a thread on metaphysics. Be there or be square.

 

[Angra Mainyu]

That said, I think I can argue quite well that there's something blatantly platonistic in the language used by modern day classical mathematicians, while there is something blatantly intuitionistic about the language used by the ancient Greeks.

Regarding the first part, I know that plenty of mathematicians are Platonists, but I don't think the language is committed to Platonism. Could you briefly point out why you think it is? (examples, etc.)

I plan for computer science to eat the entire world, and when we come to eat maths, we'll be making it intuitionistic, because it will force you guys to think in a way that always has a computational interpretation. I'm offering you a head-start before we begin the invasion in full.

That's an argument in support of a claim that intuitionistic math is more effective for the computer science, but it does not show that mathematicians should be doing that. Some people may just like an enjoy classical mathematics - just like they can enjoy, say, chess. Why should they give up what they like?

But aside from pure mathematics for the fun of it (and getting paid to, as professional chess players do), as far as I know much of modern science is based on mathematics that were not done using intuitionistic logic, but full classical logic (e.g., in physics). Now, computers in the future probably will have much better theories, but it's what we have now, and the development was thanks to classical logic. Even if all of that can be translated into a theory that only uses intuitionistic logic, that is not how it was developed, and I would say unsurprisingly so - it would have been much more difficult, given the restrictions imposed by intuitionistic logic.

 

[Phil Scott]

That said, I think I can argue quite well that there's something blatantly platonistic in the language used by modern day classical mathematicians, while there is something blatantly intuitionistic about the language used by the ancient Greeks.

Regarding the first part, I know that plenty of mathematicians are Platonists, but I don't think the language is committed to Platonism. Could you briefly point out why you think it is? (examples, etc.)

Here's my favourite example. This is the same geometric axiom rendered by Euclid and Hilbert:

Euclid: To draw a line between two points.

Hilbert: For every two points, A, B, there exists a line a that contains each of the points A, B.

Euclid tells us there is something that can be done. Hilbert tells us that something exists (where?).

Note that I am not saying that Hilbert's language commits him to Platonism. I am merely saying that the language is Platonistic. Hilbert himself ended up being a finitist/formalist who would have regarded this talk as syntax sugar over a very concrete finitist claim, while a fictionalist might just say that a Platonistic world is a useful fiction.

I plan for computer science to eat the entire world, and when we come to eat maths, we'll be making it intuitionistic, because it will force you guys to think in a way that always has a computational interpretation. I'm offering you a head-start before we begin the invasion in full.

That's an argument in support of a claim that intuitionistic math is more effective for the computer science, but it does not show that mathematicians should be doing that. Some people may just like an enjoy classical mathematics - just like they can enjoy, say, chess. Why should they give up what they like?

Because those that don't will be the first against the wall when my revolution comes, with a few maybe put into camps to solve ever harder busy beaver problems.

I'm not sure why you confused my tongue-in-cheek faux-belligerence for an argument.

But aside from pure mathematics for the fun of it (and getting paid to, as professional chess players do), as far as I know much of modern science is based on mathematics that were not done using intuitionistic logic, but full classical logic (e.g., in physics). Now, computers in the future probably will have much better theories, but it's what we have now, and the development was thanks to classical logic. Even if all of that can be translated into a theory that only uses intuitionistic logic, that is not how it was developed, and I would say unsurprisingly so - it would have been much more difficult, given the restrictions imposed by intuitionistic logic.

The classical/intuitionistic divide is barely a hundred years old, and mostly arose out of what were considered to be excesses in the logic being used to reason about infinite sets. Physicists (and indeed, most mathematicians) have been getting along happily for centuries without worrying about such things.

Physics isn't anywhere close to being formalised to a standard where foundational issues are going to come up. And the video I linked earlier is not the first to make the claim that, if you look at physics as practised, you might just come away with the impression that it's implicitly intuitionist and most faithfully formalised that way. Do you know of any restrictions you think intuitionism presents to a modern physicist? Do physicists need to well-order the reals in order to model Helium atoms?

 

[Angra Mainyu]

Euclid tells us there is something that can be done. Hilbert tells us that something exists (where?).

In the space (or maybe class of spaces) Hilbert is talking about, perhaps R^3, or some other space, or a general class of spaces (in a way, R^3 can be seen as a general class, but we don't need to get into that for now I think). I would need more context to tell for sure, but that is not a problem, because when two or more people (properly) talk about abstract scenarios - or, for that matter, concrete hypothetical scenarios in philosophy -, they have enough information about them to communicate what each is trying to say. For example, if students take an introductory algebra course and the professor begins to talk about prime numbers in the set of positive integers, the students generally understand the domain of discourse - namely, the set of positive integers. And if they're talking an introductury linear algebra course, they will also understand (well, generally they will; there are exceptions always) that the lines are in such-and-such spaces. The amount of information required to get one's interlocutors to understand the domain of discourse (i.e., roughly "where" those things exist, etc.) depends on the subject matter, but again, that does not appear to be problematic.

This, by the way, is not exclusive of mathematics, but of talk of hypothetical scenarios in general. Some of them (in math, logic, and sometimes though less frequently philosophy) are abstract, others concrete (e.g., scenarios in ethics, or in stragegic planning, etc.), but other than that, it's a matter of conveying enough information for one's interlocutors to know what one is talking about, which of course depends on the context.

Yet, I don't think this language is committed to Platonism. As far as I can tell, it is silent on the matter of Platonism.

Let's take, for example a standard Trolley case in philosophy. Philosophers talk about whether it's permissible to press the button, push the fat man, etc., and they debate about what is true or false in that particular scenario. But they don't seem at all committed to the existence of some sort of Platonic realm where that scenario exists - though some philosophers do believe that, others do not, and I don't see why one would think the latter are somehow committed to Platonism.

Similarly, and for example also, biologists can speculate about how life might have evolved on a planet with such-and-such characteristics, which were different from those of Earth. I recall I watched a TV program like that. And the presenter talked about the sort of animal-like organisms that lived there, predators, prey, etc., and a lot of other things. But they weren't committed to the existence of such planet. In fact, while most of them may well have belived there are extrasolar planets with lots of weird things, it's very unlikely that most of them believed that there was an actual, concrete planet with exactly those things. But similarly, they did not seem to believe in some Platonic realm of possible planets, either. And they weren't making any mistakes.

We could give examples in physics and other stuff as well.

Note that I am not saying that Hilbert's language commits him to Platonism. I am merely saying that the language is Platonistic. Hilbert himself ended up being a finitist/formalist who would have regarded this talk as syntax sugar over a very concrete finitist claim, while a fictionalist might just say that a Platonistic world is a useful fiction.

But what do you mean by saying that "the language is Platonistic", if you don't mean that it commits him to Platonism?
My impression from the claim that the language is Platonistic is that the meaning of the words is such that a Platonic realm is implicit. If that's what you think, I would ask why. If not, I would ask for clarification as to what you mean by saying that the language is Platonistic.

Because those that don't will be the first against the wall when my revolution comes, with a few maybe put into camps to solve ever harder busy beaver problems.

I'm not sure why you confused my tongue-in-cheek faux-belligerence for an argument.

I took your tongue-in-cheek faux-belligerence as such (i.e., I knew it was not actually belligerant ), but I thought that implicitly you were saying that because it is more effective for computer science (or something along those lines), then mathematicians should do that. I thought so because you made that tongue-in-cheek comment in a reply to beero1000, and particularly to a part of a post where he asked why you thought classical mathematicians should switch to intuitionism. Sorry if I misinterpreted. So, I'd like to ask whether you think classical mathematicians should switch to intuitionism, and if you do, then why you think so.

The classical/intuitionistic divide is barely a hundred years old, and mostly arose out of what were considered to be excesses in the logic being used to reason about infinite sets. Physicists (and indeed, most mathematicians) have been getting along happily for centuries without worrying about such things.

They didn't worry about the divide, but they did accept things like ¬¬P->P, or (P v ¬P) in their reasoning. That's also unsurprising, since that's very...intuitive

Do you know of any restrictions you think intuitionism presents to a modern physicist? Do physicists need to well-order the reals in order to model Helium atoms?

No, though in order to get rid of the well order, it's enough (for example) to get rid of AC, not of (P v ¬P). But that's not the issue. I'm not saying that current results can't be done with intuitionistic logic. I'm saying that that's not how people seem to think, or how they got their results. It would be weird (indeed, very counterintuitive) to get rid of (P v ¬P)), but even the AC is implicitly or explicitly used in the math studied by actual phycisists. For example, physics students learn calculus and use sequences (and implicitly, AC) just as math students do. They also work on manifolds, learn topological concepts, etc., and all of that is done in a context of classical proofs, without even considering eliminating (P v ¬P) or things like that. And phycisists go on with that learned background (they also implicitly understand, accept and use AC when they do math, though that's another issue).

Might it be done without (P v ¬P), and other stuff?
I think so, but I think it would probably be considerably more difficult, since intuitionism seems to be considerably counterintuitive. That would make studying physics (for example) likely considerably more difficult than it already is, and I don't see the advantage - there may well be one in computer science, though. But then, that suggests using intuitionistic logic in those contexts in which it has practical benefits, and classical logic in context where it's the other way around.

 

[Phil Scott]

But what do you mean by saying that "the language is Platonistic", if you don't mean that it commits him to Platonism?

I mean that, on the face of it, it is Platonistic. By contrast, Euclid's language is absolutely not Platonistic. I'm confused as to why you ignored the Euclid quote, given that my whole point here is to contrast the two ways of saying what purports to be the same thing.

As for whether you can desugar the Hilbert quote to something else, and since you used an example from moral philosophy: I'm pretty sympathetic to non-cognitivist metaethics, while freely admitting that our moral language is, on the face of it, cognitivist and realist. I am perfectly happy to think that the surface realism and cognitivism of "factory farming is wrong" does not in any way commit me to a realist or cognitivist metaethics, and that it might just be a useful way to say "Boo factory farming." But I will readily admit that the language is both realist and cognitivist.

If you're interested, there is no context to the Hilbert quote. These are foundational axioms for geometry, as Euclid's are. He explicitly does not want to be building on top of something like R3. There is philosophical commentary that he was trying to axiomatise Kantian a priori notions of space.

Because those that don't will be the first against the wall when my revolution comes, with a few maybe put into camps to solve ever harder busy beaver problems.

I'm not sure why you confused my tongue-in-cheek faux-belligerence for an argument.

I took your tongue-in-cheek faux-belligerence as such (i.e., I knew it was not actually belligerant ), but I thought that implicitly you were saying that because it is more effective for computer science (or something along those lines), then mathematicians should do that. I thought so because you made that tongue-in-cheek comment in a reply to beero1000, and particularly to a part of a post where he asked why you thought classical mathematicians should switch to intuitionism. Sorry if I misinterpreted. So, I'd like to ask whether you think classical mathematicians should switch to intuitionism, and if you do, then why you think so.

Mathematics and computer science require lots of mathematicians and lots of computer scientists: manpower. My preference for intuitionism means that I want more folk on the ground doing intuitionistic maths. However, I didn't arrive at my preferences by way of a philosophical argument, and I'd be suspicious of those preferences if I had, so I'm not going to attempt anything of the sort here. I think I'd get the job done quicker by gunboat.

The classical/intuitionistic divide is barely a hundred years old, and mostly arose out of what were considered to be excesses in the logic being used to reason about infinite sets. Physicists (and indeed, most mathematicians) have been getting along happily for centuries without worrying about such things.

They didn't worry about the divide, but they did accept things like ¬¬P->P, or (P v ¬P) in their reasoning. That's also unsurprising, since that's very...intuitive

Intuitionistists accept ¬¬P->P and (P v ¬P) when reasoning in any finite domain, and they'll often accept that reasoning in simple finite domains. Infinite domains of infinite domains only seriously appear in the 1800s, and immediately kick-off controversy as to whether reasoning about such objects goes awry. And it does, as shown by the set theory paradoxes. Intuitionism is one solution to those paradoxes. ZF set theory is another. I will happily argue that intuitonism is a more intuitive take on infinite domains than the hack-job that is the ZF axioms.

Do you know of any restrictions you think intuitionism presents to a modern physicist? Do physicists need to well-order the reals in order to model Helium atoms?

No, though in order to get rid of the well order, it's enough (for example) to get rid of AC, not of (P v ¬P). But that's not the issue. I'm not saying that current results can't be done with intuitionistic logic. I'm saying that that's not how people seem to think, or how they got their results. It would be weird (indeed, very counterintuitive) to get rid of (P v ¬P)), but even the AC is implicitly or explicitly used in the math studied by actual phycisists. For example, physics students learn calculus and use sequences (and implicitly, AC) just as math students do. They also work on manifolds, learn topological concepts, etc., and all of that is done in a context of classical proofs, without even considering eliminating (P v ¬P) or things like that. And phycisists go on with that learned background (they also implicitly understand, accept and use AC when they do math, though that's another issue).

Might it be done without (P v ¬P), and other stuff?
I think so, but I think it would probably be considerably more difficult, since intuitionism seems to be considerably counterintuitive. That would make studying physics (for example) likely considerably more difficult than it already is, and I don't see the advantage - there may well be one in computer science, though. But then, that suggests using intuitionistic logic in those contexts in which it has practical benefits, and classical logic in context where it's the other way around.

Classical foundations of analysis require AC. Intuitionistic foundations do not. We don't know which the physicists are using, because they largely don't care about mathematical foundations. It is a point up for debate, and mentioned in the video I linked, as to whether physicists implicitly assume classical or intuitionistic mathematics. I currently have little evidence either way.

Here's an anecdote: I have a friend who got his physics degree from Cambridge, and one evening, I was berating physicists for not minding their epsilons and deltas, and thus ignoring potential pathological edge cases that are counterexamples to their reasoning. He responded quite fairly: we don't care about being too rigorous about epsilons and deltas, because we assume that, once you get to a small enough scale, the epsilons are literally 0. This isn't intuitionistic: it's basically finitistic.

It could be that the leaps of logic that are made here, that would horrify a classical real analyst, implicitly assume the very restrictions that must be imposed by intuitionistic and finitistic analysis, where they don't have to worry about Weierstrass functions and Banach Tarski.

As for the intuitiveness of excluded-middle, it's absolutely absent if you phrase things like Euclid does. What sense is there to this:

1. To draw a line between two points or to fail to draw a line between two points.

If I wanted to emphasise the counterintuitiveness of excluded middle in reasoning, I might just encourage a physicist to adopt this sort of style, and phrase their claims with a mind to what they can measure. Though I suspect the average physicist would be utterly patronised by this, since I assume that's how they're thinking all the time anyway.

 

[Angra Mainyu]

I mean that, on the face of it, it is Platonistic.

But I'd like to ask what you mean by "it is Platonistic". I thought it might be something like:

Platonistic-1: Assertions using that language, and in particular the assertion that "there exists a line", etc., imply (by the meaning of the words + classical? logic) that Platonism is true.

However, given that you're saying that the assertion does not commit Hilbert to Platonism, I'm not sure. Are you saying that it does not commit him because he said otherwise elsewhere, or because Platonistic 1 is not what you have in mind by "Platonistic"?
At any rate, I don't see anything Platonistic-1 about the language on the face of it, so if you mean Platonistic-1, I would like to ask why you find it so.

If you did not mean Platonistic-1 (and now it seems more likely that you did not), I'm not sure how to construe "it is Platonistic" in your sentence above. I can try some options, such as:

Platonistic-2: Assertions using that language, and in particular the assertion that "there exists a line", etc., provide strong evidence that the person is a Platonist.

But in that case (i.e., if you mean that), I disagree that it's Platonistic-2, and I would like to ask why you think it is so on the face of it. If you did not mean that, I could try:

Platonistic-3: Assertions using that language, and in particular the assertion that "there exists a line", etc., provide at least some evidence that the person is a Platonist.

Given that the language is pretty standard, and appears clearly compatible with Platonism, it seems extremely improbable that a Platonist would refrain from using that language. On the other hand, some non-Platonists think that the language is or in some sense looks "Platonistic", and refrain from using it for that reason.
Hence (by Bayes' Theorem), if the only new piece of information that we add to our background evidence (as just described) is that a person uses the language in question, one should update upwards the probability that the person in question is a Platonist. So, given this background, it does seem to provide at least some evidence that the person is a Platonist. However, the language is definitely not Platonistic-3 on the face of it. Rather, in order to realize it's Platonistic-3, one needs information about the existence of non-Platonists who prefer to avoid the language, no Platonists (very probably, or at least a minuscule percentage) who prefer to avoid it, etc.
In short, if by "Platonistic" you mean "Platonistic-3", then I would agree it's Platonistic, but disagree that it is so on the face of it. Moreover, if that's what you mean, then I think the use of the expression "Platonistic" to describe the language is misleading.

That said, I think it's very improbable that you meant that, in particular because of your assessment that it's Platonistic "on the face of it", so I'm still in doubt as to what you mean.

Granted, usually we don't need to define the expressions we use in a stipulative manner. Rather, we do so ostensively. So, if you prefer to define "Platonistic" ostensively, I'm okay with that.

By contrast, Euclid's language is absolutely not Platonistic. I'm confused as to why you ignored the Euclid quote, given that my whole point here is to contrast the two ways of saying what purports to be the same thing.

I did not address Euclid's quote because I did not see how you were trying to get to that. I suppose one could say that Euclid's language is incompatible with Platonism, since in a Platonic realm, one can't draw a line if one takes that literally. Is that what you meant?

As for whether you can desugar the Hilbert quote to something else, and since you used an example from moral philosophy: I'm pretty sympathetic to non-cognitivist metaethics, while freely admitting that our moral language is, on the face of it, cognitivist and realist. I am perfectly happy to think that the surface realism and cognitivism of "factory farming is wrong" does not in any way commit me to a realist or cognitivist metaethics, and that it might just be a useful way to say "Boo factory farming." But I will readily admit that the language is both realist and cognitivist.

Without further input, I would have understood the claim that the language is cognitivist as implying that it is not a useful way of saying "Boo factory farming". Clearly, either you mean by "cognitivist" something I'm not familiar with (in which case, I'd like to ask for a definition), or you are saying that the language appears on the face of it cognitivist (in the sense I would understand "cognitivist"), but it might turn out not to be so. That, I can understand. But then, translated to the Platonism case, it would be something like Platonistic-1, and by saying that the language is Platonistic, you're saying that on the face of it, it is Platonistic-1, though it might turn out not to be so for some other reason. If that's what you're saying, as I mentioned, I don't see anything Platonistic-1 about the language on the face of it (but if I misunderstood what you mean by "Platonistic", please let me know what you mean and I will address that).

If you're interested, there is no context to the Hilbert quote. These are foundational axioms for geometry, as Euclid's are. He explicitly does not want to be building on top of something like R3. There is philosophical commentary that he was trying to axiomatise Kantian a priori notions of space.

When one gives axioms, there is still a context. If - say - someone explains Peano's Postulates, the context generally is the set of natural numbers, already intuitively grasped at least to some extent by the people the explanation is meant to. The same applies to Hilbert.

Mathematics and computer science require lots of mathematicians and lots of computer scientists: manpower. My preference for intuitionism means that I want more folk on the ground doing intuitionistic maths. However, I didn't arrive at my preferences by way of a philosophical argument, and I'd be suspicious of those preferences if I had, so I'm not going to attempt anything of the sort here. I think I'd get the job done quicker by gunboat.

Fair enough (I don't understand the "by gunboat" expression, and I would also suggest the argument would not need to be philosophical, but I think those are details in context; I get you aren't making that argument).

Intuitionistists accept ¬¬P->P and (P v ¬P) when reasoning in any finite domain, and they'll often accept that reasoning in simple finite domains. Infinite domains of infinite domains only seriously appear in the 1800s, and immediately kick-off controversy as to whether reasoning about such objects goes awry. And it does, as shown by the set theory paradoxes. Intuitionism is one solution to those paradoxes. ZF set theory is another. I will happily argue that intuitonism is a more intuitive take on infinite domains than the hack-job that is the ZF axioms.

What I mean is that they don't accept ¬¬P->P or (P v ¬P) in all domains, which is very counterintuitive. The set theory paradoxes show that some reasoning goes awry, not that all reasoning about infinite domains goes awry. ZF theory is an answer to the paradoxes in the limited context of set theory - not in all infinite domains -, and it has some advantages and disadvantages. But I think I will take you up on that offer (i.e., to argue that intuitionism is a more intuitive take on infinite domains that the ZF axioms are). At any rate, I'm saying that it's counterintuitive to reject (P v ¬P), etc., in general mathematic contexts, such as analysis (in many, or rather most of its variants).

Classical foundations of analysis require AC. Intuitionistic foundations do not. We don't know which the physicists are using, because they largely don't care about mathematical foundations. It is a point up for debate, and mentioned in the video I linked, as to whether physicists implicitly assume classical or intuitionistic mathematics. I currently have little evidence either way.

They do not care about the foundations, but physics students, like math students, (at least implicitly) use AC when they learn math, and - more to the point -, they also use classical logic. That seems to come intuitively to them. It's not that they require it for physics, but rather, that it would seem weird to reject it (i.e., classical logic; I prefer to leave AC aside since it's another matter, though I also think it's intuitive).

Here's an anecdote: I have a friend who got his physics degree from Cambridge, and one evening, I was berating physicists for not minding their epsilons and deltas, and thus ignoring potential pathological edge cases that are counterexamples to their reasoning. He responded quite fairly: we don't care about being too rigorous about epsilons and deltas, because we assume that, once you get to a small enough scale, the epsilons are literally 0. This isn't intuitionistic: it's basically finitistic.

I don't think it's finitistic (based on that quote; of course, you might have a lot more evidence about your friend), because physicists do not seem to assume that the number of galaxies, stars, etc., is finite. If anything, it's an assumption (perhaps justified, though I wouldn't bet on it) that space is discrete, there is a minimum actual distance, etc.

As for the intuitiveness of excluded-middle, it's absolutely absent if you phrase things like Euclid does.

But it's intuitive when you don't restrict the language in that manner. What is counterintuitive is to reject it, as far as I can tell.

 

[fast]

The assertion that there is in fact a line at all can be spooky to the trusting person hearing the assertion who can yet use no instruments of science to positively identify the line's presence.

In character, a student at the front row may chuckle and say: oh teacher(!), why in the world would you even tell us such a thing (that there is a line) when we can all plainly see what is so blindingly obvious for ourselves. But then as he turns to point to this line that exists, his face turns pale upon recognizing that the other students can't see it either. He so very much trusts the teacher. Where? Where or where can it be?

It must be in Platonic land. Platonicville?

But wait, although the language we use is most helpful in explaining our points, the literalists become quite conflicted. To say that there is something when there is no detection of it is quite bothersome for some.

 

[Treedbear]

The assertion that there is in fact a line at all can be spooky to the trusting person hearing the assertion who can yet use no instruments of science to positively identify the line's presence.

In character, a student at the front row may chuckle and say: oh teacher(!), why in the world would you even tell us such a thing (that there is a line) when we can all plainly see what is so blindingly obvious for ourselves. But then as he turns to point to this line that exists, his face turns pale upon recognizing that the other students can't see it either. He so very much trusts the teacher. Where? Where or where can it be?

It must be in Platonic land. Platonicville?

But wait, although the language we use is most helpful in explaining our points, the literalists become quite conflicted. To say that there is something when there is no detection of it is quite bothersome for some.

It just goes to show how pervasive and accepted Platonic thinking is in western culture.

 

[Horatio Parker]

The assertion that there is in fact a line at all can be spooky to the trusting person hearing the assertion who can yet use no instruments of science to positively identify the line's presence.

In character, a student at the front row may chuckle and say: oh teacher(!), why in the world would you even tell us such a thing (that there is a line) when we can all plainly see what is so blindingly obvious for ourselves. But then as he turns to point to this line that exists, his face turns pale upon recognizing that the other students can't see it either. He so very much trusts the teacher. Where? Where or where can it be?

It must be in Platonic land. Platonicville?

But wait, although the language we use is most helpful in explaining our points, the literalists become quite conflicted. To say that there is something when there is no detection of it is quite bothersome for some.

It just goes to show how pervasive and accepted Platonic thinking is in western culture.

Yes. A geometric point is my favorite example. No mass, dimension, location, nothing but intelligibility. But it exists, as do the rules of baseball.

In the Parmenides, Plato explores the reality of the One, and IIRC, more or less concludes that it's a postulate. Our reality is based on working assumptions, guesses.

Popper argued that because math has its own structure, it must it some represent or reflect reality even if numbers are not real.

To my mind, the intelligible is obviously real. What that means is of course another question.

 

[Phil Scott]

I did not address Euclid's quote because I did not see how you were trying to get to that.

Well I only care to contrast between Euclid and Hilbert, so the previous stuff was a bit of a puzzling meander to me. Besides which, please note that I automatically switch off if people mention "Bayes' Theorem" outside of a mathematical or scientific context

They do not care about the foundations, but physics students, like math students, (at least implicitly) use AC when they learn math, and - more to the point -, they also use classical logic. That seems to come intuitively to them. It's not that they require it for physics, but rather, that it would seem weird to reject it (i.e., classical logic; I prefer to leave AC aside since it's another matter, though I also think it's intuitive).

If they're not using AC explicitly, how do you know they're using it at all? Maybe they're implicitly using intuitionistic analysis. How would you tell (please be specific)?

I don't think it's finitistic (based on that quote; of course, you might have a lot more evidence about your friend), because physicists do not seem to assume that the number of galaxies, stars, etc., is finite. If anything, it's an assumption (perhaps justified, though I wouldn't bet on it) that space is discrete, there is a minimum actual distance, etc.

It's a finitistic take on the infinitesimal calculus. That's all I need to claim it's not based on classical analysis.

As for the intuitiveness of excluded-middle, it's absolutely absent if you phrase things like Euclid does.

But it's intuitive when you don't restrict the language in that manner. What is counterintuitive is to reject it, as far as I can tell.

Why do you call it a restriction? Was Euclid restricting himself?

 

[Treedbear]

The assertion that there is in fact a line at all can be spooky to the trusting person hearing the assertion who can yet use no instruments of science to positively identify the line's presence.

In character, a student at the front row may chuckle and say: oh teacher(!), why in the world would you even tell us such a thing (that there is a line) when we can all plainly see what is so blindingly obvious for ourselves. But then as he turns to point to this line that exists, his face turns pale upon recognizing that the other students can't see it either. He so very much trusts the teacher. Where? Where or where can it be?

It must be in Platonic land. Platonicville?

But wait, although the language we use is most helpful in explaining our points, the literalists become quite conflicted. To say that there is something when there is no detection of it is quite bothersome for some.

It just goes to show how pervasive and accepted Platonic thinking is in western culture.

Yes. A geometric point is my favorite example. No mass, dimension, location, nothing but intelligibility. But it exists, as do the rules of baseball.

In the Parmenides, Plato explores the reality of the One, and IIRC, more or less concludes that it's a postulate. Our reality is based on working assumptions, guesses.

Popper argued that because math has its own structure, it must it some represent or reflect reality even if numbers are not real.

To my mind, the intelligible is obviously real. What that means is of course another question.

The alternative to Platonism and the existence of absolutes is the reality that things only exist in relation to other things. That would include mathematics and geometry. A point is only meaningful if it symbolizes something. Anything more than that and it becomes a crutch, imho.

 

[Angra Mainyu]

Well I only care to contrast between Euclid and Hilbert, so the previous stuff was a bit of a puzzling meander to me.

But I don't see how the contrast supports the view that Hilbert's language (or similar language, as used by others) is Platonistic.

That aside, I'm still not sure what you mean by "Platonistic" (yes, I do know you mean it's Platonistic on the face of it, but that does not tell me what it is that you think the language, on the face of it, is ). I'm not sure why you find my trolley problem example unpersuasive, but I used the trolley problem merely as an example of a widespread phenomenon. For instance, let's say that military strategists are considering war scenarios, planning for contingencies, etc. They devise scenarios, make assessments about what happens in them (or probably happens, etc.), and so on. But that does not indicate a commitment to those scenarios existing "somewhere" in either a Platonic realm, or a concrete possible world (under modal realism), or anything like that. It's run of the mill talk about hypothetical scenarios. And military scenarios are only one example. Economists also speculate about different hypothetical scenarios in which different stuff happens, and the same goes for people in many other disciplines. We also do that in our daily lives, intuitively. Someone might ask "but where does that happen?". Well, it happens in the scenario in question (or, to be more precise, one could call them a class of scenarios, since they're not fully specified). I don't see anything in their language suggesting a committment to Platonism about those scenarios, and/or to modal realism. The fact that mathematical scenarios involve abstract objects only does not seem to be relevant to the question of whether the people talking about them assume that the scenarios exist in some kind of Platonic realm, as far as I can tell. If you think otherwise, I would like to ask why, but at any rate, that would require argumentation, and it would not be a case of the language being Platonistic on the face of it, at least in any of the senses of "Platonistic" I have considered (and almost certainly in any other, unless you're using "Platonistic" in a way that makes run-of-the-mill talk about hypoethetical scenarios Platonistic on its face).

Besides which, please note that I automatically switch off if people mention "Bayes' Theorem" outside of a mathematical or scientific context

That's too bad. You're switching off much of philosophy (yes, there is much of philosophy that is pretty bad, but not because of that!). Philosophers aside, the fact is that our intuitive probabilistic statements do not require formal knowledge of mathematics or logic, and predate it by, well, since there is language - very probably! Bayes' Theorem, while developed in hypothetical and simplified scenarios, seems to capture some of the way we can and do properly reason about probability in the usual context, so it seems proper to use it - at least, it very probably is proper!

Anyway, I'd like to ask: have you already switched me off on the points I'm most interested in, namely my points and questions about your claim that the language is Platonistic on the face of it? (I'm asking because you're not addressing them, right after you tell me you switch off people who do what I just did).
In any case, I will address the rest of your points, though I have to admit I consider the matters less interesting.

If they're not using AC explicitly, how do you know they're using it at all? Maybe they're implicitly using intuitionistic analysis. How would you tell (please be specific)?

Fair enough (and let's include classical logic in the process).

Suppose Alice is a math or physics student, and there is an exercise in which she's asked to prove that a function f on a metric space X is continuous. Then, she may reason: "Suppose otherwise. Then, there is y \in X, \epsilon >0, and a sequence \{x_n\}_{n\in\mathbb{N}} such that...", etc. Similarly, if she wants to prove that a set is closed, that a normed space is Banach, etc. In general, the axiom of choice is implicitly used when choosing sequences, and that's fairly common. It seems to me AC makes learning math easier, for math students, for physics students, and so on.

It's a finitistic take on the infinitesimal calculus. That's all I need to claim it's not based on classical analysis.

I would need more background information. My impression was that he thought that the world has a mininum distance, etc., but wasn't talking about the math (i.e., the deltas, etc., were zero because of that apparent limit). Regardless, the physicists and physics students I know, at least in the cases I'm familiar with their doing math, they just do it in the usual manner, without assuming they're zeros or anything. In fact, they work with infinitesimals in a way some mathematicians dislike (though nonstandard analysis can handle it), and they surely don't consider them zero (else, they would be dividing by zero very frequently). Now, you could say that's not classical analysis. But that's not the issue: they use the tools of classical analysis (including AC, and of course excluded middle), and more stuff too.

Why do you call it a restriction? Was Euclid restricting himself?

It would be a restriction of the language to use it only as Euclid did as a means of avoiding the intuitiveness of excluded middle. And it's a restriction because something is removed from the language. Now, was Euclid restricting himself?
As near as I can tell, very probably not. I'm not so familiar with his work, but - again as near as I can tell - he was using the language he found useful for what he wanted to achieve, without removing anything. Other people use other language, which includes other stuff. Restricting the language to the language Euclid chose to use would be, well, a restriction. It wasn't (very probably) for Euclid, since he was using the language he wanted to in order to achieve what he wanted.