I just think you're working yourself up over nothing. I'm losing interest. Can you write less? This is tedious, I find nothing of interest here. — fishfry

If you desire to avoid the long posts, I think, by the end of my reply here, that I have isolated the primary point of disagreement between us. It is exposed in how you and I each relate to what is referred to by "the real numbers", and what is referred to with "5" in the context of "the real numbers". And further, how this relates to the extension/intension distinction.

Therefore, I think you might just read through my post and reply to the aspects which are related to this issue. However, the issue of what mathematics is, how you and I would each describe "what mathematicians do", might also be important and relevant.

Pure math is math done without any eye towards contemporary applications. That's a decent enough working definition. Mathematicians know the difference. — fishfry

The issue though, is that even supposedly "pure" mathematicians work toward resolving problems, and problems always have a real world source or else they are really not problems, but more like amusements. A mathematician working in pure abstractions works with abstractions already produced, and may not even know how real world problems have shaped the already exist abstract structure. Even if we attempt to step aside from existing conceptions, and 'start from scratch' as philosophers often do, we are guided by our intuitions which have been shaped and formed by life in the world. And intuition comes from the subconscious into the mind, so we cannot get our minds beneath it, to free ourselves from that real world base. And since it is from the subconscious, we have no idea of how the real world effects it.

Mathematics is whatever mathematicians do in their professional capacity. — fishfry

I agree, but the description of what mathematicians do, is very difficult to get an agreement on. It's not a circular definition, but a proposal of how to produce a definition. So to actually provide the definition of mathematics, we need that description. It will be very difficult for you and I to agree on such description. You will probably place as the primary defining feature, (the essential aspect), of what mathematicians do, as working with abstractions. I will say, that description is problematic because then we need some understanding of what an abstraction is, and what it means to "work" with this type of thing. This almost certainly will lead to Platonism because we've already assumed as a premise, the existence of things called "abstractions".

Therefore I look at what mathematicians are doing as "solving problems". That's what they do, and there is a specific type of problem which they deal with. You are most likely not going to like this proposal for a description of what mathematicians are doing, because it eliminates the distinction between "pure" mathematics and "applied" mathematics. In the way described above, there is no such thing as "pure" mathematics. However, my starting point has the advantage of applying equally to all mathematicians, by applying the initial assumption of pragmaticism. Instead of saying "mathematicians are working with abstractions", we say "mathematicians are working with symbols (language), to solve problems. This way we avoid the messy ontological problem of "abstractions" It is only when we start sorting out the different types of problems which mathematicians work on, do we get the divisions within mathematics.

This is a standard complaint. If math follows from axioms, then all the theorems are tautologies hence no new information is added once we write down the theorems. But that's like saying the sculptor should save himself the trouble and just leave the statue in the block of clay. Or that once elements exist, chemists are doing trivial work in combining them. It's a specious and disingenuous argument. — fishfry

This is not the point at all, and you are not paying respect to the difference between the two distinct fields, mathematics, and mathematical logic, so your analogy is not well formed. If the field of mathematics is represented by the sculptor, then the field of mathematical logic is represented by the critic. Whenever the critic mistakenly represents what the sculptor is doing, then the critic is wrong. When mathematical logic represents mathematicians as using = to symbolize identity, the logic is wrong.

We agreed long ago that 1 + 1 and 2 are not the same string; and many people have explained the difference between the intensional and extensional meanings of a string. Morning star and evening star and all that. — fishfry

Fishfry, wake up! Was it getting late there or something? There is no physical object involved! There is no star! I think we've been through this before. The intensional/extensional distinction is completely irrelevant in this case because everything referred to is meaning (intensional). There is nothing extensional, no objects referred to by "1+1", or "2". That is the heart of the sophistic ruse. This intensional/extensional rhetoric falsely persuades mathematicians. It wrongly misleads them due to their tendency to be Platonist, and to think of mathematical abstractions as objects. As soon as meaning is replaced by objects, then "extensional" is validated, the sophist has succeeded in misleading you, and down the misguided route you go. In reality, there is only meaning referred to by "1+1", and by "2", everything here is intensional, and there is nothing extensional.

This is why I was very steadfast on the previous issue, to explain that "5" is not "an instance of a real number". It is that type of nomenclature, that type of understanding, which leads one into allowing that there is a place for extensional definitions in mathematics. Really, "5" in that example is just a part of that conception called "the real numbers". It receives it's meaning as part of that conception. there are no extensional objects referred to by "the real numbers", and "5" is just an intensional aspect of that conception. When you apprehend "the real numbers" as referring to a collection of things, instead of as referring to a conception, then you understand "5" as referring to an instance of a real number, instead of understanding it as a specific part of that conception. Then you may be misled into the "extensionality" of real numbers, instead of understanding "the real numbers" as completely intensional.

What math teacher hurt your feelings, man? Was it Mrs. Screechy in third grade? I had Mrs. Screechy for trig, and she all but wrecked me. It's over half a century later and I can still hear her screechy voice. I hated that woman, still do. When I'm in charge, I'm sending all the math teachers to Gitmo first thing. — fishfry

Again, you are not distinguishing between "mathematics", and the "mathematical logic" which the head sophist preaches. One is the artist, the other the critic. My beef is not with mathematics (the art), it is with mathematical logic (the critic). I see mathematical logic as sophistry intended to deceive. And I will explain the reason why i say there is an intent to deceive.

Mathematics has a long history of exposing us to problems which we just cannot seem to solve. These are issues such as Zeno's paradoxes, and other apparent paradoxes discussed at TPF, which generally amount to problems with the conception of infinity, the continuity of space and time, etc.. What mathematical logic does, is create the illusion that such problems have been solved. So, the intent to deceive is inherent within the conceptual structure, which makes these problems solvable. It deceives mathematicians into thinking that they have solved various problems, by allowing them to work within a structure which makes them solvable. The problem though is that the basic axioms (extensionality for example) are blatantly wrong, and designed specifically so as to make a bunch of problems solvable, regardless of the fact that incorrect axioms are required to make the problems solvable.

Whatevs. I can't follow you. And I've already noted that the difference between pure and applied math is often a century or two, or a millennium or two. — fishfry

Future application is not the issue here. The issue is that mathematicians work toward problem solving, by the very nature of what mathematics is. The problems are preexistent. Therefore mathematics by its very nature is fundamentally "applied". If you remove problem solving from the essence of mathematics, then it would be random fictions. But mathematics is not random fictions, the mathematicians always follow at least some principles of "number", already produced.

Now what do I mean by "essentially the same?" Well now we're into structuralism and category theory. Sameness in math is a deep subject. I'll take your point on that. — fishfry

What I think, is that there is really no such things as sameness in math, and this is better described as a misleading subject. Mathematics actual deals with difference, and ways of making difference intelligible through number. Similarity is not sameness, but difference which can be quantified. To me, "essential the same" just means similar, which is different.

Even so, 5 is one of the real numbers. What do you call it if not an instance? What WOULD be an instance of a real number? — fishfry

This appears to be the substance of our difference, or disagreement. If you do not like long posts, we could just focus on this specific issue. The issue is whether "the real numbers" refers to a conceptual structure, or whether it refers to a group of things, numbers. I believe the former, and the fact that "numbers" is plural is just a relic of ancient tradition. From my perspective, "5", in the context of "a real number" is just a specific part of that conception. Then the relations are purely intensional, and there is nothing extensional here. If however, you apprehend "the real numbers" as referring to a group of things called "numbers", then "5" refers to one of those things, and there is the premises required for extensionality.

It is bizarre to suggest there's any arguing the point, when the point has been so profusely documented. Your retraction and your offer to retract the bizarre qualifier in the retraction are a self-serving and sneaky way to put the ball back in my court where it doesn't belong. — TonesInDeepFreeze

If you reject my retraction and apology, that is your right.

It's not a matter of whether I accept or reject. I said what I had to say in my post. If you wish not to address what I said it in, that is your right.

What do you mean by "put on"? I only said that Frege's system is one attempt to derive mathematics solely from logic, and the system is inconsistent.

/

I don't know what you have in mind about "throwing away everything finite"?

Frege's approach to even defining the number 0 from logic alone requires an infinite class.

/

Frege's system was proven to be inconsistent.

First order PA has not been proven to be inconsistent. I don't see a reason to believe that first order PA is inconsistent. And accepting the premises of Gentzen's proof, first order PA is proven not to be inconsistent.

So I don't know why you would ask.

Therefore I look at what mathematicians are doing as "solving problems". That's what they do, and there is a specific type of problem which they deal with. . . . Instead of saying "mathematicians are working with abstractions", we say "mathematicians are working with symbols (language), to solve problems. This way we avoid the messy ontological problem of "abstractions" It is only when we start sorting out the different types of problems which mathematicians work on, do we get the divisions within mathematics. — Metaphysician Undercover

It is true that some mathematicians are "problem solvers", perhaps the majority. But for the others, myself included, a mathematician is an explorer trying to find a path extending knowledge in a particular direction or discovering new directions. Creation and discovering are two sides of the same coin: we create, for instance, simply by virtue of defining and we discover where those creations lead.

As to describing what mathematicians do as "solving problems", that's fine as long as "solving problems" includes proving theorems, because mostly what mathematicians do is prove theorems.

And the evening star (which is the morning star) is not a star, it's a planet, and exists as a physical object.

And the crank adds to displaying his lack of intellectual capability by showing that he cannot comprehend intensionality and extensionality, which is not surprising since he is incapable of comprehending use and mention.

If you desire to avoid the long posts, I think, by the end of my reply here, that I have isolated the primary point of disagreement between us. It is exposed in how you and I each relate to what is referred to by "the real numbers", and what is referred to with "5" in the context of "the real numbers". And further, how this relates to the extension/intension distinction. — Metaphysician Undercover

Looking ahead, you wrote a long post. To which, in my own verbose style, I will reply to at length para by para, increasing the overall length of the thread.

I see an out. In this para you have stated your aim about the real numbers and the number 5. I don't think I have any interest in this topic. I know it's important and meaningful to you, but it isn't to me. Perhaps I'm to dim to grasp all these philosophical subtleties such as you raise. If so, so be it.

But secondly, and I'd be remiss if I didn't add, that I have formally studied the real numbers and the number 5. That doesn't make me right and you wrong, by any means. What it does mean is that I'm not likely to ever defer to your opinions about the real numbers or the number 5.

And if, as you say, that's all you want me to know, then now I know it. We disagree on the real numbers and the number 5. Ok. I am a pluralist. It doesn't bother me when people have different opinions than I do. I don't have to convert you nor you me. We can let the matter rest. I'm for that.

Therefore, I think you might just read through my post and reply to the aspects which are related to this issue. However, the issue of what mathematics is, how you and I would each describe "what mathematicians do", might also be important and relevant. — Metaphysician Undercover

I have no strong or absolute opinion about "what math is," nor do I feel any need to argue for or against any particular interpretation of that question. The history of math is the evolution of the answer to that question! "What math is," is always changing. It literally is what mathematicians do.

The issue though, is that even supposedly "pure" mathematicians work toward resolving problems, and problems always have a real world source or else they are really not problems, but more like amusements. — Metaphysician Undercover

If your claim is that by definition, what pure mathematicians do amounts to amusements, in the sense that they solve problems that were only inspired by their own meaningless work; and not by anything that we currently know about in the world.

If so, it's perfectly and trivially true, if that's your definition. What of it? Doesn't mean anything.

Read Hardy's A Mathematician's Apology. He'd have been insulted if you told him his work was useful. The irony is that he did number theory, which had been a beautiful but utterly useless branch of math for over 2000 years. And then finally in the 1980s, people invented public key cryptography, and Hardy's work was at the heart of grubby world commerce.

So you just never know.

But still. Amusement? Ok. Whatever. Like Picasso, he made amusements too.

A mathematician working in pure abstractions works with abstractions already produced, and may not even know how real world problems have shaped the already exist abstract structure. Even if we attempt to step aside from existing conceptions, and 'start from scratch' as philosophers often do, we are guided by our intuitions which have been shaped and formed by life in the world. And intuition comes from the subconscious into the mind, so we cannot get our minds beneath it, to free ourselves from that real world base. And since it is from the subconscious, we have no idea of how the real world effects it. — Metaphysician Undercover

Such a triviality. Everyone knows that math started when some caveman put a mark in the ground when he killed a wooly mammoth, and then put another mark next to it when he killed another one. That was the first mathematical abstraction, and it is the paradigm for all others.

Everybody knows this. You think you just discovered it. Just like someone could say that abstract art is an evolution of representational art, or is influence by it or is a reaction to it.

It's just the process of abstraction. And abstraction is always based on our own experience of the world. Nobody is denying that. It's your strawman.

I agree, but the description of what mathematicians do, is very difficult to get an agreement on. — Metaphysician Undercover

Correct. Mathematics is a historically contingent human activity that changes every day as new papers are published. And every few decades new ideas come out that change our very conception of what math is.

Even in contemporary practice, there is professional disagreement about what is mathematics. I refer to the amazing dispute over the work of Shinichi Mochizuki, about which mathematicians have been arguing for over a dozen years, and illustrating the fact that mathematical truth is subject to social agreement.

In view of the history of math, we see that this has always been so.

It's not a circular definition, but a proposal of how to produce a definition. So to actually provide the definition of mathematics, we need that description. It will be very difficult for you and I to agree on such description. You will probably place as the primary defining feature, (the essential aspect), of what mathematicians do, as working with abstractions. I will say, that description is problematic because then we need some understanding of what an abstraction is, and what it means to "work" with this type of thing. This almost certainly will lead to Platonism because we've already assumed as a premise, the existence of things called "abstractions". — Metaphysician Undercover

Trying to nail down a definition of mathematics is like a cat chasing its own tail. Not as cute though. Why do you persist? Why do you even think it matters? It changes throughout history, and it's not even agreed on by all professional practitioners today!

But who said I'm not a Platonist? I am? When it suits my argument. I'm a formalist as well at times. Mathematical philosophies are tools, nothing more. Conceptual tools, frameworks for thinking about the development and structure of math. They aren't "true" or "false," they're just models, if you will.

Therefore I look at what mathematicians are doing as "solving problems". That's what they do, and there is a specific type of problem which they deal with. You are most likely not going to like this proposal for a description of what mathematicians are doing, because it eliminates the distinction between "pure" mathematics and "applied" mathematics. — Metaphysician Undercover

I don't like it because it's factually false. There's a famous essay on that, about the kind of mathematicians who solve problems, and the kind of mathematicians who build theories. To the Internet! Yes here it is, The Two Cultures of Mathematics by Timothy Gowers.

Problem solvers and theory builders. The theory builders don't solve problems at all. They create conceptual frameworks in which others can solve problems.

In the way described above, there is no such thing as "pure" mathematics. However, my starting point has the advantage of applying equally to all mathematicians, by applying the initial assumption of pragmaticism. Instead of saying "mathematicians are working with abstractions", we say "mathematicians are working with symbols (language), to solve problems. This way we avoid the messy ontological problem of "abstractions" It is only when we start sorting out the different types of problems which mathematicians work on, do we get the divisions within mathematics. — Metaphysician Undercover

Ok, there's no such thing as pure mathematics. So what? Why do you care? Why should I? I've already explained that not only is math historically contingent, there's not even universal agreement today about what math is.

You are arguing against a strawman of your own imagination.

This is not the point at all, and you are not paying respect to the difference between the two distinct fields, mathematics, and mathematical logic, so your analogy is not well formed. If the field of mathematics is represented by the sculptor, then the field of mathematical logic is represented by the critic. Whenever the critic mistakenly represents what the sculptor is doing, then the critic is wrong. When mathematical logic represents mathematicians as using = to symbolize identity, the logic is wrong. — Metaphysician Undercover

Ok you didn't like my sculptor analogy.

Fishfry, wake up! Was it getting late there or something? There is no physical object involved! There is no star! I think we've been through this before. The intensional/extensional distinction is completely irrelevant in this case because everything referred to is meaning (intensional). There is nothing extensional, no objects referred to by "1+1", or "2". That is the heart of the sophistic ruse. This intensional/extensional rhetoric falsely persuades mathematicians. It wrongly misleads them due to their tendency to be Platonist, and to think of mathematical abstractions as objects. As soon as meaning is replaced by objects, then "extensional" is validated, the sophist has succeeded in misleading you, and down the misguided route you go. In reality, there is only meaning referred to by "1+1", and by "2", everything here is intensional, and there is nothing extensional. — Metaphysician Undercover

LOL. 1 + 1 and 2 are each representations of the same set in ZF, with "1" and "2" interpreted as defined symbols in the inductive set given by the axiom of infinity; and likewise "+" is formally defined.

But we've been having this conversation for years, and I don't think today is the day for any more.

This is why I was very steadfast on the previous issue, to explain that "5" is not "an instance of a real number". It is that type of nomenclature, that type of understanding, which leads one into allowing that there is a place for extensional definitions in mathematics. Really, "5" in that example is just a part of that conception called "the real numbers". It receives it's meaning as part of that conception. there are no extensional objects referred to by "the real numbers", and "5" is just an intensional aspect of that conception. When you apprehend "the real numbers" as referring to a collection of things, instead of as referring to a conception, then you understand "5" as referring to an instance of a real number, instead of understanding it as a specific part of that conception. Then you may be misled into the "extensionality" of real numbers, instead of understanding "the real numbers" as completely intensional. — Metaphysician Undercover

Well.

I will stipulate that the real number 5 has a relatively shake ontological status. The mathematical real numbers are a very strange gadget, and I genuinely doubt that they are instantiated or exemplified by anything in the physical world. That is, the real world is not a continuum in the sense of being isomorphic to the mathematical real numbers. The real numbers are far too weird to be real.

BUT! Are you telling me that you don't believe in the physical instantiation of the natural number 5? Just look at the fingers on your hand. I rest my case.

Again, you are not distinguishing between "mathematics", and the "mathematical logic" which the head sophist preaches. One is the artist, the other the critic. My beef is not with mathematics (the art), it is with mathematical logic (the critic). I see mathematical logic as sophistry intended to deceive. And I will explain the reason why i say there is an intent to deceive. — Metaphysician Undercover

Mathematical logic. Anyone in particular? Do you go back to Aristotle, or are you annoyed with Russell, or Godel, or what? What is the specific nature of your beef, as you put it.

Mathematics has a long history of exposing us to problems which we just cannot seem to solve. These are issues such as Zeno's paradoxes, and other apparent paradoxes discussed at TPF, which generally amount to problems with the conception of infinity, the continuity of space and time, etc.. — Metaphysician Undercover

Your level of mathematical knowledge is so naive, that you actually think that Zeno's paradox is a mathematical problem; and that the extremely naive conceptions of mathematics exhibited by innumerate philosophers that drive these inane discussions of supertasks, has any relationship whatsoever to the professional activities of mathematicians. You just have no idea. In argument, you wield your ignorance like a club. You have no idea what you are talking about.

What mathematical logic does, is create the illusion that such problems have been solved. So, the intent to deceive is inherent within the conceptual structure, which makes these problems solvable. It deceives mathematicians into thinking that they have solved various problems, by allowing them to work within a structure which makes them solvable. The problem though is that the basic axioms (extensionality for example) are blatantly wrong, and designed specifically so as to make a bunch of problems solvable, regardless of the fact that incorrect axioms are required to make the problems solvable. — Metaphysician Undercover

Instead of addressing these bitter complaints to me, have you thought about going down and picketing the math department at your local university? "Lying corrupt sophists all! Should not drink and derive!" Get some press for sure.

Why me?

Future application is not the issue here. The issue is that mathematicians work toward problem solving, by the very nature of what mathematics is. The problems are preexistent. Therefore mathematics by its very nature is fundamentally "applied". If you remove problem solving from the essence of mathematics, then it would be random fictions. But mathematics is not random fictions, the mathematicians always follow at least some principles of "number", already produced. — Metaphysician Undercover

Yeah ok, all math is applied. I have no strong opinion. It's all an abstraction of the first caveman who made a bijection between marks on the ground and wooly mammoths he killed. I have no problem with that. But actually you have zero idea what pure mathematicians do, and why they are so regarded by their peers. You're just making stuff up that you don't know anything about. You keep saying mathematicians do this and mathematicians do that, and you have repeatedly demonstrated to me that you have no understanding of mathematics nor mathematicians.

What I think, is that there is really no such things as sameness in math, and this is better described as a misleading subject. Mathematics actual deals with difference, and ways of making difference intelligible through number. Similarity is not sameness, but difference which can be quantified. To me, "essential the same" just means similar, which is different.[/quotem

You keep talking about mathematics as if you forget that you're speaking to someone who has been observing for over five years that you don't know anything about mathematics. You are just making up strawman to have an argument that only you care about.

There is quite a lot of mathematical thought about what "sameness" is in math. I'm thinking of the work in Univalent foundations, in which there's a univalent axiom that sort of says that "things that are isomorphic are the same." It's based on intuitionist math and the denial of LEM. It's all the rage in proof assistants and the formalization of math. A lot of philosophically inclined mathematicians have worked n that area.


— Metaphysician Undercover

This appears to be the substance of our difference, or disagreement. If you do not like long posts, we could just focus on this specific issue. The issue is whether "the real numbers" refers to a conceptual structure, or whether it refers to a group of things, numbers. I believe the former, and the fact that "numbers" is plural is just a relic of ancient tradition. From my perspective, "5", in the context of "a real number" is just a specific part of that conception. Then the relations are purely intensional, and there is nothing extensional here. If however, you apprehend "the real numbers" as referring to a group of things called "numbers", then "5" refers to one of those things, and there is the premises required for extensionality. — Metaphysician Undercover

If I'm understanding you, I agree. I don't think the mathematical real numbers refer to anything in the world at all. They describe the idealized continuum, something that we have no evidence can exist.

a mathematician is an explorer trying to find a path extending knowledge in a particular direction or discovering new directions. — jgill

I would say that this is a type of problem solving, wouldn't you? The problem being worked on is not necessarily a practical issue. Philosophy is like this too, as well as speculative theorizing, there is a wide range to the types of problems. Sometimes, problems are being worked on without any obvious practical implications.

I see an out. In this para you have stated your aim about the real numbers and the number 5. I don't think I have any interest in this topic. I know it's important and meaningful to you, but it isn't to me. Perhaps I'm to dim to grasp all these philosophical subtleties such as you raise. If so, so be it.

But secondly, and I'd be remiss if I didn't add, that I have formally studied the real numbers and the number 5. That doesn't make me right and you wrong, by any means. What it does mean is that I'm not likely to ever defer to your opinions about the real numbers or the number 5. — fishfry

Well, "the real numbers", and "5" being an instance of a real number, was your example. I agree that by some accepted principles of mathematics, the axioms of set theory, etc., 5 is an instance of a real number. This I believe to be the influence of Platonism which assumes that a number is an object. I disagree with this, and think that a number is a concept, and conceptions are quite different from objects. The way that one concept relates to another for example is completely different from the way that one object relates to another.

You might think that it doesn't matter whether a number is an object or not. You might think that within the confines of the logical system of "the real numbers", a number can be whatever the mathematician who states the axiom wants it to be. My argument is that numbers are used billions of times a day by human beings, and according to that usage there is some truth and falsity about what a number is. Therefore when an axiom makes a statement about what a number is, and it's not consistent with how numbers are actually used, the axiom can be judged as false.

When it suits my argument. I'm a formalist as well at times. — fishfry

Like I explained earlier, formulism is just a specific type of Platonism. It takes Platonist principles much deeper in an attempt to realize the ideal within the work of human beings, while other Platonists allow the ideal to be separate from human beings.

Mathematical philosophies are tools, nothing more. Conceptual tools, frameworks for thinking about the development and structure of math. They aren't "true" or "false," they're just models, if you will. — fishfry

Do you not look at mathematics, and mathematicians as real human beings, carrying out activities in the real world? If so, then don't you think that there is such a thing as true and false propositions about what those mathematicians are doing? If you follow, and agree so far, then why wouldn't you also agree that mathematical philosophies, as tools, or models, ought to be judged for truth and falsity? If a mathematical philosophy provides false propositions about what mathematicians are doing, offering this philosophy as a tool for understanding the structure and development of math, it is likely to mislead.

Problem solvers and theory builders. The theory builders don't solve problems at all. They create conceptual frameworks in which others can solve problems. — fishfry

As I explained to jgill above, theory building is a form of problem solving, it just involves a different type of problem. There are many different types of problems which can be categorized in different ways.

LOL. 1 + 1 and 2 are each representations of the same set in ZF, with "1" and "2" interpreted as defined symbols in the inductive set given by the axiom of infinity; and likewise "+" is formally defined. — fishfry

Yes, this is the problem, axioms of set theory are false, in the way described above.

BUT! Are you telling me that you don't believe in the physical instantiation of the natural number 5? Just look at the fingers on your hand. I rest my case. — fishfry

I said that 5 is not an instance of a real number. Also, I would say that the fingers on my hand are not an instance of the number 5, they are an instance of a quantity of five. You see, this is the problem of mixing up the ideal with the physical. "The natural number 5" is an ideal, a type of Platonic object called "a number". There is no physical instantiation of numbers, they are by definition ideal. So we need to refer to the use of "5" to see its meaning, and then we can find a physical representation for its meaning. In the context of usage of the natural numbers my understanding is that 5 represents a specific quantity, and the fingers on my hand provide an example of this specific quantity.

If we say that the numeral 5 represents a number, which goes by that name, 5, we have no meaning indicated to assist us in finding a physical example of the number five. All we have is that there is a type of thing called a number, and one of them is named 5. In order for numbers such as 5 to be used in practise, we need to provide something more, otherwise we're stuck with the interaction problem of idealism, these ideal things have no bearing on the real world. But if we give the number 5 further meaning, such as "a specific quantity", to allow it to be useful in the world, then the ideal, the number 5 becomes redundant, and completely useless. Why not just say that the numeral "5" means a specific quantity, and be done with it. Well I'll tell you why not. The numeral "5" is assumed to represent a number, 5, which is an abstract, Platonic object, for another purpose. The other purpose is mathematical philosophy, building structures and frameworks to be used as tools for understanding the development of math. However, as explained above, rather than assisting understanding, it misleads.

Why me? — fishfry

You are free to abandon me anytime you want.

If I'm understanding you, I agree. I don't think the mathematical real numbers refer to anything in the world at all. They describe the idealized continuum, something that we have no evidence can exist. — fishfry

If you truly believe this, then how would you validate your claim that the number 5 is an instance of a real number. Do you see that when you talk about "a real number", and "the real numbers", you validate the claim that "the real numbers" refers to a collection of individual objects? And that is contrary to what you say here. And do you see that in set theory, "numbers" also must refer to individual things, and this is contrary to being a description of "the idealized continuum".

But who said I'm not a Platonist? I am? When it suits my argument. I'm a formalist as well at times. — fishfry

Are the two really mutually exclusive?

Is Hilbertian formalism incompatible with platonism? I'd like to see an argument that it is.

That is one of the questions I ask in my Grundlagenkrise thread :^)

b. The article associates formalism with nominalism, logicism with realism, and intuitionism with conceptualism. The last one seems uncontroversial, but how true are the first two? Couldn’t a logicist also be a nominalist? Why does reduction of mathematics to logical propositions have to imply numbers as abstract objects? — Lionino

So you are asking "couldn't a formalist not be a nominalist?"

I'll try to check it out.

It does, just indirectly.

Right, I caught that a moment later, and edited mine.

So you are asking "couldn't a formalist not be a nominalist?" — TonesInDeepFreeze

That, and also "Couldn't a logicist not be a platonist?".

Right.

I wonder about the categories. The schools could be something like:

realist

logicist

formalist

structuralist

constructivist

Couldn't a logicist not be a platonist? — Lionino

Why not? Maybe if 'logical truth' was regarded as a property of formal semantics? I mean, can't we regard 'logical axiom' as merely a logical notion without ontological commitment?

realist

logicist

formalist

structuralist

constructivist — TonesInDeepFreeze

I think the first and fourth are about the metaphysics of mathematics, while the second, third, and fifth are about the foundations of mathematics, so then they would be grouped in two separate sets. Some of the questions in my thread are also about what arrows we are supposed to draw between those two supposed sets.

I mean, can't we regard 'logical axiom' as merely a logical notion without ontological commitment? — TonesInDeepFreeze

I would imagine so, there is nothing about logical statements to me that imply an ontological commitment. On the other hand, my problem is that I am very unpersuaded by platonism, so I am not keen on spotting flaws in arguments against it.

Yes, I can see a distinction between metaphysics/ontology and foundations. Perhaps though the distinctions can be quite less than sharp.

Perhaps though the distinctions can be quite less than sharp. — TonesInDeepFreeze

Perhaps. The matter of mathematics being invented or discovered is absolutely derivative from the ontology. It is possible that the foundations are also derivative from ontology or vice versa.

Yes. Well put.

I may have messed up the order of some of the paragraphs. I don't think I can hold up my any of this any more. I will bow out now. Thank you for the chat.

I said that 5 is not an instance of a real number. Also, I would say that the fingers on my hand are not an instance of the number 5, they are an instance of a quantity of five. — Metaphysician Undercover

Meaningless word games. The fingers on your hand are a physical instantiation of the number 5. Positive integers have the property that the smaller among them may be physically instantiated. 12 as in a dozen eggs, 9 as in the planets unless an astronomical bureaucracy demotes Pluto. That's one for the philosophers, don't you agree? The number of planets turns out to be a matter of politics, not math or astrophysics.

You see, this is the problem of mixing up the ideal with the physical. "The natural number 5" is an ideal, a type of Platonic object called "a number". There is no physical instantiation of numbers, they are by definition ideal. So we need to refer to the use of "5" to see its meaning, and then we can find a physical representation for its meaning. In the context of usage of the natural numbers my understanding is that 5 represents a specific quantity, and the fingers on my hand provide an example of this specific quantity. — Metaphysician Undercover

Actually some numbers happen to have physical instantiations and some don't. 5 does.

But it's a distinction without a difference. It's something that consumes you but nobody else. Least of all me.

If we say that the numeral 5 represents a number, which goes by that name, 5, we have no meaning indicated to assist us in finding a physical example of the number five. All we have is that there is a type of thing called a number, and one of them is named 5. In order for numbers such as 5 to be used in practise, we need to provide something more, otherwise we're stuck with the interaction problem of idealism, these ideal things have no bearing on the real world. But if we give the number 5 further meaning, such as "a specific quantity", to allow it to be useful in the world, then the ideal, the number 5 becomes redundant, and completely useless. Why not just say that the numeral "5" means a specific quantity, and be done with it. Well I'll tell you why not. The numeral "5" is assumed to represent a number, 5, which is an abstract, Platonic object, for another purpose. The other purpose is mathematical philosophy, building structures and frameworks to be used as tools for understanding the development of math. However, as explained above, rather than assisting understanding, it misleads. — Metaphysician Undercover

Nah. Not buying any of it.

Well, "the real numbers", and "5" being an instance of a real number, was your example. I agree that by some accepted principles of mathematics, the axioms of set theory, etc., 5 is an instance of a real number. This I believe to be the influence of Platonism which assumes that a number is an object. I disagree with this, and think that a number is a concept, and conceptions are quite different from objects. The way that one concept relates to another for example is completely different from the way that one object relates to another.

You might think that it doesn't matter whether a number is an object or not. You might think that within the confines of the logical system of "the real numbers", a number can be whatever the mathematician who states the axiom wants it to be. My argument is that numbers are used billions of times a day by human beings, and according to that usage there is some truth and falsity about what a number is. Therefore when an axiom makes a statement about what a number is, and it's not consistent with how numbers are actually used, the axiom can be judged as false. — Metaphysician Undercover

Why do you think I take a position on any of this?

Like I explained earlier, formulism is just a specific type of Platonism. It takes Platonist principles much deeper in an attempt to realize the ideal within the work of human beings, while other Platonists allow the ideal to be separate from human beings. — Metaphysician Undercover

The swine.

Do you not look at mathematics, and mathematicians as real human beings, carrying out activities in the real world? If so, then don't you think that there is such a thing as true and false propositions about what those mathematicians are doing? If you follow, and agree so far, then why wouldn't you also agree that mathematical philosophies, as tools, or models, ought to be judged for truth and falsity? If a mathematical philosophy provides false propositions about what mathematicians are doing, offering this philosophy as a tool for understanding the structure and development of math, it is likely to mislead. — Metaphysician Undercover

Judged by who? Politicians? Academic administrators? Philosophers? How about by their fellow mathematicians? That's the standard of what counts as math.

As I explained to jgill above, theory building is a form of problem solving, it just involves a different type of problem. There are many different types of problems which can be categorized in different ways. — Metaphysician Undercover

Just change the meanings of words to suit your argument. Pigs fly, if I redefine pigs as birds.

Yes, this is the problem, axioms of set theory are false, in the way described above. — Metaphysician Undercover

False. Not just "not true," or "lacking a truth value," but literally false. If they can be false then they can also be true.

I said that 5 is not an instance of a real number. Also, I would say that the fingers on my hand are not an instance of the number 5, they are an instance of a quantity of five. You see, this is the problem of mixing up the ideal with the physical. "The natural number 5" is an ideal, a type of Platonic object called "a number". There is no physical instantiation of numbers, they are by definition ideal. So we need to refer to the use of "5" to see its meaning, and then we can find a physical representation for its meaning. In the context of usage of the natural numbers my understanding is that 5 represents a specific quantity, and the fingers on my hand provide an example of this specific quantity. — Metaphysician Undercover

Meaningless word games. The fingers on your hand are a physical instantiation of the number 5. Positive integers have the property that the smaller among them may be physically instantiated. 12 as in a dozen eggs, 9 as in the planets unless an astronomical bureaucracy demotes Pluto. That's one for the philosophers, don't you agree? The number of planets turns out to be a matter of politics, not math or astrophysics.

If we say that the numeral 5 represents a number, which goes by that name, 5, we have no meaning indicated to assist us in finding a physical example of the number five. All we have is that there is a type of thing called a number, and one of them is named 5. In order for numbers such as 5 to be used in practise, we need to provide something more, otherwise we're stuck with the interaction problem of idealism, these ideal things have no bearing on the real world. — Metaphysician Undercover

I can establish a bijection between the fingers on one hand, and the elements of the set {0, 1, 2, 3, 4}. Done.

But if we give the number 5 further meaning, such as "a specific quantity", to allow it to be useful in the world, then the ideal, the number 5 becomes redundant, and completely useless. Why not just say that the numeral "5" means a specific quantity, and be done with it. Well I'll tell you why not. The numeral "5" is assumed to represent a number, 5, which is an abstract, Platonic object, for another purpose. The other purpose is mathematical philosophy, building structures and frameworks to be used as tools for understanding the development of math. However, as explained above, rather than assisting understanding, it misleads. — Metaphysician Undercover

Because sometimes it's useful to study things in the abstract; just as one guy goes fishing, and another studies ichthyology

You are free to abandon me anytime you want. — Metaphysician Undercover

Thank you. I should avail myself of that option at the end of this post. I think I will.

If you truly believe this, then how would you validate your claim that the number 5 is an instance of a real number. Do you see that when you talk about "a real number", and "the real numbers", you validate the claim that "the real numbers" refers to a collection of individual objects? And that is contrary to what you say here. And do you see that in set theory, "numbers" also must refer to individual things, and this is contrary to being a description of "the idealized continuum". — Metaphysician Undercover

It depends on how you look at them.

Well, "the real numbers", and "5" being an instance of a real number, was your example. I agree that by some accepted principles of mathematics, the axioms of set theory, etc., 5 is an instance of a real number. This I believe to be the influence of Platonism which assumes that a number is an object. I disagree with this, and think that a number is a concept, and conceptions are quite different from objects. The way that one concept relates to another for example is completely different from the way that one object relates to another.

You might think that it doesn't matter whether a number is an object or not. You might think that within the confines of the logical system of "the real numbers", a number can be whatever the mathematician who states the axiom wants it to be. My argument is that numbers are used billions of times a day by human beings, and according to that usage there is some truth and falsity about what a number is. Therefore when an axiom makes a statement about what a number is, and it's not consistent with how numbers are actually used, the axiom can be judged as false. — Metaphysician Undercover

Suppose math is entirely fraudulent. Would it then be any less useful in the world? Would you fire all the professors? What is your suggested remedy for all these academic crimes?

Do you not look at mathematics, and mathematicians as real human beings, carrying out activities in the real world? If so, then don't you think that there is such a thing as true and false propositions about what those mathematicians are doing? If you follow, and agree so far, then why wouldn't you also agree that mathematical philosophies, as tools, or models, ought to be judged for truth and falsity? If a mathematical philosophy provides false propositions about what mathematicians are doing, offering this philosophy as a tool for understanding the structure and development of math, it is likely to mislead. — Metaphysician Undercover

You are free to judge things as you wish.

You might think that it doesn't matter whether a number is an object or not. You might think that within the confines of the logical system of "the real numbers", a number can be whatever the mathematician who states the axiom wants it to be. My argument is that numbers are used billions of times a day by human beings, and according to that usage there is some truth and falsity about what a number is. Therefore when an axiom makes a statement about what a number is, and it's not consistent with how numbers are actually used, the axiom can be judged as false. — Metaphysician Undercover

They're meta-false, as I understand you. They're not literally false. If the powerset axiom is false, you get set theory without powersets. You don't get some kind of philosophical contradiction. You are equivocating levels.

If you truly believe this, then how would you validate your claim that the number 5 is an instance of a real number. Do you see that when you talk about "a real number", and "the real numbers", you validate the claim that "the real numbers" refers to a collection of individual objects? And that is contrary to what you say here. And do you see that in set theory, "numbers" also must refer to individual things, and this is contrary to being a description of "the idealized continuum". — Metaphysician Undercover

A model, not a description. Is that better?

But who said I'm not a Platonist? I am? When it suits my argument. I'm a formalist as well at times.
— fishfry

Are the two really mutually exclusive? — Lionino

That's funny. @Metaphysician Undercover just told me that formalism is just a deeper kind of Platonism. Which actually makes sense in this context. It explains how one can be both.

That doesn't make sense automatically because formalism is a program for foundations, platonism is an ontological claim. And idk what post of MU it is.

Meaningless word games. The fingers on your hand are a physical instantiation of the number 5. Positive integers have the property that the smaller among them may be physically instantiated. 12 as in a dozen eggs, 9 as in the planets unless an astronomical bureaucracy demotes Pluto. That's one for the philosophers, don't you agree? The number of planets turns out to be a matter of politics, not math or astrophysics. — fishfry

I don't see what this all has to do with your claim that a concept like a number, 5, could have a physical instantiation . Fingers are fingers, and are therefore physical instantiations of fingers, not of numbers, not matter how many of them you have. Wittgenstein took up this issue in the Philosophical Investigations, showing why there is a lot more involved with learning a language than simple ostensive definition. Abstraction is very complex, and with complex concepts like number, an explanation of what it is about the thing which is being shown, which is being referred to with the word, is a requirement.

A person cannot simply look at the fingers on a hand and apprehend the concept 5. An explanation about quantity, or counting is required. The concept 5 is learned from the explanation, not from the ostensive hand, therefore the hand is not a physical instantiation of the number.

Judged by who? Politicians? Academic administrators? Philosophers? How about by their fellow mathematicians? That's the standard of what counts as math. — fishfry

It can be judged by anyone. The issue though, is that many, like yourself refuse to make such a judgement. You say that there is no truth or falsity to mathematical axioms, they are simply tools which cannot be judged for truth. Since mathematicians tend to think this way, they are not well suited for judging truth or falsity of their axioms. But I've shown how axioms can be judged for truth. If an axiom defines a word or symbol in a way which is inconsistent with the way that the symbol is used, then it is a false axiom.

So for example, if a mathematical axiom defines "=" as meaning "the same as", yet in applied mathematics the mathematicians use "=" to mean "has the same value as", then the axiom makes a false definition. This axiom will be misleading to any "pure mathematician" who uses it to produce a further conceptual structure with that axiom at the base, just like if anyone else working in speculative theories in other fields of science starts from a false premise. False propositions are fascinating, sometimes leading to theories which are extremely useful, because they are designed for the purpose at hand.

They're meta-false, as I understand you. They're not literally false. If the powerset axiom is false, you get set theory without powersets. You don't get some kind of philosophical contradiction. You are equivocating levels. — fishfry

Sorry, I don't understand what you mean by "meta-false". I am talking about "literally false". False to me, means not corresponding with reality. For example, if someone says that in the use of mathematics, "=" indicates "the same as", but in reality, when mathematicians use equations, "=" means "has the same value as", then the person who said that "=" indicates "the same as" has spoken a falsity. Do you agree that this would be an instance of "literally false"?

A model, not a description. Is that better? — fishfry

That doesn't help. Numbers form discrete units, and discrete units cannot model an idealized continuum. There is an inconsistency between these two, demonstrated by those philosophers who argue that no matter how many non-dimensional points you put together, you'll never get a line. The real numbers mark non-dimensional points, the continuum is a line. The two are incompatible.

That doesn't make sense automatically because formalism is a program for foundations, platonism is an ontological claim. And idk what post of MU it is. — Lionino

Ontological assumptions are what foundations are made of, and Platonism provides the assumptions required for formalism, the idea of pure form.

Ontological assumptions are what foundations are made of — Metaphysician Undercover

Some would disagree. But it is quite possible.

and Platonism provides the assumptions required for formalism, the idea of pure form. — Metaphysician Undercover

That is not true for every formalist. If you want to know why, look it up.

That is not true for every formalist. If you want to know why, look it up. — Lionino

I believe it is required to validate any formalist approach. If you think otherwise maybe you could explain.

There are formalists who are in for it exactly for the anti-platonist element.

Then I would say that they misunderstand the foundations of the principles they believe in.