The empty set comes quite naturally from two principles:
(1) The ability to state what elements sets have in common.
(1a) The elements that sets have in common must always be equal to a set.
(2) That two sets might be disjoint. — fdrake

(1a) would be better stated as, "if" two sets have elements in common this must be a set. There is no reason to interpret a situation where two sets have nothing in common, as meaning that this nothing ought to be a set. That is arbitrary, and actually illogical. If two sets have nothing in common, then why must this nothing be a set? That's nonsense, by the described situation, they have "nothing" in common. Why try to make nothing into something (a set)?

Maybe - maybe - we can sharpen this with an example. What is a warehouse? If you hold that a warehouse is a place where goods are stored, then there can be no such thing as an empty warehouse, and if you say there is, then I reject the entire idea of warehousing. — tim wood

A set is not analogous with a warehouse where things are stored, or any such container, because there is no separation between the set and its elements. Unlike a warehouse which has an identity as a warehouse, with the potential to store things regardless of what is actually stored within it, a set is identified solely by its elements, and is therefore inseparable from its elements. That's why an empty set is contradictory nonsense.

but you each end up saying that math itself is flawed therefore there's no empty set. There must be a name for such an argument. You want to argue a very narrow technical point and your only argument is to blow up the entire enterprise. — fishfry

Don't cast this in the wrong light. I don't say "math itself is flawed therefore there's no empty set". I say "there's no empty set therefore math is flawed". I point out the specific flaws to justify the more general claim, that math is flawed. So it's not the case that I am arguing that math by its very nature is inherently flawed (what you call nihilism), I think the very opposite, that math by its very nature is perfect, "ideal". And, because it has this status of being ideal, we must hold it to the highest standards of perfection. Therefore we are obliged to reject any imperfections.

Math precedes foundations. Not the other way 'round. — fishfry

How could you conceive of this?

I came to this forum about 6 months ago because I wanted to present some ideas that in my opinion are original but are not completely formalized (until now, maybe they will be one day). I discovered that nobody understood what I meant. So probably that's the wrong approach. I am not interested in discussing ZFC. I don't know ZFC and I am not really interested in studying it.

I think I'll have to complete my work in a more formal way and present it to a math forum.

Sorry for bringing trouble. You can continue without me.

I came to this forum about 6 months ago because I wanted to present some ideas that in my opinion are original but are not completely formalized (until now, maybe they will be one day). I discovered that nobody understood what I meant. So probably that's the wrong approach. I am not interested in discussing ZFC. I don't know ZFC and I am not really interested in studying it.

I think I'll have to complete my work in a more formal way and present it to a math forum.

Sorry for bringing trouble. You can continue without me. — Mephist

Ah ... was it something I said? I have no sense of having suppressed any thoughts you may have. I had no idea you were presenting anything original. If I somehow crushed your creativity I apologize. Is that what I did?

Just for my understanding, can you point to a post where you presented your original ideas?

Surely you may have noticed that you can write the most simple, commonly understood things here and have many people not understand you. Happens a lot to me.

I think I'll have to complete my work in a more formal way and present it to a math forum. — Mephist

Just so I have some idea what this is about ... can you just summarize your idea? I can't for the life of me remember any interaction where you presented an original idea and I crushed your spirit. Truly sorry if anything came across that way.

ps -- I've reviewed your earliest posts from 9 months ago. I see no evidence that you ever put forth an original idea and got shot down or discouraged by anyone. What the heck is this about? I simply can find no evidence of your assertion.

Many people who come to math through foundations believe math is about the foundations. It's the other way 'round. Mathematics comes first and foundations are just our halting and historically contingent attempts to formalize accepted mathematical practice. — fishfry

The example of synthetic differential geometry was given to show that the point of alternative foundations is to shed light on problems, not to brag about which foundation is more fundamental. — fishfry

I am just pulling these out to highlight them as excellent observations. After all, Peirce--the founder of pragmatism--was decidedly non-foundationalist in his own philosophical system. SDG/SIA captures certain aspects of true continuity that modern analysis using the standard real numbers does not.

I am just pulling these out to highlight them as excellent observations. After all, Peirce--the founder of pragmatism--was decidedly non-foundationalist in his own philosophical system. SDG/SIA captures certain aspects of true continuity that modern analysis using the standard real numbers does not. — aletheist

Tell me something of pragmatism. Let me say first where I'm coming from.

I've read Maddy's great papers Believing the Axioms part I and part II. These papers provide a historical overview of how and why the various axioms of set theory came to be adopted; along with a number of pragmatic (in the everyday sense of the word) criteria for deciding whether to adopt an axiom. For example one principle is Maximize, which says that we choose the axioms that give us the richest theory.

I'll give an example. The axiom of choice (AC) may be taken or denied with equal logical consistency. Both AC and its negation are consistent with ZF.

Why do mathematicians simply accept AC? I'm not talking about specialists who investigate the consequences of the negation of AC, but rather everyday working mathematicians who never give a thought to foundations. They're taught in grad school to freely apply Zorn's lemma, the Hahn-Banach theorem, and other applications of AC in their work, and they do so as a matter of course.

Philosophically, we accept AC because it gives us more and better theorems. That is the reason. This is a very pragmatic (in the everyday sense, not necessarily the technical sense) way to view the axioms. We adopt the axioms that give us a good theory.

Is that Peircean pragmatism? Or what exactly is it, given what I already know about the practical or pragmatic (everyday sense) reasons to adopt or reject axioms?

Math precedes foundations. Not the other way 'round.
— fishfry

How could you conceive of this? — Metaphysician Undercover

Knowledge of the history of math.

But it's the same in any discipline. There's science, and then there's the philosophy of science. One can and does do science without regard for its philosophy. That's true in every field. X precedes the philosophy of X.

I say "there's no empty set therefore math is flawed". — Metaphysician Undercover

Yes I understand this. I get that you reject math and, when pressed, physical science as well. I'm happy for you. I can't take such a point of view seriously. The task of the philosopher is to explain how it comes to be that math and science are abstract yet useful. You can't solve the problem by rejecting math and science; not unless you live in a cave. Without Internet access. Even then you'd draw a square in the sand and eventually discover the square root of two; and from that, abstract algebra and group theory and modern physics. You haven't got a serious philosophy, just sophistry.

We adopt the axioms that give us a good theory. Is that Peircean pragmatism? — fishfry

As applied to mathematics, yes. Charles Peirce adopted his father Benjamin's definition of it as the science that draws necessary conclusions about hypothetical states of things. There are no restrictions on conceiving such hypothetical states, other than that they be suited to the purposes for which we wish to analyze them. Applied mathematics seeks to formulate hypothetical states that resemble reality in certain significant ways, but pure mathematics does not have that particular limitation.

As applied to mathematics, yes. Charles Peirce adopted his father Benjamin's definition of it as the science that draws necessary conclusions about hypothetical states of things. — aletheist

Ok. Not too far from Bertie's quip that math is the subject where we never know what we're talking about or if what we say is true.

There are no restrictions on conceiving such hypothetical states, other than that they be suited to the purposes for which we wish to analyze them. Applied mathematics seeks to formulate hypothetical states that resemble reality in certain significant ways, but pure mathematics does not have that particular limitation. — aletheist

Ok I perfectly well agree with that. Completely. Have explained it to many people on many forums over the years. May be coming to understand it myself!

But now Maddy's pragmatism (in the everyday sense) goes further. According to Peirce, we can accept or reject AC at will, which of course we can. But that does not explain why mainstream math accepts it and never gives it a second thought. For that you need Maddy's Maximize principle, which says that assuming AC gives a better or richer theory. I'm paraphrasing my understanding of Maddy and didn't double-check her article so I might be mangling her ideas a bit but I think I'm in the ballpark.

Point being that Maddy says that given two equally logically consistent but mutually inconsistent axioms; we adopt the one that's more fun, more interesting, gives more good theorems. That's a pragmatic justification for choosing AC over not-AC.

Is that an example of Peircean pragmatism or is that an expression of something else?

Also ... given that I'm perfectly in agreement with Peirce's point of view here, and Maddy's as well; can you help me to understand @Metaphysician Undercover's point of view? He rejects the idea of taking math on its own terms; insists that it must refer to something outside of itself. Nobody believes that, not about pure math. Why does @Metaphysician Undercover believe that? What is the basis for his ideas? Do they have a name? Is there a reference? I asked him this once and did not get a satisfactory answer. I can't tell if he is asserting the ideas of a particular school of thought, or just venting over a bad experience with his screechy third-grade math teacher.

Is that an example of Peircean pragmatism or is that an expression of something else? — fishfry

It is consistent with Peircean pragmatism in the sense that one's purpose dictates how one formulates the hypothetical state of things to be explicated--in this case, the particular set of axioms to adopt.

... can you help me to understand Metaphysician Undercover's point of view? — fishfry

I can try. As I noted several times a few weeks ago, MU employs a rigid metaphysical terminology and seeks to impose it on everyone else. "Existence" has only one meaning, and that is its ontological definition; so asserting mathematical existence is, according to MU, asserting some kind of ontological existence--no matter how many times and in how many ways we explain that this is not what anyone actually means by mathematical existence. More broadly, my impression is that MU--as the name suggests--is a philosophical foundationalist who begins with metaphysics, deriving everything else from that. Accordingly, MU has been quite dismissive of pragmatism on various occasions.

By contrast, as I said before, Peirce was a philosophical non-foundationalist, and he had harsh words for the dogmatic metaphysicians of his day. He classified the sciences in accordance with their nature and purpose, such that the more basic ones furnish principles to those above them. Mathematics comes first, because its subject matter consists entirely of hypotheses that may or may not have any basis in reality. Every other science thus relies on mathematics to some extent, not as a foundation but as a necessary tool. The first positive science is phenomenology, which deals only with what appears to the mind and identifies three irreducible elements--quality, reaction, and mediation. Then come the normative sciences of esthetics (feeling), ethics (action), and logic (thought), although Peirce generalized the last of these to semeiotic--the theory of all kinds of signs, not just symbols. Only then--after we have established the proper method for discerning truth from falsehood--do we reach metaphysics, the science of reality.

Metaphysics consists in the results of the absolute acceptance of logical principles not merely as regulatively valid, but as truths of being ... Just as the logical verb with its signification reappears in metaphysics as a quality, an ens having a nature as its mode of being, and as a logical individual subject reappears in metaphysics as a thing, an ens having existence as its mode of being, so the logical reason, or premiss, reappears in metaphysics as a reason, an ens having a reality, consisting in a ruling both of the outward and of the inward world, as its mode of being. The being of the quality lies wholly in itself, the being of the thing lies in opposition to other things, the being of the reason lies in its bringing qualities and things together. — Peirce, c. 1896

Knowledge of the history of math.

But it's the same in any discipline. There's science, and then there's the philosophy of science. One can and does do science without regard for its philosophy. That's true in every field. X precedes the philosophy of X. — fishfry

The claim was that math precedes foundations. But math consists of concepts, and concepts require foundations, so it is impossible that math is prior to foundations. It should be obvious to you, a structure is built on foundations, not vise versa. The fact that the philosophy of math follows math is irrelevant, because it is not the philosophy of math which produces the foundations. The philosophy of math might study the foundations, but it does not produce the foundations.

The task of the philosopher is to explain how it comes to be that math and science are abstract yet useful. — fishfry

You don't seem to grasp the problem. "Usefulness" is judged in relation to a goal or end, and that goal or end may be contrary to truth. When principles are used in a way which is contrary to revealing the truth (i.e. hiding the truth), this is deception. The philosopher seeks truth, so the task of the philosopher is to prevent such deception. Principles of math, axioms, might be useful for obtaining truth, or they might be useful for deception, if they are produced without any goals whatsoever. Since the philosopher seeks truth, the mathematical principles which are not useful for revealing truth, and are only useful for hiding the truth,(i.e. deception), ought to be rejected.

You seem to be under the illusion that if mathematical principles are useful they ought to be accepted by the philosopher, simply because they are useful. But you misunderstand philosophy, which seeks to distinguish truth from mere usefulness. The philosopher seeks to know, without regard for the usefulness of that knowledge, knowing for the sake of knowing.

He rejects the idea of taking math on its own terms; insists that it must refer to something outside of itself. Nobody believes that, not about pure math. Why does Metaphysician Undercover believe that? What is the basis for his ideas? — fishfry

Look, just above, you refer to the "usefulness" of math. "Usefulness" of something is determined by relating that thing to something outside itself. That's what usefulness is, it's putting the thing, as a tool, toward a further end, something outside itself. So, if philosophy to you, is explaining how something like mathematical abstractions might be "useful", you need to apprehend how mathematical abstractions relate to things outside of mathematics. This is what it means to be useful. If you assert that no one believes mathematics ought to relate to anything outside of mathematics, then all you are doing is denying that mathematics ought to be useful. Then your philosophical dilemma, of how it is that math is useful, is self-imposed by your faulty belief, that math does not relate to anything outside itself. By this belief mathematical abstractions cannot be useful because usefulness requires relations to other things..

However, true philosophers take for granted that math is useful, because the evidence is everywhere. Therefore we take for granted that mathematical abstractions relate to things outside themselves. Then we can proceed to analyze these relations, instead of denying that there is such relations and wondering how it is that math can be useful when it doesn't relate to anything outside itself.

It should be obvious to you, a structure is built on foundations, not vise versa. — Metaphysician Undercover

I happen to be a practicing structural engineer. What might not be obvious to you and others is that a structure is always designed first, followed by its foundations. We have to establish what needs to be supported before we can go about determining how it will be supported. Often there are multiple options--footings on native soil, footings on special fill materials, footings on improved ground, driven steel piles, driven precast concrete piles, auger-cast grout piles, drilled shafts filled with cast-in-place concrete, etc. Typically what we end up selecting is either the least expensive solution or the one that in our judgment properly balances cost with risk. In other words, the "best" foundation can only be determined relative to a specific purpose. Why would philosophy be any different?

The philosophy of math might study the foundations, but it does not produce the foundations. — Metaphysician Undercover

A philosophy of mathematics identifies the foundations that have already been established by the practice of mathematics. Most people who use mathematics, including most professional mathematicians, have little or no knowledge of its foundations--and little or no need to have such knowledge. Likewise, most people in general have little or no knowledge of philosophy in general--and little or no need to have such knowledge.

But you misunderstand philosophy, which seeks to distinguish truth from mere usefulness. — Metaphysician Undercover

Some philosophers define truth as mere usefulness. I am not one of them, and neither was Peirce, although many of them call themselves "pragmatists"--one reason why he eventually started calling himself a pragmaticist instead.

"Usefulness" is judged in relation to a goal or end, and that goal or end may be contrary to truth ... The philosopher seeks to know, without regard for the usefulness of that knowledge, knowing for the sake of knowing. — Metaphysician Undercover

In other words, philosophy is useful in relation to the goal or end of knowing for the sake of knowing. Well, so is pure mathematics in relation to the goal or end of knowing what follows necessarily from certain axioms, purely for the sake of knowing it.

Typically what we end up selecting is either the least expensive solution or the one that in our judgment properly balances cost with risk. — aletheist

"Risk" here means the possibility of a law suit against the engineering firm. Real risk contains unknown factors which cannot be assessed. What we really end up with "typically", is an extremely expensive building designed to protect the engineer from lawsuits. The exorbitant expense is not balanced by the building's owner being protected from risk, because the real risk, the unknown, has not been assessed.

In other words, the "best" foundation can only be determined relative to a specific purpose. Why would philosophy be any different? — aletheist

As I said already, the purpose of philosophy is truth. I don't try to disguise the fact that this is a "specific purpose". But if we justify the use of false principles on the basis that false principles may still be useful for some other purpose, this particular usefulness would be inconsistent with the specific purpose of philosophy, being truth. Therefore they would be rejected by good philosophers..

A philosophy of mathematics identifies the foundations that have already been established by the practice of mathematics. Most people who use mathematics, including most professional mathematicians, have little or no knowledge of its foundations--and little or no need to have such knowledge. Likewise, most people in general have little or no knowledge of philosophy in general--and little or no need to have such knowledge. — aletheist

That people have little knowledge of the foundations of mathematics, is not an argument supportive of fishfry's claim that mathematics precedes its foundations. As you explain, the design is prior to the structure. If people move into houses, having no knowledge of the design or blueprints, or even that there were blueprints, this does not mean that the house is prior to the design.

The practise of mathematics itself, requires that a foundation already be laid in order for that practise to occur. These are the theoretical principles employed in the practise. So it is a mistake to say that the practise produces the foundation. In reality, the theory which provides the principles for practise, is the foundation.

In other words, philosophy is useful in relation to the goal or end of knowing for the sake of knowing. Well, so is pure mathematics in relation to the goal or end of knowing what follows necessarily from certain axioms, purely for the sake of knowing it. — aletheist

Do you see what happens when you qualify "knowing" with "what follows necessarily from certain axioms"? The goal of philosophy is an unqualified sense of knowing, and truth is implied by "knowing". If the axioms involved are false, then what follows from them cannot be sound conclusions. Therefore we cannot even call this a form of "knowing". This problem becomes very evident in the use of counterfactuals in logic. Even when we recognize the counterfactual as false, it is doubtful whether the use of counterfactuals provides us with any real knowledge.

What we really end up with "typically", is an extremely expensive building designed to protect the engineer from lawsuits. — Metaphysician Undercover

Thanks for confirming that you have no idea how building design and construction actually work.

The practise of mathematics itself, requires that a foundation already be laid in order for that practise to occur. — Metaphysician Undercover

History demonstrates otherwise, as @fishfry has pointed out.

If the axioms involved are false, then what follows from them cannot be sound conclusions. — Metaphysician Undercover

The axioms of pure mathematics are neither true nor false, and claiming otherwise is a category mistake. If a certain theorem follows necessarily from a particular system of axioms, then it is true within that system.

History demonstrates otherwise, as fishfry has pointed out. — aletheist

Actually history has not demonstrated that. You and Fishfry have simply made the assertion that history has demonstrated this. I demonstrated how that assertion is illogical..

The axioms of pure mathematics are neither true nor false, and claiming otherwise is a category mistake. — aletheist

Another assertion without justification.

One assertion after the other, without explanation or justification, together with an ignorance of logic; you just reminded me of how difficult it is to hold a discussion with you.

One assertion after the other, without explanation or justification, together with an ignorance of logic; you just reminded me of how difficult it is to hold a discussion with you. — Metaphysician Undercover

LOL! Right back at you.

The practise of mathematics itself, requires that a foundation already be laid in order for that practise to occur — Metaphysician Undercover

Assuming that by "foundations" you mean the acceptable accumulated knowledge and practices up to any particular moment of mathematical history. Weierstrass and Cauchy laid the critical foundations for my interests, above and beyond what came before.

Formal foundations, such as PAs and ZFCs are another matter. Apart from certain mathematical specialties, they are dispensable. Others might differ. It's a philosophy thread. :cool:

Assuming that by "foundations" you mean the acceptable accumulated knowledge and practices up to any particular moment of mathematical history. Weierstrass and Cauchy laid the critical foundations for my interests, above and beyond what came before. — jgill

I think "foundation" refers to principles which "the acceptable accumulated knowledge and practises" are based in, built upon. So Weiersfass and Cauchy produced some new principles and also built upon some existing principles, and this would be the foundation. Notice that no conceptual structure is completely new, in an absolute sense, it's always built with some already existing principles. Because of this, there is an appearance of an infinite regress in the creation of such principles, y, as a principle, was created using x, but x was created using w, and so on.

The infinite regress of dependency was avoided by the Pythagoreans by assuming that the principles are eternal, have always existed (platonic realism), and are simply discovered rather than created. But since these ideas or Forms are necessarily dependent on a mind for their existence, this conceptual structure is closely tied to the idea of an eternal soul. So Plato describes the discovery of the principles as remembering what is already known (Platonic theory of recollection, "Meno").

From the principles of modern science, we reject the eternal soul, along with the eternal Forms, and Plato's theory of recollection. This puts us back toward an ontology of mathematical principles which describes them as actively evolving. The problem though, as described above, is the appearance of infinite regress, because there must always be something which supports the changing structure, the underlying kernel (what I call the foundation). So when a new type of structure is created, such as in your example of what interests you, it is not a completely new creation, it is still supported by underlying principles which have already been tested by practise.

If we follow this type of evolutionary theory, rather than platonic realism, we might consider that there was a first mathematical idea created. We see that other animals don't use mathematics, and human beings evolved from other animals, so there must have been a time when mathematical ideas were first thought up. It might be a first group of ideas which we could call the first mathematical ideas. So at some time, mathematical ideas were thought up for the first time. Would you agree that these first mathematical principles would also have some underlying principles, being not mathematical in nature, but already tested by practise, and these non-mathematical principles would support the first mathematical ideas? We could say that these principles are the foundation for mathematics.

I can try. As I noted several times a few weeks ago, MU employs a rigid metaphysical terminology and seeks to impose it on everyone else — aletheist

Thanks for your detailed response. It helped put a lot of things into context. So MU is an anti-Peircean. That actually helps me understand Peirce. Sounds like I'm a Peircean and never knew it.

Notice that no conceptual structure is completely new, in an absolute sense, it's always built with some already existing principles. — Metaphysician Undercover

My late advisor would say that there is nothing really new in mathematics. I would disagree, but to some extent much is preordained by the past. There has long been an ongoing discussion as to whether mathematics is discovered or created. I think it is both.

" we might consider that there was a first mathematical idea created"

Counting fingers and toes by some means, perhaps. Think how much of the modern world flows from that. :chin:

So MU is an anti-Peircean. That actually helps me understand Peirce. Sounds like I'm a Peircean and never knew it. — fishfry

That's not a logical conclusion.

Furthermore, I actually agree with a lot that Peirce has said. in particular his very coherent method of laying out particular epistemological problems. What I disagree with is the metaphysics he proposes to resolve those problems. He provides no real solution, only the illusion of a solution. Aletheist resorts to contradiction in an effort to support Peirce's illusory solutions, because Peirce draws on dialectical materialism, or dialetheist principles which decline ruling out contradiction. Aletheist refuses to acknowledge this.

Furthermore, I actually agree with a lot that Peirce has said. — Metaphysician Undercover

How would you know? You have not demonstrated any familiarity with his voluminous writings.

Aletheist resorts to contradiction in an effort to support Peirce's illusory solutions ... — Metaphysician Undercover

Nonsense. As usual, bare assertion with no basis in fact.

... because Peirce draws on dialectical materialism, or dialetheist principles which decline ruling out contradiction. Aletheist refuses to acknowledge this. — Metaphysician Undercover

I refuse to acknowledge this because it is blatantly false.

That's not a logical conclusion.

Furthermore, I actually agree with a lot that Peirce has said. in particular his very coherent method of laying out particular epistemological problems. What I disagree with is the metaphysics he proposes to resolve those problems. He provides no real solution, only the illusion of a solution. Aletheist resorts to contradiction in an effort to support Peirce's illusory solutions, because Peirce draws on dialectical materialism, or dialetheist principles which decline ruling out contradiction. Aletheist refuses to acknowledge this. — Metaphysician Undercover

I'm surely unqualified to discuss Peirce. But his view on math as outlined by @aletheist appears to track mine. Mathematical existence is pragmatic. Useful/necessary for our theory. Agreed on by a preponderance of professionals.

I want to highlight the references in the linked post as being excellent. They give you a real sense for what concepts people were wrestling with at the dawn of set theory.