The actual meaning of mathematical existence is that it's whatever working professional mathematicians say it is. You don't accept that, but that is how it works. — fishfry

I hope you see the problem with this. You're saying, if we (mathematicians) agree that it exists then it exists, without any definition of what it means to exist. In any other field, no one would agree that such and such "exists", unless there was a definition of "exists" and some evidence to show that the thing actually exists. For example, would some biologist come in with a fictitious life form and ask the other biologists, can we agree that this life form exists, so that it can be a real existent life form? Or would a physicist propose the existence of a fictitious particle?

What I think is that "existence" is just a facilitator. The mathematicians realize that if they posit the existence of these things, they can treat them logically as we would treat objects, and this makes things much easier. A symbol represents an object, nice and simple. The problem though is that we can't really treat these things like objects. So the mathematicians have created a wall of illusion which separates them from reality. And now, they do not even know how to properly deal with these things which they have assumed to be objects, because they have spent so long wrongly assuming that they are objects, that they have no understanding of what they really are any more.

You have the same objection to football, baseball, Chinese checkers, and whist? You reject playing poker because the only Queen you know is Elizabeth? Nihilism. Childish rejection of the very concept of abstraction. — fishfry

What I reject, is not the concept of abstraction, but the childish notion that an abstraction is an existing object

What I reject, is not the concept of abstraction, but the childish notion that an abstraction is an existing object — Metaphysician Undercover

And what is completely absurd is an otherwise educated individual who cannot grasp that there are terms of art, and what they mean as terms of art is just what the people who use them as such say they mean, period. And besides, take care for your qualification: have you never heard of instantiation? For example, I have a chair. By your standards, that's incoherent because "chair" is an abstraction....

nd what is completely absurd is an otherwise educated individual who cannot grasp that there are terms of art, and what they mean as terms of art is just what the people who use them as such say they mean, period. — tim wood

Trying to change the subject to distract from the fact that you're wrong? If so, you're unsuccessful because you're still wrong. The critics determine the value and meaning of the piece of art, not the artist.

For example, I have a chair. By your standards, that's incoherent because "chair" is an abstraction.... — tim wood

It seems like you misunderstand "instantiation". From Wikipedia: "The instantiation principle, the idea that in order for a property to exist, it must be had by some object or substance; the instance being a specific object rather than the idea of it".

"I have a chair" is not incoherent, but we need to respect the fact that it may be false. You might not really have a chair when you say this, and then you would be deceiving us. What fishfry has finally started to realize above, this principle: "if mathematicians say it exists then it exists", is a faulty principle.

Not changing the subject. You say the mathematicians' empty set doesn't exist, and that's just ignorant nonsense. It may well not exist within your understanding, but that is just plain irrelevant.

But you're positively Trumpian in your manipulations in keeping your view "up." I have not changed the subject. I am not wrong. I am not "still" wrong. And critics do not determine the value and meaning.... They may determine a or some aspect or estimate of value or meaning.

Nor did I mention any "instantiation principle." I used a nominal form of the verb instantiate. Instantiate: verb, represent as or by an instance.

And the idea that my claim about a chair might be factually false is also completely irrelevant. Misdirection, deflection, deliberate misreading, straw men, red herring. Why?

I agree with you. If I go to the store to get some beer and chips, I'm not going to find a display of empty sets on sale next to the sllm-jims. They do not exist in that sense. On the other hand, the ideas/concepts represented under the name "empty set" certainly do exist. They're functional and purposeful. So also is my imaginary hippopotamus friend: he exists too, but also not at the store.

But

What I reject, is not the concept of abstraction, but the childish notion that an abstraction is an existing object — Metaphysician Undercover

. What do you say of a chair? "Chair" is certainly an abstract noun. Actually, all nouns expect for proper names are abstract. Where do you go with that?

Because I have not seen any resolution to these questions, I would not say that a "rule" has any existence at all. — Metaphysician Undercover

I would say that the laws of Mathematics and Logic are normative principles pertaining to conduct regulation so as to make the world easier to describe and manipulate.

These normative principles cannot be given a logical justification on pain of circularity, rather their justification stands or falls with their general overall usefulness.

What fishfry has finally started to realize above, this principle: "if mathematicians say it exists then it exists", is a faulty principle. — Metaphysician Undercover

Think of mathematicians sitting around a table and creating a game, discussing the pieces that are played, the environment in which they are played, and the rules that are agreed upon. Once done, would you then say, "The game does not exist."? You fail to recognize that math is a social endeavor, frequently deriving from observations of the physical world, but just as frequently not.

From this perspective, would you say the rules are the axioms? I would say no, there are ill-defined patterns of thought that precede the establishment of the rules, and that might be the subject of study and formalization at a later time - as is the case of the foundations of mathematics.

Just a thought. :smile:

I hope you see the problem with this. You're saying, if we (mathematicians) agree that it exists then it exists, without any definition of what it means to exist. In any other field, no one would agree that such and such "exists", unless there was a definition of "exists" and some evidence to show that the thing actually exists. For example, would some biologist come in with a fictitious life form and ask the other biologists, can we agree that this life form exists, so that it can be a real existent life form? Or would a physicist propose the existence of a fictitious particle? — Metaphysician Undercover

But yes and yes.

Physicists thought one day there must be atoms. Then they discovered the atoms are made of protons and electrons and neutrons. Then they discovered the protons are made of quarks. Now they think the quarks are made of strings. Do any of these abstractions exist? Yes they do, in the sense that they are part of an abstract mathematical theory that explains the experiments we're capable of doing at any moment in history.

Physicist invent new existing things all the time. And de-exist things to. The luminiferous aether was once regarded as existing, till Michelson and Morley couldn't find it and Einstein did away with its necessity.

A scientific entity has existence when it's a necessary ingredient of a successful physical theory. Nobody can say whether a quark or an electron "really" exists; only that positing their existence gives a good theory. That is the definition of scientific existence. And mathematical existence too. I'll go with that, since I challenged myself to define mathematical existence for you.

Biology? Once, disease was caused by ill humours in the blood. Then they came up with the germ theory of disease. Germs are an abstract thing that gives a good theory of disease. Now we can study germs under a microscope, but really, what are they? Bundles of biological material. More abstractions. In the end, they're all quarks and the properties that emerge from various organizations of quarks. But now we treat infections with antibiotics and not leeches, so there is slow progress towards the good. Our abstractions become real because they work. In the future some of the things we think are real will turn out not to be (like the force of gravity) and other things we didn't think were real will turn out to be (electrons, quarks, strings, loops ...)

I gather you call "real" only what is "really out there." But if the 20th century taught us anything, it's that the existence of such a thing as "real things out there" is an assumption and not a fact. I believe if I'm not mistaken this is called scientific realism. It's only an idea. We could kick it around. But you have no logical basis for claiming it's true and everybody else is wrong. The days of Euclidean geometry and Newtonian physics are gone. Now we know the world consists of probability waves that are everywhere at once till we measure them. What can that mean? We don't know. But you claiming that you personally know what things are real, is a delusion on your part. Since you called me delusional the other day, which I can live without.

On the other hand, the ideas/concepts represented under the name "empty set" certainly do exist. They're functional and purposeful. So also is my imaginary hippopotamus friend: he exists too, but also not at the store. — tim wood

The problem is, as I demonstrated, the concept of "empty set" is self-contradicting. Sure, a contradictory concept is functional and purposeful, but that purpose is nothing other than deception.

What do you say of a chair? "Chair" is certainly an abstract noun. Actually, all nouns expect for proper names are abstract. Where do you go with that? — tim wood

I told you already, abstractions do not exist as objects. That is the oversimplification of platonic realism which Plato himself demonstrated as false. Abstraction is an activity of individual human minds, and the proposition that there is "an abstraction" which is created by numerous human minds, is dependent on both a category mistake and a composition fallacy.

First, abstraction is a process of the human mind, there is no evidence that it produces an object, called "an abstraction". There is recollection of the image, representation with symbols, and application, but no evidence of an abstracted object. The proposition, that abstraction does produce an object, "an abstraction" is a falsity intended to simplify reasoning. It is a convenient falsity, accepted because it produces efficiency, but false because it is based in the category error that mental activity, "abstraction", can be represented as an object, an abstraction.

And, even if we are fooled by the category mistake, and accept that mental activity produces an object, "an abstraction", we have to get past the composition fallacy involved with the proposition that there is "an abstraction" common to numerous human minds.

So, the idea of "an abstraction" is supported by a double falsity. Some human beings might argue that two wrongs make a right, because a double negation is a positive, but that assumption as well, is based in faulty principles.

I would say that the laws of Mathematics and Logic are normative principles pertaining to conduct regulation so as to make the world easier to describe and manipulate.

These normative principles cannot be given a logical justification on pain of circularity, rather their justification stands or falls with their general overall usefulness. — sime

At least someone here has a reasonable perspective. Still, there is a problem basing justification in "overall usefulness", because deception is a valid intention. So unless we allow that "useful for the purpose of deception" is valid justification, we need further principles to judge "overall usefulness". I think it is necessary to exclude "useful for the purpose of deception" as a valid justification.

Think of mathematicians sitting around a table and creating a game, discussing the pieces that are played, the environment in which they are played, and the rules that are agreed upon. Once done, would you then say, "The game does not exist."? You fail to recognize that math is a social endeavor, frequently deriving from observations of the physical world, but just as frequently not. — jgill

Saying "the game exists" has the same problem as saying "the rules exist", or saying "the concepts exist". It is an over-simplification made to facilitate communication. Each of these terms, "game", "rules", "concepts", refers to a complexity of physical objects, symbols, and mental interpretations of the symbols. To make communication smooth and swift we refer to those complexities with simple words. The problem is, that common language use which is an habitual activity, clouds our minds as to what is really behind those terms. Because we use the language as if there is an existent thing referred to by "game", "rule", or "concept", we fall under the illusion that there is such existent things.

But a careful, clear, and rigorous analysis of what is actually referred to by these words reveals that there is no such existent things. Each of these words is used to refer to a massive complexity of social interactions which we do not properly understand how to represent. So, we have a word, we assume that the word represents an existent object, and we go about our business ignoring the fact that the word does not represent an existent object, it really represents a massive complexity of social interactions which is not understood. In philosophy though, we seek to unravel these mysteries of the misunderstood, and that is why I insist on recognizing the reality that there is no existent objects referred to by these names.

From this perspective, would you say the rules are the axioms? I would say no, there are ill-defined patterns of thought that precede the establishment of the rules, and that might be the subject of study and formalization at a later time - as is the case of the foundations of mathematics. — jgill

If I understand you correctly, you are suggesting that the axioms are like proposals for rules. Each axiom is presented by a mathematician as a proposition to be accepted, or rejected, by the others. The various mathematicians will then take these proposals and try them out, relate them to each other, combine them with each other, etc., in a sort of trial and error fashion, and after some time of doing this some axioms will emerge as "the rules". I accept this representation, it's similar to the way we do science, hypotheses are presented, they are related to each other, tried and tested with experimentation, until certain theories emerge as "the rules".

Here's something we ought to respect though. The proposals don't ever change their inherent nature as "proposals", despite the fact that they get accepted into the fold as "rules". Therefore we ought not change our attitude toward any proposal just because it has obtained the status of "rule". In reality, things change, human beings and their social structures evolve, so the "rules" change accordingly. Therefore we cannot allow that a proposal, long ago accepted into the status of "rule", is beyond reproach.

Physicists thought one day there must be atoms. Then they discovered the atoms are made of protons and electrons and neutrons. Then they discovered the protons are made of quarks. Now they think the quarks are made of strings. Do any of these abstractions exist? Yes they do, in the sense that they are part of an abstract mathematical theory that explains the experiments we're capable of doing at any moment in history. — fishfry

The problem here is that you assume "they are part of an abstract mathematical theory", when there is no such unified "theory". There is a multiplicity of theories, related or connected to each other in various different ways, dependent on an individual's interpretation. So this assumption is misleading. It is the belief that the abstractions are all part of one overriding theory, which gives credence to the claim that they exist. Once we recognize the falsity of this assumption, the assigned unity is lost, and the entire structure which depends on the reality of this unity, falls apart into individual ideas in individual human minds. The assumed reality of these ideas, as existing "abstractions" is dependent on this unity of "an abstract mathematical theory", which is not supported or justified.

Physicist invent new existing things all the time. And de-exist things to. The luminiferous aether was once regarded as existing, till Michelson and Morley couldn't find it and Einstein did away with its necessity. — fishfry

Again, to call these things "existing" is an over-simplification designed to facilitate communication. That we say at one time X exists, and later x does not exist is an indication of this. At one time we are comfortable using these theories, that is the convention, so we talk as if they exist, at another time we are not comfortable using them, they have become unconventional, so we talk as if they do not exist. Using the word "exist" is just a convenient fiction to refer to what is and isn't conventional. But if we take that fiction literally, and assume that because a conventional idea "exists", it is therefore an object, and we try to treat the idea as an object, we have been misled down the path of misunderstanding.

A scientific entity has existence when it's a necessary ingredient of a successful physical theory. — fishfry

Again, you are over-simplifying, referring to a "physical theory" as a unified object. Every theory is interpreted in numerous different ways, by numerous different people, and one interpretation may be demonstrated as unacceptable while another one is acceptable, so there emerges a conventional interpretation.

I gather you call "real" only what is "really out there." But if the 20th century taught us anything, it's that the existence of such a thing as "real things out there" is an assumption and not a fact. I believe if I'm not mistaken this is called scientific realism. It's only an idea. We could kick it around. But you have no logical basis for claiming it's true and everybody else is wrong. The days of Euclidean geometry and Newtonian physics are gone. Now we know the world consists of probability waves that are everywhere at once till we measure them. What can that mean? We don't know. But you claiming that you personally know what things are real, is a delusion on your part. Since you called me delusional the other day, which I can live without. — fishfry

Sorry, there was no ill intent with the word delusional, and it wasn't meant to single you out. I think we are all delusional, it's a function of where our institutions and conventions have misled us. You might think that science has guided us to the ontology of model-dependent realism, meaning that there are no "real things out there", but this necessitates also that there are no "real things in there". So this form of "realism" is not a realism at all because it cannot validate anything as real. That's why we're deluded, we base reality on usefulness and eloquence, having dismissed truth as unreal. But without truth, eloquence is useful for, and justified through, deception.

Because we use the language as if there is an existent thing referred to by "game", "rule", or "concept", we fall under the illusion that there is such existent things. — Metaphysician Undercover

Interesting metaphysical limb upon which you are perched. Nothing really exists because there are no entities of sufficient purity that they are not compositions of things, many of which fail to exist themselves. Believe me, mathematics does indeed exist, as do the peculiar thoughts that bubble up into your consciousness, having complex pedigrees which apparently do not exist as well.

Seek help, my friend. You limb is but a twig.

Nothing really exists because there are no entities of sufficient purity that they are not compositions of things, many of which fail to exist themselves. — jgill

Right, how could a thing which is composed of parts which do not exist, itself exist? This doesn't mean nothing exists, only those things composed of fictional parts do not exist.

quote="jgill;380008"]Seek help, my friend. You limb is but a twig.[/quote]

Size is irrelevant. What matters is that it provides support. Non-existent things provide no support.

Right, how could a thing which is composed of parts which do not exist, itself exist? — Metaphysician Undercover

You're made up of quarks whose position is a probability wave smeared across the universe until somebody looks, at which time you end up in the place you're most likely to be found. If that ain't fiction I don't know what is. But it works. And you exist.

It doesn't work though, because it doesn't explain how I'm here when no one's looking. So my real, true existence, is not supported by that fiction. That it is, is a delusion.

It doesn't work though, because it doesn't explain how I'm here when no one's looking. So my real, true existence, is not supported by that fiction. That it is, is a delusion. — Metaphysician Undercover

Perhaps you could publish your interpretation of quantum physics in a reputable journal. Or just explicate it here. A Nobel awaits.

It doesn't work though, because it doesn't explain how I'm here when no one's looking. — Metaphysician Undercover

You and the moon don't need the attention.

Here's my definition of infinity, and for simplicity I'm only referring to positive infinity: infinity is a number, but it has a characteristic that all real numbers do not possess. Namely, it is a number that is greater than any particular real number. All the rules of arithmetic applicable to real numbers do not carry over to use of infinity. Examples: infinity plus a real number is infinity: infinity divided by infinity is not equal to one: infinity subtracted from infinity is not equal to zero. — Michael Lee

Notice what you're doing: you're defining an extension to real numbers and real arithmetic. This doesn't magically transform infinity into something it's not (it's not a real number) it just means that the concept of infinity within your extended system, is coherent.

In ordinary arithmetic, infinity is not a number to which arithmetic operations can be applied. It is false to claim that (1/infinity) = 0. Rather, one can abstractly consider where this series leads:
1/n, for n=1,2,3...
the series never ends, but it gets increasingly closer to zero. This leads to the (extended) concept of "limit". The misconception that (1/infinity)=0 is an inexact way of saying that the limit of (1/n), as n approaches infinity, =0. It's useful math, it's fundamental to calculus, but that doesn't "prove" infinity exists in the real world - not in the sense that "7" exists in any collection of 7 objects in the real world.

The problem is, as I demonstrated, the concept of "empty set" is self-contradicting. — Metaphysician Undercover

This simply your "take" on language. It has nothing to do with the definitions, purposes, and functionality of the idea as used by its users and represented by them in this phrase as term of art. You may as well object to "parkway" as a place where drivers drive, and "driveway" as a place where drivers park. Or, you can object to these all you like, but with respect to the functionality identified, your objection is simply irrelevant and inappropriate. Are the words taken by themselves as juxtaposed self-contradictory in any sense? Sure, just not in any sense that matters.

Your attack on the term of art and its substance, through the meaning of its several words taken out of context, becomes, finally, immature and offensive.

What you have not done, near as I can tell, is demonstrate any understanding of the idea(s) represented. You're just caught up in a word game. And if you wish to refute this charge against your argument, then do it. So far, you've just been dismissive, in a bad way. And that stinks after a while.

One approach: since you wish to disqualify what others are doing, you must first make clear that you know what they're doing, and how and why and what for. Then you demonstrate the flaw in what they're doing, or how or why or what for. Fail of this and you're just wasting time and the patience of the posters here, the energy and benefits of whose posts are wasted on you.

Non-existent things provide no support. — Metaphysician Undercover

You would deem the mathematics "supporting" the moon landing and Mars' vehicles non-existent. You would also label the very thoughts you post here non-existent. :roll:

[quote=

This simply your "take" on language. It has nothing to do with the definitions, purposes, and functionality of the idea as used by its users and represented by them in this phrase as term of art. — tim wood

It seems you didn't read my earlier posts

You would deem the mathematics "supporting" the moon landing and Mars' vehicles non-existent. You would also label the very thoughts you post here non-existent. — jgill

Right, the symbols on the paper, and on this page have existence. The thoughts which were used to produce those symbols are events which are in the past, and no longer have existence.

The thoughts which were used to produce those symbols are events which are in the past, and no longer have existence. — Metaphysician Undercover

But those who read and interpret those symbols revive those thoughts and give them renewed existence. Thus, like monks reading and reciting scripture, were an order to so illuminate and pronounce mathematical works with unflagging resolve those thoughts would exist forever.

Hemingway's thoughts exist unendingly, for someone, somewhere is reading them now.

But those who read and interpret those symbols revive those thoughts and give them renewed existence. — jgill

Not quite, the readers produce new thoughts, within a new context. So if thoughts are existent things, the old thoughts of the author are different things from the new thoughts of the reader. There is no continuity of existence between a thought at one time and aa thought at a later time, so the two are not the same thing. The moon landing, along with all the thoughts involved, is a distant memory. It may be recreated with new thoughts, but the new thoughts are not the same thoughts as the old thoughts, as is evident from conspiracy theories. The thoughts which supported the moon landing are not existing.

Thus, like monks reading and reciting scripture, were an order to so illuminate and pronounce mathematical works with unflagging resolve those thoughts would exist forever.

Hemingway's thoughts exist unendingly, for someone, somewhere is reading them now. — jgill

This is a myth which science has dispelled. Symbols, which represent thoughts, might exist indefinitely, but not the thoughts themselves. And, as time passes, the context within which the symbols exist, changes. Since the meaning of any symbols is context dependent, the interpretation of the symbols changes with the passing of time.

There is no continuity of existence between a thought at one time and a thought at a later time, so the two are not the same thing. — Metaphysician Undercover

Nonsense. Take the Pythagorean Theorem: ${{a}^{2}}+{{b}^{2}}={{c}^{2}}$

The original thought occurred millennia ago, and it has been transmitted through the intervening years both by a variety of symbols and word of mouth. It remains essentially the same in Euclidean geometry, which by and large is the world in which we live, even though there are other forms of geometry.

the new thoughts are not the same thoughts as the old thoughts — Metaphysician Undercover

Probably true if you qualify your statement. But even then there is indeed an underlying structure of thought which might be analogous to a step function. The function, though discontinuous at points of time, exists throughout a long period and in effect provides a continuity of existence across time in that interpretations exist at each instant, although they may differ, changing abruptly from time to time.

Nonsense. Take the Pythagorean Theorem: a2+b2=c2a2+b2=c2

The original thought occurred millennia ago, and it has been transmitted through the intervening years both by a variety of symbols and word of mouth. It remains essentially the same in Euclidean geometry, which by and large is the world in which we live, even though there are other forms of geometry. — jgill

Symbols, and word of mouth do not transmit thoughts from one person to another. It's you who is speaking nonsense. When I write this symbol "A", do you think that there is a thought inside there, which is coming from my mind to go into your mind?

Are you familiar with the law of identity? When I read the symbols you wrote, I do not have the same thoughts as you had when you wrote those symbols. When two things are similar, like my thoughts and your thoughts, they are of the same type, they are not the same thing.

"The word 'thought' may mean: a single product of thinking or a single idea." (Wiki)

The PT is an idea transmitted down through the ages. Thought = Idea. Your definition of "thought" is far too narrow. You clearly want to keep all your thoughts to yourself. :roll:

"The word 'thought' may mean: a single product of thinking or a single idea." (Wiki)

The PT is an idea transmitted down through the ages. Thought = Idea. Your definition of "thought" is far too narrow. You clearly want to keep all your thoughts to yourself. :roll: — jgill

There is a reason why we have the law of identity. It was established to prevent the faulty arguments of sophistry. If you think my definition of "thought" is far too narrow, because I adhere to the law of identity, and you'd like to allow that thoughts in different people's minds might be "one and the same" thought, then I see no reason why you would propose this, unless you are trying to argue some trick of sophistry.

I have no problem saying that a thought is "a single idea", where I have the problem is in saying that my ideas are your ideas. As evidence of the difference, there are laws of intellectual property, based in the real separation between the ideas of distinct people. But in this case, it's not that I "want" to keep my thoughts to myself (otherwise I would not be here), but I have a healthy respect for the constraints of reality, which make my thoughts uniquely my own.

In math, cardinality is often expressed as an equivalence relation between two sets: Sets A and B are equivalent (in cardinality) exactly when there exists a one-to-one correspondence between them.
This is more primitive than the concept of number: Two children can confirm that they each have the same number of pieces of candy by pairing one child's pieces with the other's — which does not require knowing how to count.

In math, an infinite set is often defined by the condition that it has the same cardinality as some proper subset of itself (a subset that doesn't include all the members of the original set). And sure enough, the positive integers {1, 2, 3, ...} can be put in one-to-one correspondence with the set of even positive integers {2, 4, 6, ...} just by the formula n <—> 2n. (As everyone knows, you can't do this with a finite set.)

There is a very simple proof that shows that for any set X, the "set of all subsets of X" can never be in one-to-one correspondence with the original set X, no matter how big or small X may be. So the set of all subsets of the integers is strictly larger than the set of integers. And doing this again, we find that the set of all subsets of *that* set is larger than that set is. There is no limit to how many times this can be done.

But hold on to your hat! Because the operation of "taking the set of all subsets of a set" can be done infinitely many times! And the union of all the resulting sets is larger than any previous one. So there are actually infinitely many distinct infinities.

This suggests why there is no largest infinite set — because the set of its subsets is strictly larger.

I'm now convinced: chairs don't exist and we should stop teaching children about them.