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Mathematics: Reality And Infinity

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jrs:

Thanks for the proof step.

And we're on the exact same page as to all the other points you mentioned in your previous post.

/

In an earlier post I misstated a famous formula and even its translation in a way that makes them appear ludicrously trival. I should have posted instead:

~ExAy(Sxy <-> ~Syy)

There does not exist an x such that, for all y, x shaves y iff y does not shave itself.

/

Hal:

"[...] your claim that formalization allows proofs to be easily 'checked by computer' is straight out of fantasy land." [Hal]

If this is addressed to me, then you've made a strawman by adding the word 'easily'.

"Mathematics isnt really a deductive discipline in practice [...]"

Then what kind of reasoning do you think mathematicians use when they prove theorems?

"[...] and I'm unclear what advantages there are to any axiomatic/set-theoretical approaches."

Advantages: Objectivity, rigor, precision, effective decidability whether an argument does indeed establish its conclusion, concepts elucidated by seeing their pure structure, simplicity, elegance, rich results already proven, and rich and difficult open problems...

Also, would you please point to any current mathematics that rejects axiomatics?

"The problem with set theory is that it's utterly dependent on the early 20th century idea that mathematics somehow needs a 'foundation'."

Set theory does not depend on any such idea.

"There is simply no advantage that I can see coming from the axiomization of a particular branch of mathematics [...]"

Since set theory provides an axiomatization (or at least axioms for a primitive predicate from which branches may formulate their own axioms) for virtually all branches, I can only surmise that you're saying that no branch needs even its own axioms. If my surmise is correct, then I'd ask if you believe branches such as geometry, number theory, and the study of real numbers do not require axioms? And if these branches do not require axioms, then what do think a theorem is if not a propostion derived from axioms?

"[...] there is always going to be a possibility that an accepted proof can turn out to be wrong, and set theory does nothing to change this.[...]"

Set theory doesn't pretend to remedy such errors as incorrectly written computerized proof checkers, nor is it the purpose of set theory to do so.

"Mathematics managed to function quite well before ZFC despite the objections of philosophers, and I'm sure it would continue to function in its absence."

First, set theory is a branch of mathematics itself. Second, if I'm not mistaken, axiomatization was sought and practiced by mathematicians (whatever else philosophers had to say about the subject). Third, that one can perform mathematics without set theory is not disputed (but that very much mathematics can be performed with no axiomatization at all is doubtful). Fourth, results from set theory have affected results in other branches. Fifth, as to the 'C' in ZFC, there's a lot of mathematics (not just set theory) that cannot function without it.

Edited by LauricAcid

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If this is addressed to me, then you've made a strawman by adding the word 'easily'.
Ok, then I'll replace it by 'at all'. For a computer to check a proof it must be formalised as a series of step-by-step deductions from axioms , and creating such a formalisation will normally be infinitely more difficult than actually checking the proof by hand. There is far more likely to be something overlooked in the formalisation of a proof than in the proof itself. There are good reasons why the 'computer checking' approach isnt widely used and probably never will be (unless the 'reduction to basic axioms' can itself be carried out by a computer, which will only increase the degree of uncertainty that it has been done correctly).

Then what kind of reasoning do you think mathematicians use when they prove theorems?
Induction, intuition, geometric reasoning, and so on. Deduction is normally used to establish results only after they have been arrived at by non-deductive means. I doubt any research mathematician has ever sat down with a list of theorems in his field and tried to deduce things from them directly. Even most formal mathematical proofs arent 'deductive' in the strong sense of the word - most proofs fall far short of the 'rigour' of basic deductions from axioms, and there is normally a significant appeal to intuition.

Advantages: Objectivity, rigor, precision, effective decidability whether an argument does indeed establish its conclusion, concepts elucidated by seeing their pure structure, simplicity, elegance, rich results already proven, and rich and difficult open problems... 
Most of this is mythical for the reasons I stated above - 'machine checking' will never replace (or be more objective than) checking by hand. The simplicity is debateable - I would personally say that the axiomatic teaching of a subject tends to hide it's geometric/intuitive appeal (particularly in, for instance, complex analysis or group theory).

Also, would you please point to any current mathematics that rejects axiomatics?
I'm not 100% sure what you mean here. Research mathematicians dont really 'reject' or 'support' axiomatics - they just do maths. The axiomisation of a field will only normally occur once it is significantly mature, not while it is being actively pursued. In terms of philosophy of mathematics, there has been significant moves away from the axiomatic/formalist approaches of the first half of the 20th century, mainly inspired by Wittgenstein and (more recently) Putnam.

Set theory does not depend on any such idea.
It doesnt make much sense otherwise. Once we get over the idea of mathematics needing a foundation, there doesnt seem to be much point in ZFC.

If my surmise is correct, then I'd ask if you believe branches such as geometry, number theory, and the study of real numbers do not require axioms? And if these branches do not require axioms, then what do think a theorem is if not a propostion derived from axioms?
I wouldnt claim that any of these require axiomatics in the sense I suspect you mean by the term - number theory and geometry have been pursued for 2000 years and most of that time has been without axioms (in the modern sense of the word). Has our rigorous axiomisation really led to any significant results? Did Hilbert's formalism of Euclidean geometry lead to new innovation in the subject that wouldnt have been possible otherwise? Does the formalist approach to group theory help us to prove results that wouldnt have been accessible to the 'naive intuition' of Galois and Abel?

Set theory doesn't pretend to remedy such errors as incorrectly written computerized proof checkers, nor is it the purpose of set theory to do so.
Then what IS the purpose of set theory in your opinion?

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"For a computer to check a proof it must be formalised as a series of step-by-step deductions from axioms , and creating such a formalisation will normally be infinitely more difficult than actually checking the proof by hand." [Hal]

1. 'Infinitely' is an amusing hyperbole in this context.

2. There exists a proof that takes less labor to hand check than the amount of labor required to create a computer program to check proofs in general. But there exists a set of proofs that takes more labor to hand check than the amount of labor required to create a computer program to check proofs in general.

3. What is important to me is that there is an effective method to check proofs. I'm much less concerned whether people create or use computerized proof checkers.

4. Proofs that don't exceed the physical limitations of physical computers can be checked by computers. That this might be difficult to achieve or that there are even proofs that exceed the physical limitations of any physical computer is not a matter of contention.

"There is far more likely to be something overlooked in the formalisation of a proof than in the proof itself." [Hal]

This could refer to two things: 1) Errors in the proof checking program or 2) Errors in the symbolization of the proof. In either case, it is not a matter of contention with me that such errors may occur.

"There are good reasons why the 'computer checking' approach isnt widely used and probably never will be [...]" [Hal]

Fine with me.

"[...] unless the 'reduction to basic axioms' can itself be carried out by a computer [...]" [Hal]

I don't know if you have a special meaning for 'reduction to basic axioms', but it's a given that a proof to be checked by a computerized proof checker is a proof from axioms.

"[...] which will only increase the degree of uncertainty that it has been done correctly [...]." [Hal]

1. Without axioms, it's not clear to me in what sense you think any proof has any amount of certainty.

2. Yes, the more steps in a proof, the more chance for error, would seem to be a good rule of thumb. But, by analogy, complaining about the added formalization needed for axioms is like complaining that building a chassis for an automobile just adds to the chances of some flaw in the automobile.

"Then what kind of reasoning do you think mathematicians use when they prove theorems?" [LauricAcid]

"Induction, intuition, geometric reasoning, and so on." [Hal]

1. And deduction also.

2. All of those contribute to the thinking that goes into having insights that certain things are true or are provable, and discovering proofs and making conjectures. But the reasoning in the proofs themselves is deductive.

3. Whatever goes into thinking up theorems, they are made public and objective by proof. If mathematics is not just the study of individual moments of personal flashes of insight, but rather of communicated and publically verifiable theorems, then mathematics doesn't exist as such an enterprise without deductive proof.

"Deduction is normally used to establish results only after they have been arrived at by non-deductive means."

1. What is your basis for that assertion? Is there a survey of mathematicians that shows that their thoughts are usually non-deductive prior to formulating proofs?

2. Even if other forms of insight were more usual than deduction, and deduction were only used to establish the results of non-deductive thinking, then I don't see a matter of contention. To 'establish a result' is to prove a proposition. And proofs are deductive.

"I doubt any research mathematician has ever sat down with a list of theorems in his field and tried to deduce things from them directly." [Hal]

1. What do you mean by 'research mathematician' as opposed to 'mathematician'?

2. Mathematicians look at previously proved theorems and try to deduce things from them pretty much any time they're doing mathematics. Or they're formulating axioms from which to prove things. Or they're formulating logistic systems in which axioms can be formulated.

"Even most formal mathematical proofs arent 'deductive' in the strong sense of the word - most proofs fall far short of the 'rigour' of basic deductions from axioms, and there is normally a significant appeal to intuition." [Hal]

Since the advent of formalization, mathematicians almost always give informal presentations, but enough of the formalization has already been digested so that competent readers can see that the informal presentation could, with sufficient patience, be formalized. The proofs can be made rigorous, down to the very last parenthesis symbol. Intuition plays a part in understanding the thinking behind a proof, but appeal to intuition is no part of the proof itself. Mathematicians explain their thinking and motivations in ways that are intuitive, but the proofs, since the advent of formalization, themselves do not rely on acceptance of anything that is not deducible from formal axioms in a formal system. Moreover, in presenting a formalized theory from scratch, often each step is rigorous, not informal, at least until readers can have sufficient grasp of the formalism to be able to see how the subsequent informal steps can be formalized.

"Advantages: Objectivity, rigor, precision, effective decidability whether an argument does indeed establish its conclusion, concepts elucidated by seeing their pure structure, simplicity, elegance, rich results already proven, and rich and difficult open problems..." [Lauric Acid]

"Most of this is mythical for the reasons I stated above [...]" [Hal]

You've given no reasons except incorrect ones. Most specifically, your point about the laboriousness of computer programming and possibilities of human error are irrelevant. And your point about formalization masking understanding is irrelevant since formalization does not preclude anyone from looking at the subject matter in informal ways too.

"'machine checking' will never replace (or be more objective than) checking by hand." [Hal]

None of what I mentioned depends on relying on machine checking instead of hand checking.

"The simplicity is debateable - I would personally say that the axiomatic teaching of a subject tends to hide it's geometric/intuitive appeal (particularly in, for instance, complex analysis or group theory)." [Hal]

1. Simplicity and hiding intuitive appeal are separate matters.

2. Formalization does not prevent anyone from looking at a subject matter informally and with intuitions rather than calculations. There's no requirement of formalization that people not also obtain informal understanding. On the contrary, most mathematicians in the subject of foundations do give informal and intuitive explanations and do encourage understanding the subject matter not just as a formalization.

"The axiomisation of a field will only normally occur once it is significantly mature, not while it is being actively pursued." [Hal]

Please point me to a specific mathematical result, since the advent of formalization, that is not a proof from previous results, all of which are traced back to axioms at some point in the evolution of mathematics. I'm not denying that there aren't such results (and in what sense they would be 'results' is unclear to me) but I am sincerely interested in knowing about them.

"In terms of philosophy of mathematics, there has been significant moves away from the axiomatic/formalist approaches of the first half of the 20th century, mainly inspired by Wittgenstein and (more recently) Putnam." [Hal]

'formalism' can mean at least three things: (1) 'formalism' is sometimes loosely used to mean a formalization, (2) 'formalism' is used to refer to the philosophy of mathematics associated with Hilbert, (3) 'formalism' is used to refer to the idea that mathematics can or should be formalized, irrespective of any other particular tenets in a philosophy of mathematics.

So I'm interested in reading what you consider to be any critiques by Wittengstein or Putnam of axiomatics and of formalism in sense (3). Would you point me to the specific texts?

"Once we get over the idea of mathematics needing a foundation, there doesnt seem to be much point in ZFC." [Hal]

What there is "much point" of doing is a matter of individual preference. If you feel there's not much point to set theory aside from foundations, then there's probably no need to convince you otherwise, (unless, of course, you're on some university budget committee!).

"[...] number theory and geometry have been pursued for 2000 years and most of that time has been without axioms (in the modern sense of the word)." [Hal]

1. Foundations and formalization have sharpened the axiomatics and methods of mathematics. But, still mathematics has pretty much always been focused on deductively proving propositions from previously proved propositions, and where the axioms may have not been articulated, mathematicians did show propositions as proven from previously proven propositions, not just by singular flashes of intuition.

2. Is there an important result do you believe cannot be proven by axioms? If so, what is it? (Of course, the incompleteness theorem provides that there are a lot of truths not provable, but I'm interested in knowing if therer is an unprovable mathematical proposition that is important to own understanding in a subject of mathematics.)

"Has our rigorous axiomisation really led to any significant results? Did Hilbert's formalism of Euclidean geometry lead to new innovation in the subject that wouldnt have been possible otherwise?" [Hal]

1. The purpose of formalization is not to produce results in the sense you're asking about.

2. Hilbert's formalization played an important part in some quite profound results. This looks like a pretty good article about Hilbert and foundations (and it barely touches on his work in various non-foundational fields nor on results from cross-pollination among the fields, including foundations):

http://plato.stanford.edu/entries/hilbert-program/

"[...] what IS the purpose of set theory in your opinion?" [Hal]

Different mathematicians have different reasons for working with set theory, so when I say "the purpose of set theory", I'm being quite informal, at least in saying what it is not, as opposed to what it is. Purpose is for each individual to decide for himself. My own interest in set theory is an interest in foundations, certainty, and in understanding the implications of axioms, including axioms that can be couched in the primitives of set theory, as well as an interest in the concepts engendered by the axioms. Moreover, set theory is not just foundational for mathematics but for meta-mathematics also. And many notions of language, meaning, and consequence are made more precise by mathematics. This precision can be a powerful conceptual tool. And I'm interested in difficulties with the foundational efforts themselves. In order to understand these difficulties one must understand the subject. In order to even approach reading controversies about set theory and foundations, one must understand what they are. And, I benefit form the intellectual exercise, as well as the pure enjoyment of using reason in its most abstract yet rigorous manner.

Edited by LauricAcid

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Infinity

We can't say that the set of symbols is S = {x, 0, 1, 2, 3, 4 , 5, 6, 7, 8, 9} since what we'll apply the recursion theorem to is not S, but rather the set, S', generated by your notation from S. And S' is an infinite set. Also, I'm not don't know what you mean about a recursion operator. What I mentioned was not a concern about creating expressions for recursive functions, but rather the one I just mentioned about applying the recursion theorem.

1. I do not know what you mean by "the recursion theorem".

2. By a recursion operator, I mean something like "rec(F,A)" which is definition df-rdg 3129 in:

http://us.metamath.org/mpegif/mmdefinitions.html

It is an operator which applies a given function repeatedly to a starting value to get a new function.

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Roughly speaking, Megill's 3129 is an alternate route to the recursion theorem (actually any number of related and similar theorems). I'm not worried about proving the existence of these recursive functions, since that is what the recursion theorem does. Rather what I was refering to is that the recursive functions that are used to check things for predicate logic will require that the language have an infinite number of symbols or you won't get your basic metatheorems for a clean bill of health. Perhaps you could have finite number of individual variables and just redo all your metatheorems each time you add a variable to your language, but I suspect that won't work. I think you need an infinite number of individual variables from the start, but I could be wrong.

Edited by LauricAcid

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Hello everyone I am new to this forum and I am glad to find an objectivist forum. I am an electronics technician by trade and also a scientist in my spare time. I am pretty in agreement with almost all of Ayn Rand's philosophy but I have to disagree with what is considered the objectivist stand on infinity and other areas of mathematics and science.

 

For instance, I think there is a disregard of the fact that true concepts are not only abstractions or representations in one's mind but indeed are reflections and representations of relationships in reality. Here is where I think the concept of infinity comes in as an actual principle of reality as physical, logical and geometrical necessity shows.

 

 

This has been discussed to death in several prior threads. Briefly, your error, in essence, lies in attempting to apply time to the universe, but the universe does not exist within time. You are using the concept of "duration" in a context where "duration" does not apply. Time exists within the universe; the universe does not exist within time.

 

Can you please define what you mean by universe? I assume you mean all of the matter in existence? If that is the case then time does apply to the universe since when speaking of the universe's operation (not it's identification) every object, form or motion must have a history that extends infinitely into the past. So I would say that an infinite amount of time has passed in the universe.

 

There are other actual infinite series in reality I can think of but that is beyond the scope of this reply.

Edited by superfinguy

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Can you please define what you mean by universe? I assume you mean all of the matter in existence? If that is the case then time does apply to the universe since when speaking of the universe's operation (not it's identification) every object, form or motion must have a history that extends infinitely into the past. So I would say that an infinite amount of time has passed in the universe.

 

There are other actual infinite series in reality I can think of but that is beyond the scope of this reply.

 

This is certainly not true. If the universe had existed forever, the stars should have long-since burned out.

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