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I received Mr. Knapp's book,  Mathematics is About the World. It includes 'Hilbert's Game of Symbols' in the subtitle, but doesn't have much more in the body.

"At some point during my college freshman year, I realized that neither mathematicians nor philosophers of mathematics shared my perspective, offering only the alternatives of formalism (a game of symbol manipulation), Platonism (a separate world of mathematics), or, as a third, the Fregean view that mathematics is a branch of logic. I could accept none of these choices" (p. 10).

Hilbert was a Formalist.

"My specific concern will not be with counting objects, but with using numbers to measure magnitudes, such as length, weight, and speed. In this, we should not be surprised to find that our usage of numbers is indeed correct. But we will find that characterizing exactly what we are doing when we apply numbers is not as straightforward as one might have thought. Yet in laying this process bare, one creates the foundation for a similar understanding of mathematical concepts whose relationship to the world we live in may be far from obvious. It is the lack of such understanding that has led to the widespread false alternatives that mathematics is either a formal game played with symbols, a system of deduction from carefully chosen axioms such as the axioms of set theory, or an insight into a Platonic universe of mathematical concepts. On any of these views, the applicability of mathematics to reality must be viewed as a happy accident" (p. 101-2)

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19 hours ago, merjet said:

Hilbert was a Formalist.

[and]

"the widespread false alternatives that mathematics is either a formal game played with symbols, a system of deduction from carefully chosen axioms such as the axioms of set theory, or an insight into a Platonic universe of mathematical concepts." [Knapp]

Formalism has variants. An extreme variant is that mathematics is nothing but manipulation of symbols with no other meaning. But there are other variants that do not ascribe to that. Indeed, I have explained how Hilbert's variant does not ascribe to that extreme variant, and we have seen no evidence that Hilbert ever advanced that variant, and especially when the Wikipedia article just mentioned itself cites an SEP article that mentions the distinction. It is possible (though I doubt it) that Hilbert did say such a thing, but it is not properly justified to claim he did after request for citation, in context, from his works.

The Wikipedia page starts with a citation to an SEP article, and in that SEP article https://plato.stanford.edu/archives/spr2015/entries/formalism-mathematics/ we find:

"The Hilbertian position differs because it depends on a distinction within mathematical language between a finitary sector, whose sentences express contentful propositions, and an ideal, or infinitary sector"

And that is just as I mentioned previously. 

And the Wikipedia article on Hilbert says

"According to the formalist, mathematics is manipulation of symbols according to agreed upon formal rules. It is therefore an autonomous activity of thought. There is, however, room to doubt whether Hilbert's own views were simplistically formalist in this sense."

Again, that is not a proper description of formalism since there are variants that are not that extreme, and not only is there room to doubt, but, as I've said, without citing Hilbert himself, it is incumbent to not just take it as a given that Hilbert ascribed to that extreme variant.  

And the Wikipedia article on formalism itself says

"[According to formalism [...] mathematical statements are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation (or semantics)."

Indeed, unless. And reasonable variants of formalism may allow interpretations (the meanings) of the formal systems.

[and]

Indeed, it would be a false alternative to claim that mathematics is only of the two: either a mere game of symbols or a study of platonic truths. And upon even a cursory look at the literature of the philosophy of mathematics we find variants on those two approaches (with these variants not necessarily properly summarized as just now) and also quite different approaches from those two.

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1 hour ago, GrandMinnow said:

Formalism has variants.

Of course, Formalism designates a group of people who don't all hold identical views. Hilbert was a major figure of the school. I suggest a more charitable reading of Kline and Knapp.  "Formalism -- a major figure being Hilbert -- holds that ....."

Note that in both quotes I gave from Knapp's book, Knapp does not even use Hilbert's name. 

Edited by merjet

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Sure, since I haven't read Knapp, I'm not saying that he has himself made a certain claim about Hilbert. 

Regarding philosophy of mathematics, it's okay to mention schools of thought broadly, as long as when we get down to actual critiques we are careful not to ascribe positions to people who do not hold those positions. 

Not necessarily you personally, but it is common to find people saying such things as "Hilbert held  that mathematics is merely a game of symbols and has no other meaning."

To say (as some people do) "Hilbert held  that mathematics is purely a game of symbols and has no other meaning" would be like saying (as some people do) "Ayn Rand held that virtue is purely a matter of doing anything that is best for oneself and other people don't matter". Both are quite incorrect. 

 

 

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One fine point, just to be clear:

You mentioned "David Hilbert's opinion that the most reliable way to treat mathematics is to regard it not as factual knowledge, but as a purely formal discipline that is abstract, symbolic, and without reference to meaning."

I said that it is not fair to ascribe that view to Hilbert without citing it in his writings. And, so far, we not been given such a citation. 

However, your statement is not quite as strong as the claim that Hilbert viewed mathematics as purely a game of symbols without meaning. So I did not claim that you yourself made that stronger claim.

Here's the comparison [bold added]: 

(1) The most reliable way to treat mathematics is to regard it not as factual knowledge, but as a purely formal discipline that is abstract, symbolic, and without reference to meaning.

and the stronger claim:

(2) Mathematics is to be regarded not as factual knowledge, but as a purely formal discipline that is abstract, symbolic, and without reference to meaning.

I don't know of evidence that Hilbert made either of those claims. But, aside from Hilbert,  it is less implausible to claim (1). To say that the most reliable way to do something is such and such is not to say that it is the only way to do it, or even that it is the only correct way to do it, or that in some other senses or contexts one doesn't also do it other ways.

As I mentioned, Hilbert took finitary mathematics to be contentual and reliable beyond reasonable dispute. So I don't know in what sense he would regard finitary mathematics as most reliable when viewed as divorced from content, or even in what sense he would regard finitary mathematics as having more reliability when viewed as divorced from content. But those would be less implausible than saying he took finitary mathematics as reliable only when divorced from content. 

But as to infinitary mathematics, I probably wouldn't quibble with saying that Hilbert took it to be reliable only in terms of formal symbol rules. Indeed, it would be fair to say that, more or less, formalists don't accept that infinite mathematical objects (such as infinite sets, infinite sequences, et. al) can be taken as reliable concepts other than as informal notions as extrapolations from formal systems. 

But even this does not imply that Hilbert didn't recognize that infinitary mathematics is useful for the sciences. Hilbert, like just about any mathematician, was steeped in infinitary mathematics and would recognize that, say, infinite sequences for calculus are used for framing the mathematics for the physical sciences. 

 

Edited by GrandMinnow

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13 hours ago, GrandMinnow said:

One fine point, just to be clear:

You mentioned "David Hilbert's opinion that the most reliable way to treat mathematics is to regard it not as factual knowledge, but as a purely formal discipline that is abstract, symbolic, and without reference to meaning."

I said that it is not fair to ascribe that view to Hilbert without citing it in his writings. And, so far, we not been given such a citation. 

[snip]

As I mentioned, Hilbert took finitary mathematics to be contentual and reliable beyond reasonable dispute. So I don't know in what sense he would regard finitary mathematics as most reliable when viewed as divorced from content, or even in what sense he would regard finitary mathematics as having more reliability when viewed as divorced from content. But those would be less implausible than saying he took finitary mathematics as reliable only when divorced from content. 

But as to infinitary mathematics, I probably wouldn't quibble with saying that Hilbert took it to be reliable only in terms of formal symbol rules. Indeed, it would be fair to say that, more or less, formalists don't accept that infinite mathematical objects (such as infinite sets, infinite sequences, et. al) can be taken as reliable concepts other than as informal notions as extrapolations from formal systems. 

But even this does not imply that Hilbert didn't recognize that infinitary mathematics is useful for the sciences. Hilbert, like just about any mathematician, was steeped in infinitary mathematics and would recognize that, say, infinite sequences for calculus are used for framing the mathematics for the physical sciences. 

 

Fair enough. I did write, "David Hilbert's opinion that the most reliable way to treat mathematics is to regard it not as factual knowledge, but as a purely formal discipline that is abstract, symbolic, and without reference to meaning" twice. The second time was a direct quote from Kline. The first was copied from something I wrote several years ago, and was likely influenced by Kline.

I once also wrote elsewhere following the above quote: "The symbols may represent intuitively meaningful percepts or concepts, but they are not to be so interpreted in pure mathematics." That raises the possibility that the symbols are not always meaningless, but only that they should be so regarded at times. 

I can't remember ever reading anything by Hilbert himself. There is a risk in that, but relying on secondary sources is very hard to avoid due to limitations of time and interest.

Page 48 here is Philip Kitcher regarding Hilbert's formalism. Kitcher regards Hilbert as an apriorist. I think that Hilbert's epistemology has some bearing on Hilbert's view of meaningful/meaningless. Do you agree with that? This referenced by Kitcher might be an interesting read. I didn't find the full article anywhere.

Maybe we can more agree on our views of Hilbert's view of meaningful/meaningless if you will cite Hilbert himself or at least a secondary source you judge to be better than, or as good as, Kline and Kitcher. Maybe you have relied on Hilbert himself or a secondary source with the distinction you make between finitary and infinitary mathematics. I won't quibble with what you have said about that.

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I see the link I made to Kitcher's Hilbert's Epistemology doesn't work. It was only to an abstract anyway. The full text can be seen on JSTOR.org with a free subscription.

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Hilbert, Godel, and Cantor have been cast as bete noires by many people, especially on the Internet. Typically the absurdly stubborn and dogmatic criticisms and denunciations are from people who have not read even an utterly basic introductory textbook on symbolic logic. So we find grave misunderstandings and misrepresentations of the subject. 

My motivation responding in this thread was to point out that we should not take it that Hilbert claimed that mathematics consists merely of rules for symbols without meaning, unless we find that he wrote that. I am not a scholar, but I do have enough introductory understanding to provide at least some basic explanations, roughly formulated though they may be. And my purpose is not necessarily to defend Hilbert's philosophical views, but rather primarily at least to warn against misrepresentations of him. 

I'm not lumping you in with those stubborn people, since you seem amenable to reasonable discussion. Indeed, you mention that you wrote elsewhere:

"'The symbols may represent intuitively meaningful percepts or concepts, but they are not to be so interpreted in pure mathematics.' That raises the possibility that the symbols are not always meaningless, but only that they should be so regarded at times."

That is well put. 

I would expand to say that there is the formal syntax of the system (when we evaluate the formal syntax, we divorce from meaning), then there are formal semantics (formal meanings) for the system, then there are understandings of the formal semantics in terms of our mathematical notions not yoked to formalisms and in terms of our ideas about the world, then there is application of those ideas to practical tasks including science and technology. 

------  

My sources [I've condensed some of the quotes here] are mostly secondary, pretty much from books and articles by mathematicians and writers who I take to be professionally responsible, and whose writings are stated in correct mathematical formulations or terminology, and you and I already mentioned the SEP. 

* But at least one primary source is:

Grundlagen der Mathematik I - Foundations of Mathematics I - Part A

This book provides the first part of the famous work by Hilbert and Bernays, with English translation. From page 2:

"We call the form of mathematics where the subject matter is ignored 'formal axiomatics'. In contrast, contentual axiomatics introduces its basic notions by referring to common experience and presents its first truths either as evident facts or formulates them as extracts from experience-complexes. Thus, contentual mathematics conveys the belief that we have actually discovered laws of nature and intend to support this belief by the success of the theory. Formal axiomatics requires contentual axiomatics as a necessary supplement. It is only the latter that provides us with guidance for choosing the right formalisms, and with some instructions on how to apply a given formal theory to a domain of actuality."

For showing that Hilbert did not take mathematics to be merely a meaningless symbol game, such a quote is QED, don't you agree?

 * Perhaps an okay secondary source:

A Philosophy of Mathematics - Louis Kastoff

From page 117-118:

"[For Hilbert] mathematics is a pure calculus [of symbols], and can be replaced by a method, entirely mechanical, for deducing formulae. This does not mean, as is frequently supposed, however, that mathematics for Hilbert is a game with meaningingless symbols. The symbols may be arbitrary but not meaningless. It does mean, however, that the formalist may ignore the application of his system. Only in this sense are the symbols meaningless. And to be like a game is not to be a game. Pure axiomatics presupposes a sphere of objects which it presents in idealized form. [Hilbert writes,] 'My theory of proof is actually nothing more than the description of the innermost processes of our understanding and it is a protocol of the rules according to which our thought actually proceeds.' As a consequence, all criticisms of Hilbert based on the idea that he treats of meaningless symbols must be abandoned." 
 
(A qualification is needed: Yes, there is a mechanical method for producing all possible proofs (the set of proofs is recursively enumerable). But it would be impractical, especially given human mortality, to simply wait for proofs to be mechanically generated. So mathematicians have to use insight to complete the proofs, and usually the proofs are not presented in pure formal syntax, though the proofs could be re-written in pure formal syntax if we wanted to do so. Then, there is a mechanical method (and usually not impractical to perform) for checking whether a purported purely formal proof actually is one; and that checkability is what provides the ultimate objectivity in judging whether a mathematical proposition has actually been proven.)

* From section 3.3 here:

https://people.ucalgary.ca/~rzach/static/hptn.pdf

"Hilbert makes [the distinction] between the finitary part of mathematics and the non-finitary rest. The finitary part Hilbert calls 'contentual,' i.e., its propositions and proofs have content. The infinitary part, on the other hand, is not meaningful from a finitary point of view."

* Look at sections 2 and 3 here:

https://books.google.com/books/about/Routledge_Encyclopedia_of_Philosophy_Gen.html?id=5m5z_ca-qDkC

------

I don't have the Kline book, so I can't cite specifics at this time, but years ago I read it and I found it to be one of those books you want to throw across the room. I found it be be verging on a diatribe rather than a reasoned look at the subject. As I recall, his arguments often depend on blatantly misconstruing of some of the key specifics of the mathematics itself. 

------

I would have to study the Kitcher article to comment on it very much. In any case, I don't know how it bears on the question of whether Hilbert took mathematics to be merely a game of symbols with no meaning. 

Edited by GrandMinnow

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PS. There might be some very gifted people who can grok discussions about mathematics without first studying the actual mathematics. But I'm not one. My understanding is based on starting with the basics, through a systematic (usually quite meticulous) study of mathematical logic (and some upper division undergraduate mathematics). For an excellent start, I always recommend:

Logic: Techniques of Formal Reasoning (second edition) - Kalish, Montague and Mar

This is the very best introduction to how to work in the first order predicate calculus that I have found. And understanding how the first order predicate calculus works is crucial for any reasonable discussion of such things as Hilbert's view of mathematics, except for those very gifted, very lucky people who can just jump straight into the advanced subjects without preparation. 

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14 hours ago, GrandMinnow said:

....

"'The symbols may represent intuitively meaningful percepts or concepts, but they are not to be so interpreted in pure mathematics.' That raises the possibility that the symbols are not always meaningless, but only that they should be so regarded at times."

That is well put.

....

* But at least one primary source is:

Grundlagen der Mathematik I - Foundations of Mathematics I - Part A

....

For showing that Hilbert did not take mathematics to be merely a meaningless symbol game, such a quote is QED, don't you agree?

....

* From section 3.3 here:

https://people.ucalgary.ca/~rzach/static/hptn.pdf

"Hilbert makes [the distinction] between the finitary part of mathematics and the non-finitary rest. The finitary part Hilbert calls 'contentual,' i.e., its propositions and proofs have content. The infinitary part, on the other hand, is not meaningful from a finitary point of view."

.... 

Thank you.

The quote is not clear or extensive enough for me to accept as proof Hilbert regarded no part of mathematics to be merely a meaningless symbol game.’ You even quoted Zach: “The infinitary part, on the other hand, is not meaningful from a finitary point of view."

How did you get an English translation of Grundlagen der Mathematik? According to this page, some of Volume 1 has been translated to English. Anyway, it’s difficult for me to cite Hilbert himself about meaningful/meaningless without a full English translation available.

Returning to the SEP entry Formalism in the Philosophy of Mathematics again: "The Hilbertian position differs because it depends on a distinction within mathematical language between a finitary sector, whose sentences express contentful propositions, and an ideal, or infinitary sector. Where exactly Hilbert drew the distinction, or where it should be drawn, is a matter of debate. Crucially, though, Hilbert adopted an instrumentalistic attitude towards the ideal sector. The formulae of this language are, or are treated as if they are, uninterpreted, having the syntactic form of sentences to which we can apply formal rules of transformation and inference but no semantics."

"No semantics" means no meaning.

Edited by merjet

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The question is not whether 

(1) Hilbert regarded no part of mathematics to be a merely meaningless symbol game. 

No one denies that (putting it a bit roughly) Hilbert regards evaluation of syntax onto itself as a meaningless symbol game. The point (that I have amply explained and shown by now) is that it is not the case that 

(2) Hilbert regarded mathematics to be a merely meaningless symbol game. 

Again, the Hilbert quote itself plainly denies that mathematics is a merely meaningless symbol game:

"Contentual axiomatics introduces its basic notions by referring to common experience and presents its first truths either as evident facts or formulates them as extracts from experience-complexes. Thus, contentual mathematics conveys the belief that we have actually discovered laws of nature and intend to support this belief by the success of the theory."

Truths, facts, notions, experience, conveys, belief, discovered laws, success of the theory.

Yes, the infinitary part is meaningless from a finitary point of view. But the finitary part is not meaningless. Moreover, even though the infinitary part is literally meaningless, it still plays an instrumental role in the mathematics for the sciences. So mathematics is not a just a meaningless symbol game. 

And from the Zach article you mentioned, we have: "(finitarily) meaningful" .

And, yes, the syntax in one regard is treated without semantics. But in other regards of course we may apply semantics. I've quite explained this already.

And from the SEP quote you just adduced: "the finitary sector, whose sentences express contentual propositions". 

express, contentual.

------

The Hilbert volume is edited by Claus-Peter Wirth, et. al - published by College Publications 2011. ISBN 978-1-890-033-2.

But I strenuously recommend that it is folly to read a volume such as this without first learning the basics of symbolic logic and then at least introductory mathematical logic.

On the other hand, a person can merely skim over the technical terminology and mathematical formulas, ignoring or misconstruing the technical context, thus burdening oneself with half-baked misunderstandings that only engender even more falsehoods and confusions about these mathematicians and their mathematics and philosophies. 

Most particularly, just for starters, it is only by systematically working through a textbook that one sees how mathematical logic presents syntax as separate but then also links it with semantics.

Edited by GrandMinnow

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On 7/21/2019 at 11:51 AM, GrandMinnow said:

But I strenuously recommend that it is folly to read a volume such as this without first learning the basics of symbolic logic and then at least introductory mathematical logic.

I have no plan to do so. The above also indicates how far this thread has strayed. The title is Math and Reality. Mr. Knapp’s book’s title, sans subtitle, is Mathematics Is About the World. I agree it is very much about the world, but think it’s a little more than that. More concretely, Knapp’s thesis is that arithmetic and geometry, especially analytic geometry, pertain to the world. He defines mathematics as the science of measurement. (Analytic geometry and calculus enable indirect measurement.) I think mathematics is a little broader than that, but measurement is a big part. Functions and vector spaces also pertain to the world. His book is not about symbolic logic, mathematical logic, predicate calculus, or finitary vs. infinitary.

His book presents an alternative view of mathematics that is very different from formalism, logicism, Platonism, and others. Regarding the philosophy of mathematics schools of thought surveyed here, his is most similar to Aristotelian realism or empiricism. My view is much like Knapp’s.

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I appreciate the information in this thread.

There are a couple of terms “ideal number” and “ideal proposition” in the text below that can be googled or are provided in the SEP article Merlin and GM have mentioned. Otherwise I think the following from Hilbert’s paper “The Foundations of Mathematics” (1927) is pretty self-contained:

“If we now begin to construct mathematics, we shall first set our sights upon elementary number theory; we recognize that we can obtain and prove its truths through contentual intuitive considerations. The formulas that we encounter when we take this approach are used only to impart information. Letters stand for numerals, and an equation informs us of the fact that two signs stand for the same thing.

“The situation is different in algebra; in algebra we consider the expressionsformed with letters to be independent objects in themselves, and the propositions of number theory, which are included in algebra, are formalized by means of them. Where we had numerals, we now have formulas, which themselves are concrete objects that in their turn are considered by our perceptual intuition, and the derivation of one formula from another in accordance with certain rules takes the place of the number-theoretic proof based on content.

“Thus algebra already goes considerably beyond contentual number theory. Even the formula (1 + a) = (a+ 1), for example, in which is a genuine number-theoretic variable, in algebra no longer merely imparts information about something contentual but is a certain formal object, a provable formula, which in itself means nothing and whose proof cannot be based on content but requires appeal to the induction axiom. 

. . .

“We have an urgent reason for . . . extending the formal point of view of algebra to all of mathematics. For it is the means of relieving us of a fundamental difficulty that already makes itself felt in elementary number theory. Again I take as an example the equation (+ 1) = (1 + a); if we wanted to regard it as imparting the information that (A + 1) = (1 + A), where A stands for any given number, then this communication could not be negated, since the proposition that there exists a number A for which (A + 1) not= (1 + A) holds has no finitary meaning; one cannot, after all, try out all numbers. Thus, if we adopted the finitist attitude, we could not make use of the alternative according to which an equation like the one above, in which an unspecified numeral occurs either is satisfied for every numeral or can be refuted by a counterexample. For, as an application of the ‘principle of excluded middle’, this alternative depends essentially on the assumption that it is possible to negate the assertion that the equation in question always holds.

“But we cannot relinquish the use either of the principle of excluded middle or of any other law of Aristotelian logic expressed in our axioms, since the construction of analysis is impossible without them.

“Now the fundamental difficulty that we face here can be avoided by the use of ideal propositions. For, if to the real propositions we adjoin the ideal ones, we obtain a system of propositions in which all the simple rules of Aristotelian logic hold and all the usual methods of mathematical inference are valid. Just as, for example, the negative numbers are indispensable in elementary number theory and just as modern number theory and algebra become possible only through the Kummer-Dedekind ideals, so scientific mathematics becomes possible only through the introduction of ideal propositions.

“To be sure, one condition, a single but indispensible one, is always attached to the use of the method of ideal elements, and that is the proof of consistency; for, extension by the addition of the ideal elements is legitimate only if no contradiction is thereby brought about in the old, narrower domain, that is, if the relations that result for the old objects whenever the ideal objects are eliminated are valid in the old domain.

“In the present situation, however, this problem of consistency is perfectly amenable to treatment. . . . “

The translators are Stefan Bauer-Mengelberg and Dagfinn Follesdal, as reprinted in From Frege to Godel (1967).

GM, I think I see some of Hilbert’s motivation for a view of algebraic symbols as free of content here, although one puzzle I have about Hilbert’s remarks here is: An algebraic equation such as = (Mx+ B) can be, via coordinate geometry, plotted into a particular line when merely the M and B are given definite values, leaving the algebraic variables as algebraic variables. I mean why isn’t analytic geometry yielding facts about synthetic geometry enough to give some meaning to algebraic symbols? Wouldn’t it be sensible to say that algebraic variables mean just any and all magnitude relations (spatial or otherwise) that might be captured in such analytic relations as we have in this case? Do you think Hilbert would object to that sort of picture of algebraic symbols?

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Hilbert per Stephen: “Thus algebra already goes considerably beyond contentual number theory. Even the formula (1 + a) = (a+ 1), for example, in which is a genuine number-theoretic variable, in algebra no longer merely imparts information about something contentual but is a certain formal object, a provable formula, which in itself means nothing and whose proof cannot be based on content but requires appeal to the induction axiom."

Why not and why? Let a = 4. The formula tells me that I can (1) start with 1 dime and add 4 dimes, or (2) start with 4 dimes and add 1 dime. Either way, the result is 5 dimes.  Also, if a equals some other integer > 1, then I can (1) start with 1 dime and add a dimes, or (2) start with a dimes and add 1 dime. Either way, the result is the same count of dimes.

Edited by merjet

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On 7/21/2019 at 11:51 AM, GrandMinnow said:

Hilbert

Merlin, we should stick to A for your example, not go back up to his a. We do not succeed in showing that for all A, where A is any given number, that (A + 1) = (1 + A) by showing that it works for any A we try. That is not enough, and we have methods for going absolutely all the way. How might we show it works for all given A? Hopefully, GrandMinnow can explain what further Hilbert means here, and hopefully respond to my questions to him in the final paragraph of my previous post. 

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3 hours ago, Boydstun said:

Wouldn’t it be sensible to say that algebraic variables mean just any and all magnitude relations (spatial or otherwise) that might be captured in such analytic relations as we have in this case? Do you think Hilbert would object to that sort of picture of algebraic symbols?

I have to emphasize that I am not a scholar on Hilbert, mathematics, or philosophy, so my explanations are not necessarily always perfectly on target, and at a certain depth, I would have to defer to people who have studied more extensively than I have. And I don't mean necessarily to defend Hilbert's philosophical notions in all its aspects.

That said, however, here's a stab at answering your question:

I think what Hilbert has in mind is the distinction between a) reasoning with symbols that are taken as representing particular numbers and b) making generalizations about an infinite class of numbers.

For example, if 'a' is a token for a particular number, then the truth of 'a+1 = 1+a' cannot be reasonably contested as it can be concretely verified - it is finitistic. For example, for the particular numeral '2', the truth of '2+1 = 1+2' cannot be reasonably contested as it can be concretely verified.  

On the other hand, where 'A' stands for any undetermined member of entire infinite class of numbers, then 'A+1 = 1+A' (which is ordinarily understood as 'for all numbers A, we have A+1 = 1+A') cannot be verified concretely because it speaks of an entire infinite class that we can't exhaustively check. Therefore, some other regard must be given the formula. And that regard is to take it as not "contentual" but as "ideal" but formally provable from formal axioms (which are themselves "ideal"). And it is needed that there is an algorithm that can check for any purported formal proof that it actually is a formal proof (i.e., that its syntax is correct and that every formula does syntactically "lock" in sequence in applications of the formal rules); this is what Hilbert has in mind as the formal "game". Then Hilbert hoped that there would be found a formal proof, by using only finitistic means, that the "ideal" axioms sufficient for ordinary mathematics are consistent. Godel, though, proved that Hilbert's hope cannot be realized. 

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Thank you, GM, for the stab at it. I appreciate this thread's info, as I mentioned earlier. I'll try to add to this thread in weeks ahead. It is pertinent for getting arms around the relation of Hilbert ideas to Godel and to Carnap---for my book in progress and for my treatment at this site of Peikoff's dissertation and S/A essay for Rand's phi (1960's)---concerning analyticity and the nature of knowledge of pure mathematics.

Edited by Boydstun

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Why would induction be necessary?

If “+” is well defined, and if each of the LHS and RHS of the equation are meaningful, how can it ever not be true?

1 is 1

A is A (we do NOT need induction for this)

The only possible problem then is an ill defined “+”, but we get to define what “+” means .. we have to ... for either side of the equation to be meaningful.

and doesn’t the definition of  “+” satisfy a+b = b+a anyway?  Certainly if it means “summation”... 

 

Edit:  The other possible problem is perhaps “=“ doesn’t really mean “equals” ie it does not signify identity of the left and right hand sides with each other.

If “+” or “=“ are i’ll defined and not really “summation” and “equals” induction won’t work anyway.

Edited by StrictlyLogical

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7 hours ago, StrictlyLogical said:

Why would induction be necessary?

The nature of mathematical induction assures that it won't be derailed by infinite cardinals or non-commutative ordinals (link).   😲

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2 hours ago, merjet said:

The nature of mathematical induction assures that it won't be derailed by infinite cardinals or non-commutative ordinals (link).   😲

I do not see how anything I have written could be derailed.

A is a natural number as are 1, a, and b.

Will the "infinite cardinals" or "non-commutative ordinals" invade the equation? or either the RHS or LHS? or maybe infect "+" or "=" somehow?

 

I still don't see why induction (when possible) is necessary here.

and for clarity, yes, A does stand for any natural number i.e. is not restricted to any particular natural number.

 

I don't want to distract the discussion but as an aside:

If the simple equation with natural numbers 1+A=A+1 could be "derailed", "somehow", or if "+" or "=" are unreliable then induction itself, in principle, is not safe either as it relies upon "=", "+" and natural numbers.

Induction relies on some premise being true for X implying (necessitating) that the premise is also true for X+1... therefore the premise is true for all X  (oops there is the "+" sign... so that cant work for an ill defined "+" with a natural number on either side).

Perhaps we need two inductions...

one showing P is true for X implies it is true for X+1

AND

P is true for X implies it is true for 1+X...

It also seems almost nearly impossible to avoid use of "=" in a proof by induction, so any problem with "=" as such is not solved by induction.

and if a natural number is not a natural number... (can be infected by non-commutative ordinal or infinite cardinal) then its not a natural number...

 

 

 

 

Edited by StrictlyLogical

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It seems to me that there is an equivocation of the term "equals." In some cases it means that two things "are the same presently", and in other cases it means they "are the same once calculated or solved." In the equation "one plus 'a' equals 'a' plus one", equals here means that both sides are the same once calculated. It doesn't mean that both sides are presently equal. So it doesn't make sense that the concept of infinite numbers would render the equation unverifiable. The equation is verified every time it's solved by plugging in a particular number and doing the calculations. The mistake is treating both sides as if they are supposed to be equal presently, which they are not. They represent different possible calculations that result in the same number. 

Edited by MisterSwig

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16 minutes ago, MisterSwig said:

"are the same presently", and in other cases it means they "are the same once calculated or solved."

This would appear to render any mathematical expression as meaningless... 

an expresison stands for what it means, an expression cannot stand literally for itself (as an expression)... it is itself (an expression) and stands for what it means.

1+3

is not "presently" different from "once calculated"

1+3 MEANS the result of adding 1 and 3.

The expression does not mean "the expression 1+3 before it is calculated"...  the meaning of the expression is not the expression itself, the meaning of the expression is what the expression means.

 

Equality of the RHS and the LHs means the MEANING of the LHS and the RHS are identical.  It is a given that the expression or the calculation on either side is not the same.

The expression "1+A" IS not the expression "A+1".

but what 1+A IS, is identical to what A+1 IS, given what 1+A means and what A+1 means.

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