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GrandMinnow

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  1. No one dispenses with the ordinary notion of natural numbers. But regarding in terms of a function allows mathematics to characterize and handle the natural numbers in an exact way, and without having to say each time "what we do when we start with zero and then add one, and keep adding one after another", not just from the outset (such as the PA axioms) but when the study gets much more complicated. The use of the notion of the successor function is utterly basic even to the study of computablity that has enabled the advent of such things as the computer you are using now. Also note: The successor function is not used to build or define the natural numbers with PA. And in set theory, there are equivalent ways of defining 'is a natural number', some of them not using a successor operation. Then there is also proving that there is the set whose members are all and only the natural numbers, and that proof "basically" amounts to pointing out that there is the set that has 0 and then the next after 0, and so on, and no other members*. And, to me, that seems pretty close to the common, informal, everyday sense too. * Though, to be fair, this is accomplished by using "surrogate" notions in terms of sets. It would be a rare mathematician or philosopher of mathematics who would advocate anything that extreme or even the essence of it. On the contrary. Meanwhile, mathematicians have found that "idealizations" in mathematics are useful (or perhaps even essential) for developing the mathematics for the sciences. And formalization offers the ultimate objectivity in settling any question whether a purported mathematical proof (when put formally) is indeed a proof, as formalization entails that there is an algorithm to determine of any purported proof (when put formally) whether it is indeed a proof. Some mathematicians find this to be useful, or philosophically welcome, or just interesting. Other mathematicians don't care about it. And, of course, any non-mathematician is free to disregard it or disdain it.
  2. What’s wrong with the successor function? It’s basic arithmetic. It’s basic to the computer programs that are running your computer. I don’t know what your objection is.
  3. Quite so. I fixed it now. Thanks.
  4. Mathematicians, and different mathematicians, mean different things depending on context. The context is either stated explicitly or reasonably gleaned per a given book or article. So, just to narrow down, let's look at just two of the different contexts. (They are different but they support each other anyway.) To avoid getting too complicated for the purposes of brief posting, I'll give only a sketch, leaving out a lot of details, and not explain every concept (such as 'free variable') and taking some liberties with the notation and concepts, and for ease of reading, I won't always include quote marks to distinguish mention as opposed to use. (So this is not as accurate as a more authoritative treatment). So two contexts: (1) General, informal (or informal mixed with formal) discussion in mathematics about natural numbers. (2) Formal first order Peano arithmetic [I'll just call it 'PA' here]. (1) In general mathematics, we might taken commutativity of addition to be obvious and thus a given. Or one might say: "Okay, I'm going to state some truths about natural numbers from which I can prove a whole bunch of other truths, even though they're obvious anyway. The truths about addition I want to mention are: 0 added to any number is just that number. In symbols: x+0 = 0. The sum of a number and the successor of another (or same) number is just the successor of the sum of the number and the other number. In symbols: x+Sy = S(x+y), or, put another way (where 'S' is defined as '+1'), x+(y+1) = (x+y)+1. The induction rule. Now, with those three truths, one of the many truths I can prove, without assuming anything about natural numbers or what they are, other than those three truths, is the commutativity of addition. In whatever way you conceive the natural numbers, as long that conception includes those three truths I just mentioned, then the commutativity of addition is proven true." Notice that we can't do this with the real numbers, because the induction rule does not work for the real numbers. So, for real numbers, we would take commutativity as an axiom (or in set theory, we would prove commutativity from the properties of the real numbers as they are set theoretically "constructed"). (2) PA, as a system, has a formal first order language, with the primitive logical symbols (including '=' as a logical symbol) and certain primitive non-logical symbols. The logical symbols are: Infinitely many variables: x, y, etc. -> (interpreted as the material conditional) ~ (interpreted as negation) and, from '->' and '~' we can define: & (interpreted as conjunction) v (interpreted as inclusive disjunction) A (so that, where P(x) is any formula with 'x' occurring free, AxP is always interpreted as "for all x, P(x)") and, from 'A' and '~' we can define: E (so that ExP(x) is always interpreted as "there is an x such that P(x)") The non-logical symbols are : 0 S + * We define S(0) =1 S(1) = 2 etc. When the language is interpreted: '0' is assigned to a particular member of the domain of the interpretation; 'S' is assigned to a 1-place function (operation) on the domain, '+' and '*' are each assigned to 2-place functions on the domain. With the "intended" ("standard") interpretation: the domain is the set of natural numbers, '0' is assigned to the number zero, 'S' is assigned to the successor operation, and '+' and '*' are assigned to the addition and multiplication operations respectively. And, since '=' is a logical primitive, we assign it to the identity (equality) relation on the domain. So for any interpretation (such that each variable, in its role as a free variable, is assigned to some member of the domain): x+y is assigned to the value of the '+' operation applied to the ordered pair: <the assigned value of x, the assigned value of y>. And x+y = y+x holds in the interpretation if and only if the value of x+y is identical with (is equal to) the value of y+x. So, to answer your question, in the syntax of the formal system itself, nothing is assumed as to what 'x' and 'y' stand for. But with a formal interpretation of the system, 'x', as a free variable stands for some member of the domain and 'y', as a free variable, stands for some member of the domain. And with the standard interpretation, the domain is the set of natural numbers. However, often we tacitly understand that when formulas such as x+y = y+x are asserted, we take that assertion to be the universal closure: AxAy x+y = y+x (abbreviated Axy x+y = y+x) And so, with the standard interpretation, that asserts that addition is commutative. And we prove it from the PA axioms (we only need the three I mentioned in a previous post, which correspond to the three truths I mentioned in this post).
  5. We prove that order doesn't matter (that addition is commutative). Strictly Logical is asking why (mathematical) induction is needed to prove that addition is commutative. So the answer to 'why is induction needed to prove the commutativity of addition?' is not 'to prove the commutativity of addition'. The answer is that the commutativity of addition can't be proven from the Peano axioms without the axiom of induction (or, in set theory it is the inductive property of the natural numbers that permits the proof of commutativity of addition). The relevant axioms in Peano arithmetic ('S' stands for successor; and the formulas are tacitly taken to be the universal closures): x+0 = x x+Sy = S(x+y) From that, you can't prove x+y = y+x You need also the axiom schema of induction: For all formulas P, If P(0) and (for all x, P(x) implies P(x+1)), then for all x, P(x). ------ Or one could take x+y = y+x as an axiom. But it is not needed to do that, since it is already provable from the other axioms.
  6. Yes, exactly, the word 'apple' is not an apple. And '1+3' is not a number, it is a word (or term). And 1+3 is not a word, it is a number, and it is the number 4. John Goodman is a person, not a name. 'John Goodman' is a name, not a person. 1+3 is a number, not a name. '1+3' is a name, not a number.
  7. An argument is valid if and only if if its premises are true then its conclusion must be true. An argument is sound if and only if it is valid and its premises are true.
  8. 1+3 is the same as 10-6. It is a number. ‘1+3’ is different from ‘10-6’. They refer to the same number, but they are different strings of symbols This is an example of the distinction between use and mention.
  9. Some topics that have been mentioned: (1) MATHEMATICAL INDUCTION on NATURAL NUMBERS Induction [by 'induction' in such contexts, I mean mathematical induction] is ordinarily used in these contexts: * Proofs from the axioms of Peano arithmetic [by 'Peano arithmetic' I mean first order Peano arithmetic] in which induction is an axiom. Induction is needed because there are many things you can't prove about natural numbers from the Peano axioms without the induction axiom. * Proofs from the axioms of set theory in which there is the set of all and only the natural numbers and that set admits induction. Induction is used because it is the induction property of the natural numbers that permits many of the proofs about natural numbers. * Proofs historically before Peano arithmetic or set theory. (But such proofs can be put in Peano arithmetic or set theory retroactively.) * Proofs in general mathematics in instances where Peano arithmetic or set theory are not necessarily explicitly mentioned. (But said mathematics can be formulated in Peano arithmetic or set theory.) And none of this stems from any supposed need to avoid "derailment" from infinite cardinals or ordinal addition. (2) USE/MENTION There is a distinction between a) symbols, or sequences of symbols that are terms, to stand for objects or range as a variables over objects and b) the objects that are symbolized. Single quote marks indicate that a linguistic object - a symbol or sequence of symbols - is referred to. (Actually, more exactly, for sequences we would use a concatenation marker, but that is too pedantic for this discussion.) '2' is a symbol (a linguistic object), it is not a number. However, 2 is a number. But this has really nothing to do with stating the commutativity of addition. (3) IDENTITY x = y means x and y are the same object. So '=' stands for the identity (equality) relation. If T and S are terms, then T = S means that T and S both name the same object. Equivalence was mentioned. The identity relation is an equivalence relation, but there are equivalence relations other than identity. But there is nothing gained in this discussion by mentioning a warning against confusion with equivalence relations. There is no mistaking that '=' stands for identity. (4) An article titled 'Infinity plus one' was linked to. The title of that article is misleading. In regards to cardinals, we don't use 'infinity' as a noun, but rather 'is infinite' as an adjective. (This is different from such things as "points of infinity" in the extended reals system, as such points don't refer to cardinality but rather to ordering.) (5) This comment was posted: "It was posited that the equation (1+a=a+1) could not be verified, because we would need to check it against every possible number, which is impossible to do because infinity." Just to be clear, that is not necessarily my own view, but rather it was part of a brief explanation of Hilbert's views, and even in that regard, the statement needs important qualifications such as those I mentioned.
  10. 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.
  11. 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.
  12. 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.
  13. 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.
  14. 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.
  15. 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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