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> formalism: the idea that mathematics can be reconstructed in a content-free manner, or more strongly put, that mathematics itself is nothing but “a formal gam

by pron 7y ago

> formalism: the idea that mathematics can be reconstructed in a content-free manner, or more strongly put, that mathematics itself is nothing but “a formal game of symbols”. This is a common misrepresentation of Hilbert's Formalism. It doesn't state that no mathematical sentence has content beyond its mere symbols, but that not every mathematical sentence has content beyond its mere symbols. Formalism was meant as a compromise between the two dueling philosophies of mathematics of the time: Russel's Logicism, which essentially said that all of mathematics can be derived from some self-evident, obviously true, logical axioms, and Brouwer's Intuitionism, which essentially said that the truth of mathematics stems from empirical confirmation that breaks down when infinities are involved, and therefore some logical axioms cannot be taken to be true for infinites. Hilbert tried to accept Brouwer's finitist philosophical position without rejecting any useful axioms. Formalism, then, separates mathematical sentences into two kinds: "real" -- those dealing with finite quantities and can be tied to physical observation and prediction, and "ideal" -- those dealing with infinities. Instead of rejecting the latter, Formalism says that as long as they don't introduce inconsistencies that invalidate the former, one could view them -- the ideal sentences -- as having no content beyond symbols, i.e. as not referring to anything "real".

neel_k 7y ago
Brouwer was absolutely not a finitist! Indeed, he completely rejected the idea that you could use formal logic to bound what mathematics could talk about. A better (but still not entirely accurate) description would be that Brouwer had an epistemic conception of mathematical truth: mathematicians become able to assert mathematical truths as they engage in the process of constructing mathematical objects. This is indeed a constructive process, but it isn't finitistic -- for Brouwer mathematical reasoning is open-ended rather than a closed set of rules. So much like Goedel (who disagreed with Brouwer in many other respects), Brouwer was totally committed to the idea that mathematical statements actually mean things. And because they carry meaning, the mathematical process and the properties of the mathematical objects created by a mathematician take precedence over the rules of logic. This is why Brouwer rejected classical logic -- not because of any finitistic reason, but because (for example) classical logic permits disassembling the continuum into its set of points. Since (for Brouwer) the point of a continuum is that it is continuous, it followed that any logical principles (such as classical logic plus set theory) that permitted you to break up the continuum were fallacious. This is why he spent a lot of time on his so called "strong counterexamples", such as his proof that intuitionistically all total functions on the reals are continuous. Notice that the direction of reasoning here is exactly the opposite of what a finitist would accept: Brouwer used the properties of the continuum to justify various finitary logical rules, rather than the other way around. The continuum was real, and logic just a façon-de-parler. I find this kind of thing really interesting because it illustrates that the pretheoretical idea of truth can be formalized in many different ways, and that the various formalizations are not all compatible with each other. We are used to informal ideas like space having many different mathematical realizations -- metric spaces, topological spaces, vector spaces, etc -- but it is a little disconcerting (and a lot fascinating) to realize that logic and truth exhibit the same plural nature.
- pron 7y ago
  Hey Neel! Brouwer was a finitist in the sense that he did not accept infinitist axioms. His epistemology indeed was one of "mathematicians become able to assert mathematical truths as they engage in the process of constructing mathematical objects," but the justification was empiricism. I.e., his concept of establishing truth was by empirical, and therefore finitist. Here he is in "On the significance of the excluded middle in mathematics", 1923 (in Heijenoort): > Within a specific finite "main system" we can always test (that is, either prove or reduce to absurdity) properties of systems, that is, test whether systems can be mapped, with prescribed correspondences between elements, into other systems; for the mapping determined by the property in question can in any case be performed in only a finite number of ways, and each of these can be undertaken by itself and pursued either to its conclusion or to a point of inhibition [here is his mention of induction that I provide below]. > ... On the basis of the testability just mentioned, there hold, for properties conceived within a specific finite main system, the principle of excluded middle > ... It is a natural phenomenon, now, that numerous objects and mechanisms of the world of perception, considered in relation to extended complexes of facts and events, can be mastered if we think of them as (possibly partly unknown) finite discrete systems that for specific known parts are bound by specific laws of temporal concatenation. Hence the laws of theoretical logic, including the principle of excluded middle, are applicable to these objects and mechanisms in relation to the respective complexes of facts and events, even though here a complete empirical corroboration of the inferences drawn is usually materially excluded a priori and there cannot be any question of even a partial corroboration in the case of (juridical and other) inferences about the past. Note his use of "testability", "natural phenomenon" and "empirical corroboration." So the source of truth is nature by way of empiricism, and that demands finitism. Even his reasoning about the continuum was finitist: no determination on order/separation can be made after examining a finite prefix of the decimal expansion of a real number (see ibid §2). If he were alive today, I have little doubt he would have been an ultra-finitist -- not any constructivist! -- because, as it turns out, there is no empirical justification (even in principle) to accept inifinite induction. The very arguments he uses to reject LEM must also reject unbounded induction: it is not verified by nature. (Here's what he says on induction: > Here the principle of mathematical induction often furnishes the means of carrying out such tests without individual consideration of every element involved in the mapping or of every possible way in which the mapping can be performed; consequently the test even for systems with a very large number of elements can at times be performed relatively rapidly. After the discovery of computational complexity he would have had no choice but to be an ultra-finitist.) As to the notion of truth, I completely agree, but here Alan Turing came up with a completely different perspective. Whereas both Brouwer and Russel were looking for Truth in mathematics (and Hilbert, too, although he was content to Truth in the contentual "real propositions" and the finitist rules of inferene), Turing rejected that notion altogether. To him, mathematics was a means whose intrinsic truth is irrelevant. It is valuable inasmuch as it is useful, not because it is true (see https://mdetlefsen.nd.edu/assets/201037/jf.turing.pdf https://mdetlefsen.nd.edu/assets/201037/jf.turing.pdf).
dwohnitmok 7y ago
Hilbertian formalism is only one particular brand of formalism. The definition in the article is the one put forth by Frege, perhaps the man most responsible for launching the debate about formalism into the wider philosophical community. It seems okay to me for an informal article to define formalism in the same way (certainly the game aspect of it is the thing that ties the various schools of formalism together).
- pron 7y ago
  Frege started Logicism, so I don't think he would say that mathematical sentences lack content. Where are you getting this?
  - dwohnitmok 7y ago
    Frege was certainly arguing against formalism; he was the one responsible for taking its defenders to task. But the formalism he presented and argued against (and the version held by its defenders of the day) is clearly the form described in the article and the discussions he sparked later all circled around the same. Frege quotes Heine as > Suppose I am not satisfied to have nothing but positive rational numbers. I do not answer the question, What is a number?, by defining number conceptually, say by introducing irrationals as limits whose existence is presupposed. I define from the standpoint of the pure formalist and call certain tangible signs numbers. and Thomae > The formal conception of arithmetic accepts more modest limitations than does the logical conception. It does not ask what numbers are and what they do, but rather what is demanded of them in arithmetic. For the formalist, arithmetic is a game with signs, which are called empty. That means they have no other content (in the calculating game) than they are assigned by their behavior with respect to certain rules of combination (rules of the game). within the Grundgesetze der Arithmetik. And indeed Frege defines the position he is arguing against as > Arithmetic is concerned only with the rules governing the manipulation of the arithmetical signs, not, however, with the denotation of the signs These are from Max Black's translation of the Grundgesetze der Arithmetik in his three part series "Frege Against the Formalists." These quotes are from part 1 in The Philosophical Review, Vol. 59, No. 1 (Jan., 1950), pp. 79-93

by pron 7y ago

neel_k 7y ago

Brouwer was absolutely not a finitist! Indeed, he completely rejected the idea that you could use formal logic to bound what mathematics could talk about. A better (but still not entirely accurate) description would be that Brouwer had an epistemic conception of mathematical truth: mathematicians become able to assert mathematical truths as they engage in the process of constructing mathematical objects. This is indeed a constructive process, but it isn't finitistic -- for Brouwer mathematical reasoning is open-ended rather than a closed set of rules. So much like Goedel (who disagreed with Brouwer in many other respects), Brouwer was totally committed to the idea that mathematical statements actually mean things. And because they carry meaning, the mathematical process and the properties of the mathematical objects created by a mathematician take precedence over the rules of logic. This is why Brouwer rejected classical logic -- not because of any finitistic reason, but because (for example) classical logic permits disassembling the continuum into its set of points. Since (for Brouwer) the point of a continuum is that it is continuous, it followed that any logical principles (such as classical logic plus set theory) that permitted you to break up the continuum were fallacious. This is why he spent a lot of time on his so called "strong counterexamples", such as his proof that intuitionistically all total functions on the reals are continuous. Notice that the direction of reasoning here is exactly the opposite of what a finitist would accept: Brouwer used the properties of the continuum to justify various finitary logical rules, rather than the other way around. The continuum was real, and logic just a façon-de-parler. I find this kind of thing really interesting because it illustrates that the pretheoretical idea of truth can be formalized in many different ways, and that the various formalizations are not all compatible with each other. We are used to informal ideas like space having many different mathematical realizations -- metric spaces, topological spaces, vector spaces, etc -- but it is a little disconcerting (and a lot fascinating) to realize that logic and truth exhibit the same plural nature.

pron 7y ago
Hey Neel! Brouwer was a finitist in the sense that he did not accept infinitist axioms. His epistemology indeed was one of "mathematicians become able to assert mathematical truths as they engage in the process of constructing mathematical objects," but the justification was empiricism. I.e., his concept of establishing truth was by empirical, and therefore finitist. Here he is in "On the significance of the excluded middle in mathematics", 1923 (in Heijenoort): > Within a specific finite "main system" we can always test (that is, either prove or reduce to absurdity) properties of systems, that is, test whether systems can be mapped, with prescribed correspondences between elements, into other systems; for the mapping determined by the property in question can in any case be performed in only a finite number of ways, and each of these can be undertaken by itself and pursued either to its conclusion or to a point of inhibition [here is his mention of induction that I provide below]. > ... On the basis of the testability just mentioned, there hold, for properties conceived within a specific finite main system, the principle of excluded middle > ... It is a natural phenomenon, now, that numerous objects and mechanisms of the world of perception, considered in relation to extended complexes of facts and events, can be mastered if we think of them as (possibly partly unknown) finite discrete systems that for specific known parts are bound by specific laws of temporal concatenation. Hence the laws of theoretical logic, including the principle of excluded middle, are applicable to these objects and mechanisms in relation to the respective complexes of facts and events, even though here a complete empirical corroboration of the inferences drawn is usually materially excluded a priori and there cannot be any question of even a partial corroboration in the case of (juridical and other) inferences about the past. Note his use of "testability", "natural phenomenon" and "empirical corroboration." So the source of truth is nature by way of empiricism, and that demands finitism. Even his reasoning about the continuum was finitist: no determination on order/separation can be made after examining a finite prefix of the decimal expansion of a real number (see ibid §2). If he were alive today, I have little doubt he would have been an ultra-finitist -- not any constructivist! -- because, as it turns out, there is no empirical justification (even in principle) to accept inifinite induction. The very arguments he uses to reject LEM must also reject unbounded induction: it is not verified by nature. (Here's what he says on induction: > Here the principle of mathematical induction often furnishes the means of carrying out such tests without individual consideration of every element involved in the mapping or of every possible way in which the mapping can be performed; consequently the test even for systems with a very large number of elements can at times be performed relatively rapidly. After the discovery of computational complexity he would have had no choice but to be an ultra-finitist.) As to the notion of truth, I completely agree, but here Alan Turing came up with a completely different perspective. Whereas both Brouwer and Russel were looking for Truth in mathematics (and Hilbert, too, although he was content to Truth in the contentual "real propositions" and the finitist rules of inferene), Turing rejected that notion altogether. To him, mathematics was a means whose intrinsic truth is irrelevant. It is valuable inasmuch as it is useful, not because it is true (see https://mdetlefsen.nd.edu/assets/201037/jf.turing.pdf https://mdetlefsen.nd.edu/assets/201037/jf.turing.pdf).

dwohnitmok 7y ago

Hilbertian formalism is only one particular brand of formalism. The definition in the article is the one put forth by Frege, perhaps the man most responsible for launching the debate about formalism into the wider philosophical community. It seems okay to me for an informal article to define formalism in the same way (certainly the game aspect of it is the thing that ties the various schools of formalism together).

pron 7y ago
Frege started Logicism, so I don't think he would say that mathematical sentences lack content. Where are you getting this?
- dwohnitmok 7y ago
  Frege was certainly arguing against formalism; he was the one responsible for taking its defenders to task. But the formalism he presented and argued against (and the version held by its defenders of the day) is clearly the form described in the article and the discussions he sparked later all circled around the same. Frege quotes Heine as > Suppose I am not satisfied to have nothing but positive rational numbers. I do not answer the question, What is a number?, by defining number conceptually, say by introducing irrationals as limits whose existence is presupposed. I define from the standpoint of the pure formalist and call certain tangible signs numbers. and Thomae > The formal conception of arithmetic accepts more modest limitations than does the logical conception. It does not ask what numbers are and what they do, but rather what is demanded of them in arithmetic. For the formalist, arithmetic is a game with signs, which are called empty. That means they have no other content (in the calculating game) than they are assigned by their behavior with respect to certain rules of combination (rules of the game). within the Grundgesetze der Arithmetik. And indeed Frege defines the position he is arguing against as > Arithmetic is concerned only with the rules governing the manipulation of the arithmetical signs, not, however, with the denotation of the signs These are from Max Black's translation of the Grundgesetze der Arithmetik in his three part series "Frege Against the Formalists." These quotes are from part 1 in The Philosophical Review, Vol. 59, No. 1 (Jan., 1950), pp. 79-93