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> It is (more or less) the observation that given a set S with two elements, you can make a subset of S that has one element. But your formal definition and yo

> It is (more or less) the observation that given a set S with two elements, you can make a subset of S that has one element. But your formal definition and your explanation do not match. Effectively you are conflating defining the "cardinality of a set is 2" with "2." To illustrate this, your formal definition can be satisfied by a set that only has one member ({{0}} with 0 as the empty set)! It has no strict subset with one element! Your formal definition is but a numeral system, your informal definition is the number itself. Indeed nothing changes in spirit if we start with "{0}" as 0 instead of the empty set itself, but your formal definition fails in another way. Another more dramatic illustration is just to define this entirely with natural number objects in category theory, which do away with properties of sets and collections entirely. It leads to statements which are also true, but cannot even be phrased in the language of set theory. > I picked the ZF definition of two as "this definition" but I didn't have to. I could have picked something else, and then the phraseology would have been different. But it all eventually bottoms out in the theory of computability. You can start with ZF or Turing Machines, or the lambda calculus, or Godel numbers, or Brainfuck. It doesn't matter. All starting points lead to the same destination. This is the entire point, Turing Machines, lambda calculus, Godel numbers are all different sets of rules and symbols. Yet we recognize that they are all the "same destination" somehow and that destination is the thing we want, not the rules and symbols themselves, which are, as you say, only the "starting point." In effect, none of these definitions of natural number can fully capture what a natural number is. They each come with their own infelicities and ultimately end up as numeral systems when fully formalized rather than the natural numbers themselves. I agree that computability is one of the key insights behind mathematics, but moving symbols around is only one encoding of computability. EDIT: Another thing you still haven't addressed is the issue of the reality of natural numbers and Godel's incompleteness theorems and how we are allowed to talk about "natural numbers" as mathematical objects in that scenario.

lisper 5y ago
> that destination is the thing we want Agreed. But... > moving symbols around is only one encoding of computability I think we may be using different definitions of the word "symbol" then. Can you give me an example of an instantiation of this "destination" that is not moving symbols around?
- dwohnitmok 5y ago
  > Can you give me an example of an instantiation of this "destination" that is not moving symbols around? Visual proofs are a good example of this: https://mathoverflow.net/questions/8846/proofs-without-words https://mathoverflow.net/questions/8846/proofs-without-words The main objection against them is that diagrams can be subtly manipulated to lie or that they are overly specific in what they prove. Which leads me to... A clearer but unfortunately unrealistic example of this would be if we could have direct mind-to-mind communication in the future. I think the bulk of mathematicians would argue we would still have mathematics in that world and any symbol written down would be only an encoding of the mathematical ideas rather than the mathematics themselves, which are transmitted directly as abstract thought. Put another way, symbolic proofs are a crutch for mathematics born out of the frailties of the human body. If humans were capable of perfect, repeatable actions or otherwise directly transmit information, then symbols would not be needed to do mathematics. We could perform computations "directly" as it were. We use symbols because that is not possible, but that is a contingent concern, not a necessary one. But again the most thorny question of what mathematics is about comes from the reality of the natural numbers and Godel's incompleteness theorem. Most mathematicians would agree we study the natural numbers as they exist in our reality, and yet those can't be fully captured by symbols, so what exactly are we studying then? Now this whole line of argumentation feels a bit unnatural to me (I don't believe all of what I've been saying in these replies). As I mentioned at the beginning of all this that I'm actually quite a bit of a formalist myself despite all this and I'm really hiding behind a realist costume while I dance around here. In the absence of direct mind-to-mind communication, formalism is a great practical benchmark for rigor in mathematics (can you see how to turn this prose description into a symbol manipulation game? No? Well then the description is probably not precise enough). And that's good enough for me to put on a formalist hat most of the time when I do mathematics (there are certain intuitions in mathematical logic that are harder to grasp with a formalist hat on so that's when I take it off). I think it's as good a backup to have as "shut up and calculate" is in physics. But as a philosophical understanding of what mathematics "really is all about," I still find it wanting, especially if I believe that there is only one "true" set of natural numbers in our universe (which sometimes I don't, but if you believe that the natural numbers are physically realized by real objects, then it's a compelling conclusion), because if you believe that, then Godel's Incompleteness Theorems are a philosophical death blow for formalism. But... speaking for myself personally and leaving aside the persona I've adopted for the purposes of this discussion, I don't usually care what mathematics "really is all about." I do have a formalist counterargument for Godel's Incompleteneess theorems as well, but I've already written too much already and will only reply to that if anybody is interested. EDIT: This is an even better example of visual proofs: https://moodle.tau.ac.il/2018/pluginfile.php/403616/mod_resource/content/1/Nelsen%201993%20Proofs%20without%20Words.pdf https://moodle.tau.ac.il/2018/pluginfile.php/403616/mod_reso... again the fact that they are not 100% trustworthy is more a problem of physical deficiencies in humans and materials than anything else. EDIT EDIT: Another example of non-symbolic computation would be using physical processes for computation. Again as a practical matter, this is an inversion of results, since we do not have a perfect understanding of physical principles, but assuming we were perfect physicists and there are indeed entirely featureless pure particles (which it seems like there are), then we could do computation via physics directly instead of symbols. EDIT EDIT EDIT: Although this can be rephrased in a formalist framework of symbols and manipulation rules, a great practical example of this is diagram chasing in category theory. Although, unlike the case of pure particles, you can argue that this is just a symbol game in 2-d, it certainly doesn't feel like one in the heat of the moment and comes closest to approximating what rigorous, but non-symbolic proofs would "feel" like.

lisper 5y ago

> that destination is the thing we want Agreed. But... > moving symbols around is only one encoding of computability I think we may be using different definitions of the word "symbol" then. Can you give me an example of an instantiation of this "destination" that is not moving symbols around?

dwohnitmok 5y ago
> Can you give me an example of an instantiation of this "destination" that is not moving symbols around? Visual proofs are a good example of this: https://mathoverflow.net/questions/8846/proofs-without-words https://mathoverflow.net/questions/8846/proofs-without-words The main objection against them is that diagrams can be subtly manipulated to lie or that they are overly specific in what they prove. Which leads me to... A clearer but unfortunately unrealistic example of this would be if we could have direct mind-to-mind communication in the future. I think the bulk of mathematicians would argue we would still have mathematics in that world and any symbol written down would be only an encoding of the mathematical ideas rather than the mathematics themselves, which are transmitted directly as abstract thought. Put another way, symbolic proofs are a crutch for mathematics born out of the frailties of the human body. If humans were capable of perfect, repeatable actions or otherwise directly transmit information, then symbols would not be needed to do mathematics. We could perform computations "directly" as it were. We use symbols because that is not possible, but that is a contingent concern, not a necessary one. But again the most thorny question of what mathematics is about comes from the reality of the natural numbers and Godel's incompleteness theorem. Most mathematicians would agree we study the natural numbers as they exist in our reality, and yet those can't be fully captured by symbols, so what exactly are we studying then? Now this whole line of argumentation feels a bit unnatural to me (I don't believe all of what I've been saying in these replies). As I mentioned at the beginning of all this that I'm actually quite a bit of a formalist myself despite all this and I'm really hiding behind a realist costume while I dance around here. In the absence of direct mind-to-mind communication, formalism is a great practical benchmark for rigor in mathematics (can you see how to turn this prose description into a symbol manipulation game? No? Well then the description is probably not precise enough). And that's good enough for me to put on a formalist hat most of the time when I do mathematics (there are certain intuitions in mathematical logic that are harder to grasp with a formalist hat on so that's when I take it off). I think it's as good a backup to have as "shut up and calculate" is in physics. But as a philosophical understanding of what mathematics "really is all about," I still find it wanting, especially if I believe that there is only one "true" set of natural numbers in our universe (which sometimes I don't, but if you believe that the natural numbers are physically realized by real objects, then it's a compelling conclusion), because if you believe that, then Godel's Incompleteness Theorems are a philosophical death blow for formalism. But... speaking for myself personally and leaving aside the persona I've adopted for the purposes of this discussion, I don't usually care what mathematics "really is all about." I do have a formalist counterargument for Godel's Incompleteneess theorems as well, but I've already written too much already and will only reply to that if anybody is interested. EDIT: This is an even better example of visual proofs: https://moodle.tau.ac.il/2018/pluginfile.php/403616/mod_resource/content/1/Nelsen%201993%20Proofs%20without%20Words.pdf https://moodle.tau.ac.il/2018/pluginfile.php/403616/mod_reso... again the fact that they are not 100% trustworthy is more a problem of physical deficiencies in humans and materials than anything else. EDIT EDIT: Another example of non-symbolic computation would be using physical processes for computation. Again as a practical matter, this is an inversion of results, since we do not have a perfect understanding of physical principles, but assuming we were perfect physicists and there are indeed entirely featureless pure particles (which it seems like there are), then we could do computation via physics directly instead of symbols. EDIT EDIT EDIT: Although this can be rephrased in a formalist framework of symbols and manipulation rules, a great practical example of this is diagram chasing in category theory. Although, unlike the case of pure particles, you can argue that this is just a symbol game in 2-d, it certainly doesn't feel like one in the heat of the moment and comes closest to approximating what rigorous, but non-symbolic proofs would "feel" like.