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> Humans can create notation and formalisms, but they do not invent the truths those mathematics represent. The land represented by a map exists independently

> Humans can create notation and formalisms, but they do not invent the truths those mathematics represent. The land represented by a map exists independently of humanity. Another intelligent species would have to come up with a roughly isomorphic representation if they wanted a similar tool. Maps, to be clear, are just invented tools. They can be more or less right or wrong, but they are not the territory. Moving up the meta stack does sort of confuse this initially if you don't stay grounded. I wonder if there is a field of meta-map-making and whether map makers sometimes confuse maps with territories when they start meta-map-making work. > I frequently encounter this perspective and worry there's a fundamental problem with how mathematics is taught if so many people walk away believing this. I didn't walk away (undergrad, PhD, PI, editorial committees, grant reviewer, ...). I'm about as far from walking away as is possible. But I suppose it is possible I'm an imposture :)

whatshisface 5y ago
Any species would prove the exact same theorems given the same axioms, and since mathematicians only claim that their axioms imply their theorems, I think they are right to claim absolute truth.
- thwayunion 5y ago
  This is either an extremely obvious and boring observation or the basis for a metaphysical trip, depending on how pre-disposed you are to "mathematical spiritualism" :)
  - whatshisface 5y ago
    Nothing can be objectively interesting, only objectively true. Just because math is objectively true does not mean you've been robbed of your license to decide if you think it's interesting. :)
    - thwayunion 5y ago
      Agreed. Similarly, just because I don't "get" church doesn't mean I can prove God doesn't exist. And it certainly doesn't mean I should stand in the way of others enjoying the experience of going to church regardless of their beliefs. It just means I don't "get" it.
      - whatshisface 5y ago
        Religion is slightly different though, they claim actual direct truth (not mere truth of implication given certain assumptions) which makes their claims more interesting but prevents them from claiming automatic objective truth. The Formal Gospel would go, "If God so loved the world that he gave his only begotten son, ..." ;)
        thwayunion 5y ago
        That's why I used words like "spirituality" and "faith" instead of "religion" and "dogma". I'm describing a shared social/psychological phenomenon, not a shared epistemological status. Read my posts less as works of the philosophy of mathematics and more as anthropological musings.
- pfortuny 5y ago
  Oh, dear. Truth being the operating word… There is no truth in a set of axioms we cannot even conceive properly (any infinite set has properties beyond what seems reasonable, even “just” the Natural numbers). From that comes arithmetic, the “most elementary” form of mathematics which cannot be proved consistent… We (I am a working mathematician) do not understand our objects, we can just make do. Only finite graph theory has a chance of being “real”. And it stops being finite very soon. And we certainly should be honest enough to admit that our “science” says very little about the “real” world, where truth lies. Maths is just a tool. Funny, exciting and even in some sense beautiful. But “truth” does it not contain. Except, I insist, in very specific finite constructions. Statements hold but they are not “true” because they do not relate to the real world (otherwise, Frodo reaching Mount Doom would also be “true”). There are no continuous functions out there. Bolzano’s theorem is not “true”.
  - whatshisface 5y ago
    I would contend that A -> B can be true even if A is not true or more relevantly to this discussion if A is unknown. That's math's version of objective truth, where "A" is filled by our various axioms and rules of inference.
    - macrolocal 5y ago
      In ~300 BCE, Euclid decides to work with morphisms not objects.
  - meroes 5y ago
    How can you explain appealing to these “unreal objects” (real numbers, set theory, arithmetic) * does* help science? (Effectiveness maybe) I see you are also a non realist about science. But even the methodological naturalist (one who takes natural empirical science to be the best method but not an ontology) must wonder how we are uncovering and putting more precision to more and more of the world. I don’t think we can currently explain why this made up tool “works”.
  - zmgsabst 5y ago
    The enduring appeal of both triangles and Frodo is that they relate a truth that is hard to see, because it’s diffuse and abstract. But that “abstract truth” is why your pizza doesn’t flop if you fold it or Amazon can’t make a good LOTR.
- JackFr 5y ago
  > Any species would prove the exact same theorems given the same axioms Assume there exists at least one species....
- housecarpenter 5y ago
  Any species could prove the same theorems given the same axioms, but (besides the fact that they might not choose the same axioms) I'm not sure if they would prove the same subset of theorems that we have proven/will prove. Perhaps they'd have different ideas about what is interesting.
  - whatshisface 5y ago
    Human mathematicians are already fanning out into other systems of deduction (constructive mathematics being a great example), and given enough time the mathematicians of each galaxy will eventually discover the other galaxy's mathematics, even if it perhaps happens in a different order.
    - ganzuul 5y ago
      Surely intergalactic mathematicians already know that the only time is now? =) > As Prigogine explains, determinism is fundamentally a denial of the arrow of time. With no arrow of time, there is no longer a privileged moment known as the "present," which follows a determined "past" and precedes an undetermined "future." All of time is simply given, with the future as determined or as undetermined as the past. With irreversibility, the arrow of time is reintroduced to physics.
- stonemetal12 5y ago
  That begs the question. Would any other species pick the exact same axioms? Why would they have the exact same theorems? Are you suggesting there is only one way to think logically?
  - dane-pgp 5y ago
    Any species would face a selective pressure towards theorems which help them understand the world around them (if they have any motivation to prove theorems at all), and they will similarly face a pressure towards choosing the smallest/simplest set of axioms which allows all those theorems to be proven (and new ones to be discovered). In fact, if we assume that neural networks are the only sorts of intelligence that can occur naturally in the universe and be sophisticated enough for arbitrary abstract calculation[0], then we might be able to infer things about the sorts of concepts they will develop and in what order. For example, having the concept of finite sums would likely occur before having the concept of infinite sums. [0] I know that cellular automata can emulate a universal Turing machine, but I can't imagine a situation existing in nature where the cells evolve into an arrangement that produces a Turing machine, much less a machine running a program of instructions that lead to it generating mathematical theorems.
marcosdumay 5y ago
> I wonder if there is a field of meta-map-making There certainly is, although I'm not sure it has a name. Kids gets introductions to it on schools, when they have classes about how to read a map in Geography.
VirusNewbie 5y ago
'roughly isomorphic' would be saying 'not isomorphic', so I'm not sure what you're trying to say. Actually I'm frustrated with most physicists/math folks misusing this term 'isomorphism' to mean "a bijection". Which gets to the second point, if there is a true isomorphism between the map and the land, it doesn't matter that one isn't the other. That would mean that the land is constrained by the same axioms as the 'map', which gives some significance to them.
- thwayunion 5y ago
  It's not a bijection either. If we take your level of pedantry seriously, it's nothing. There are no mathematical structures in play. I'm using natural language words to describe informal ideas outside of any mathematical system. Let's use bloogidy-blop to avoid silly arguments over sequences of characters that have clear contextual meaning which you refuse to acknowledge for some reason :) I'm not sure what your second point is supposed to be. Here's the relevant quotes: >>> Humans can create notation and formalisms, but they do not invent the truths those mathematics represent. >> The land represented by a map exists independently of humanity. Another intelligent species would have to come up with a roughly isomorphic representation if they wanted a similar tool. > Which gets to the second point, if there is a true isomorphism between the map and the land, it doesn't matter that one isn't the other. That would mean that the land is constrained by the same axioms as the 'map', which gives some significance to them. Right... but I didn't say that there was a bloogidy-blop between maps and land. I said that a useful alien map would have to be roughly bloogidy-blop to our maps. So I'm not really sure what you're trying to say here that is relevant to my original post.
  - VirusNewbie 5y ago
    Well, we can say 'an equivalence' (or adjoint) which would be accurate and still using natural language, no? Onto my second point, you don't need an isomorphism, an equivalence works fine if you find that the axioms hold in both cases, no? So now you're not just working with a 'formalism' that has no basis in reality. Otherwise, if reality wasn't constrained by axioms (or even meta axioms) we'd use it to do things we couldn't with our formalisms.

whatshisface 5y ago

Any species would prove the exact same theorems given the same axioms, and since mathematicians only claim that their axioms imply their theorems, I think they are right to claim absolute truth.

thwayunion 5y ago
This is either an extremely obvious and boring observation or the basis for a metaphysical trip, depending on how pre-disposed you are to "mathematical spiritualism" :)
- whatshisface 5y ago
  Nothing can be objectively interesting, only objectively true. Just because math is objectively true does not mean you've been robbed of your license to decide if you think it's interesting. :)
  - thwayunion 5y ago
    Agreed. Similarly, just because I don't "get" church doesn't mean I can prove God doesn't exist. And it certainly doesn't mean I should stand in the way of others enjoying the experience of going to church regardless of their beliefs. It just means I don't "get" it.
    - whatshisface 5y ago
      Religion is slightly different though, they claim actual direct truth (not mere truth of implication given certain assumptions) which makes their claims more interesting but prevents them from claiming automatic objective truth. The Formal Gospel would go, "If God so loved the world that he gave his only begotten son, ..." ;)
      - thwayunion 5y ago
        That's why I used words like "spirituality" and "faith" instead of "religion" and "dogma". I'm describing a shared social/psychological phenomenon, not a shared epistemological status. Read my posts less as works of the philosophy of mathematics and more as anthropological musings.
pfortuny 5y ago
Oh, dear. Truth being the operating word… There is no truth in a set of axioms we cannot even conceive properly (any infinite set has properties beyond what seems reasonable, even “just” the Natural numbers). From that comes arithmetic, the “most elementary” form of mathematics which cannot be proved consistent… We (I am a working mathematician) do not understand our objects, we can just make do. Only finite graph theory has a chance of being “real”. And it stops being finite very soon. And we certainly should be honest enough to admit that our “science” says very little about the “real” world, where truth lies. Maths is just a tool. Funny, exciting and even in some sense beautiful. But “truth” does it not contain. Except, I insist, in very specific finite constructions. Statements hold but they are not “true” because they do not relate to the real world (otherwise, Frodo reaching Mount Doom would also be “true”). There are no continuous functions out there. Bolzano’s theorem is not “true”.
- whatshisface 5y ago
  I would contend that A -> B can be true even if A is not true or more relevantly to this discussion if A is unknown. That's math's version of objective truth, where "A" is filled by our various axioms and rules of inference.
  - macrolocal 5y ago
    In ~300 BCE, Euclid decides to work with morphisms not objects.
- meroes 5y ago
  How can you explain appealing to these “unreal objects” (real numbers, set theory, arithmetic) * does* help science? (Effectiveness maybe) I see you are also a non realist about science. But even the methodological naturalist (one who takes natural empirical science to be the best method but not an ontology) must wonder how we are uncovering and putting more precision to more and more of the world. I don’t think we can currently explain why this made up tool “works”.
- zmgsabst 5y ago
  The enduring appeal of both triangles and Frodo is that they relate a truth that is hard to see, because it’s diffuse and abstract. But that “abstract truth” is why your pizza doesn’t flop if you fold it or Amazon can’t make a good LOTR.
JackFr 5y ago
> Any species would prove the exact same theorems given the same axioms Assume there exists at least one species....
housecarpenter 5y ago
Any species could prove the same theorems given the same axioms, but (besides the fact that they might not choose the same axioms) I'm not sure if they would prove the same subset of theorems that we have proven/will prove. Perhaps they'd have different ideas about what is interesting.
- whatshisface 5y ago
  Human mathematicians are already fanning out into other systems of deduction (constructive mathematics being a great example), and given enough time the mathematicians of each galaxy will eventually discover the other galaxy's mathematics, even if it perhaps happens in a different order.
  - ganzuul 5y ago
    Surely intergalactic mathematicians already know that the only time is now? =) > As Prigogine explains, determinism is fundamentally a denial of the arrow of time. With no arrow of time, there is no longer a privileged moment known as the "present," which follows a determined "past" and precedes an undetermined "future." All of time is simply given, with the future as determined or as undetermined as the past. With irreversibility, the arrow of time is reintroduced to physics.
stonemetal12 5y ago
That begs the question. Would any other species pick the exact same axioms? Why would they have the exact same theorems? Are you suggesting there is only one way to think logically?
- dane-pgp 5y ago
  Any species would face a selective pressure towards theorems which help them understand the world around them (if they have any motivation to prove theorems at all), and they will similarly face a pressure towards choosing the smallest/simplest set of axioms which allows all those theorems to be proven (and new ones to be discovered). In fact, if we assume that neural networks are the only sorts of intelligence that can occur naturally in the universe and be sophisticated enough for arbitrary abstract calculation[0], then we might be able to infer things about the sorts of concepts they will develop and in what order. For example, having the concept of finite sums would likely occur before having the concept of infinite sums. [0] I know that cellular automata can emulate a universal Turing machine, but I can't imagine a situation existing in nature where the cells evolve into an arrangement that produces a Turing machine, much less a machine running a program of instructions that lead to it generating mathematical theorems.

marcosdumay 5y ago

> I wonder if there is a field of meta-map-making There certainly is, although I'm not sure it has a name. Kids gets introductions to it on schools, when they have classes about how to read a map in Geography.

VirusNewbie 5y ago

'roughly isomorphic' would be saying 'not isomorphic', so I'm not sure what you're trying to say. Actually I'm frustrated with most physicists/math folks misusing this term 'isomorphism' to mean "a bijection". Which gets to the second point, if there is a true isomorphism between the map and the land, it doesn't matter that one isn't the other. That would mean that the land is constrained by the same axioms as the 'map', which gives some significance to them.

thwayunion 5y ago
It's not a bijection either. If we take your level of pedantry seriously, it's nothing. There are no mathematical structures in play. I'm using natural language words to describe informal ideas outside of any mathematical system. Let's use bloogidy-blop to avoid silly arguments over sequences of characters that have clear contextual meaning which you refuse to acknowledge for some reason :) I'm not sure what your second point is supposed to be. Here's the relevant quotes: >>> Humans can create notation and formalisms, but they do not invent the truths those mathematics represent. >> The land represented by a map exists independently of humanity. Another intelligent species would have to come up with a roughly isomorphic representation if they wanted a similar tool. > Which gets to the second point, if there is a true isomorphism between the map and the land, it doesn't matter that one isn't the other. That would mean that the land is constrained by the same axioms as the 'map', which gives some significance to them. Right... but I didn't say that there was a bloogidy-blop between maps and land. I said that a useful alien map would have to be roughly bloogidy-blop to our maps. So I'm not really sure what you're trying to say here that is relevant to my original post.
- VirusNewbie 5y ago
  Well, we can say 'an equivalence' (or adjoint) which would be accurate and still using natural language, no? Onto my second point, you don't need an isomorphism, an equivalence works fine if you find that the axioms hold in both cases, no? So now you're not just working with a 'formalism' that has no basis in reality. Otherwise, if reality wasn't constrained by axioms (or even meta axioms) we'd use it to do things we couldn't with our formalisms.