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Curiously, this review (and perhaps the book being reviewed?) makes no mention of the model of mathematics as a language. This seems to be a common idea among w

Curiously, this review (and perhaps the book being reviewed?) makes no mention of the model of mathematics as a language. This seems to be a common idea among working mathematicians and physicists[1,2], and it is rather clear that the question of reality of a language is ill-formed. A rich enough language should be suitable for expressing both real and abstract statements. It is also interesting to think about mathematics from the point of view of programming language paradigms: is it functional (I think mostly), what is its type system, etc. [1]: Galileo says that, roughly, the book of nature is written in the language of mathematics https://en.m.wikipedia.org/wiki/The_Assayer https://en.m.wikipedia.org/wiki/The_Assayer [2]: A classical essay by Wigner on the "unreasonable effectiveness of mathematics" also brings up the language analogy. https://web.archive.org/web/20210212111540/http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html https://web.archive.org/web/20210212111540/http://www.dartmo...

worldsayshi 3y ago
So language isn't mathematics? Like physics "is" mathematics? Are both language and math a third thing? Like maybe automata?
- winwang 3y ago
  considering that formal languages are part of math, there's little reason to believe natural language isn't. ft. LLMs.
- dexwiz 3y ago
  Mathematics is the language best suited to communicate Physics. > The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed. Translates to > F=ma
  - candiddevmike 3y ago
    Now translate that back to the original English without any context. I don't think your strawman is as succinct as you think it is.
    - tomrod 3y ago
      F is a concept of force and has a wide variety of English words that apply. Mass is represented in English by the tokens: "the property of matter resistant to acceleration" Acceleration is represented in English as the change of the change of the position through time. To your point, there are an infinite number of definitions we can apply to the equation defining a product. So Physics context + mathematical verbiage = elucidated ideas.
    - cgriswald 3y ago
      You can disagree with him that it's the best language or that he didn't prove his case, but I don't see how he's offering a strawman. I'm also not sure that translating without context is a meaningful. What language operates without context? Without context, what would the pronoun 'that' in your post point to? What does the phrase 'back to' mean without the associated physical metaphor? How would this be different from translating to and from German without context?
    - Izkata 3y ago
      It's about the same as "try translating that English sentence into Swahili without learning the language".
  - namirez 3y ago
    Except the metaphor is quite misleading. Aristotle believed: Force = velocity * resistance
diogenes4 3y ago
> Curiously, this review (and perhaps the book being reviewed?) makes no mention of the model of mathematics as a language. This seems to be a common idea among working mathematicians and physicists[1,2], and it is rather clear that the question of reality of a language is ill-formed. Sorry, if this isn't clear to everyone, why does this result in the rejection of mathematics as a language rather than the rejection of the question of the realism of mathematics? The latter seems much easier with no downsides (that I can see), whereas the rejection of mathematics as a language has many downsides.
- fiforpg 3y ago
  Of course, I do think of math as a (meta)language. Maybe my phrasing does not convey that well. So much for using natural language instead of math... ;)
mxkopy 3y ago
They probably didn’t mention it because, like you pointed out in the sentence immediately after your question, it makes the philosophy ill-formed and dare I say uninteresting. Are languages real? Sure, I guess. Why not. They’re made of concrete sounds and symbols that everyone (by design) can point to - they might even be the realest thing there is. Are (math) concepts real? As in like, if I rearranged all the matter that exists out there, would the value of root(2) stop being irrational? Or maybe they don’t have to be physical to be real; maybe being “real” has more to do whether or not something has an effect on something else than its physical existence. Then, there’s a difference between how something exists and how real it is? Etc. Thinking of math as a language as opposed to a set of concepts isn’t interesting to the discussion of its reality IMO
dimask 3y ago
1. I do not think Galileo is either a characteristic example of a physicist or mathematician, or a particularly good example at all [1]. 2. While I would say that most would trivially agree that mathematics is a form of language [2], it is less common to argue that mathematics is _just that_. And, crucially, it is not supported by neuroscience [3]. It seems that there are distinct brain areas involved specifically in mathematical thinking outside of those that are language-specific, which would mean that mathematical thinking is more than language [3]. [1] For a criticism regarding galileo's mediocrity and historically unfounded idealisation https://intellectualmathematics.com/blog/the-case-against-galileo-s01-overview/ https://intellectualmathematics.com/blog/the-case-against-ga... [2] Actually, a collection of languages, as different parts of mathematics form different languages or dialects that are not easy to communicate in between even if they refer to similar objects, eg category theory vs set theory, analysis vs probability theory. [3] https://anthonybonato.com/2017/09/19/this-is-your-brain-on-mathematics-2/ https://anthonybonato.com/2017/09/19/this-is-your-brain-on-m...
- ramblenode 3y ago
  > And, crucially, it is not supported by neuroscience [3]. It seems that there are distinct brain areas involved specifically in mathematical thinking outside of those that are language-specific, which would mean that mathematical thinking is more than language [3]. I would not construe a small study of 30 people (15 math experts) as having the backing of the field of neuroscience. The steel man position of mathematics-as-language would be something along the lines of "Mathematics is mostly a language + some other stuff". The paper in question only shows there is "some other stuff"---but that is also true of almost every non-contrived use of language. The paper shows that the other stuff for sentences about history and nature appear differently in the brain than the other stuff for sentences about math, but that is far from demonstrating that math on the whole is not mostly a linguistic phenomenon. Personally, I think case studies of famous mathematicians or physicists discussing their thinking process (e.g. Einstein) is more convincing evidence against the hypothesis than fMRI studies.
sunday_serif 3y ago
I am no philosopher, and only an amateur mathematician… But to me it seems that even when you consider math as just being language… you still run into the same philosophical problem when you consider what the semantics of that language are. For example, you may believe that the practice of algebra is simply operations on stings of a language. But when you start asking what objects those sentences refer to, you end up asking what math is again.
- AndrewKemendo 3y ago
  This is where I finally landed with this question - there’s too much vagueness and importantly no agreed definitions of what is and is not “reality” Math has no fundamental non-differentiable epistemology qua mathematics. As you said there is no meaningfully discrete, bounded “object” that can be used as a universal reference - so the fact of relativity means that even if maths were “real” I’m not even sure such a thing exists (who says the rules of the universe are uniform, or static as we’re only have a few centuries of poor observation) and if it does exist, if humans have the ability to identify it
- ilyagr 3y ago
  This is a good point, but natural language has this problem to an even greater degree. What object does the word "consciousness" refer to? "Unicorn"? What makes a ship a ship? What if we include toy ships and drawn ships? "What can math describe" is an easier question than "what is reality?" or "what is the nature of all concepts a human mind can conceive of or be interested in?". Not that it's an easy question either.
  - mr_toad 3y ago
    But we don’t expect natural language to be as precise and consistent as Maths. You could argue that maths was invented because natural language was too vague.
namirez 3y ago
I don’t think Einstein could have come up with general relativity if he had to rely on any other language than math. So I suspect math is not “just” another language.
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worldsayshi 3y ago

So language isn't mathematics? Like physics "is" mathematics? Are both language and math a third thing? Like maybe automata?

winwang 3y ago
considering that formal languages are part of math, there's little reason to believe natural language isn't. ft. LLMs.
dexwiz 3y ago
Mathematics is the language best suited to communicate Physics. > The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed. Translates to > F=ma
- candiddevmike 3y ago
  Now translate that back to the original English without any context. I don't think your strawman is as succinct as you think it is.
  - tomrod 3y ago
    F is a concept of force and has a wide variety of English words that apply. Mass is represented in English by the tokens: "the property of matter resistant to acceleration" Acceleration is represented in English as the change of the change of the position through time. To your point, there are an infinite number of definitions we can apply to the equation defining a product. So Physics context + mathematical verbiage = elucidated ideas.
  - cgriswald 3y ago
    You can disagree with him that it's the best language or that he didn't prove his case, but I don't see how he's offering a strawman. I'm also not sure that translating without context is a meaningful. What language operates without context? Without context, what would the pronoun 'that' in your post point to? What does the phrase 'back to' mean without the associated physical metaphor? How would this be different from translating to and from German without context?
  - Izkata 3y ago
    It's about the same as "try translating that English sentence into Swahili without learning the language".
- namirez 3y ago
  Except the metaphor is quite misleading. Aristotle believed: Force = velocity * resistance

diogenes4 3y ago

> Curiously, this review (and perhaps the book being reviewed?) makes no mention of the model of mathematics as a language. This seems to be a common idea among working mathematicians and physicists[1,2], and it is rather clear that the question of reality of a language is ill-formed. Sorry, if this isn't clear to everyone, why does this result in the rejection of mathematics as a language rather than the rejection of the question of the realism of mathematics? The latter seems much easier with no downsides (that I can see), whereas the rejection of mathematics as a language has many downsides.

fiforpg 3y ago
Of course, I do think of math as a (meta)language. Maybe my phrasing does not convey that well. So much for using natural language instead of math... ;)

mxkopy 3y ago

They probably didn’t mention it because, like you pointed out in the sentence immediately after your question, it makes the philosophy ill-formed and dare I say uninteresting. Are languages real? Sure, I guess. Why not. They’re made of concrete sounds and symbols that everyone (by design) can point to - they might even be the realest thing there is. Are (math) concepts real? As in like, if I rearranged all the matter that exists out there, would the value of root(2) stop being irrational? Or maybe they don’t have to be physical to be real; maybe being “real” has more to do whether or not something has an effect on something else than its physical existence. Then, there’s a difference between how something exists and how real it is? Etc. Thinking of math as a language as opposed to a set of concepts isn’t interesting to the discussion of its reality IMO

dimask 3y ago

1. I do not think Galileo is either a characteristic example of a physicist or mathematician, or a particularly good example at all [1]. 2. While I would say that most would trivially agree that mathematics is a form of language [2], it is less common to argue that mathematics is _just that_. And, crucially, it is not supported by neuroscience [3]. It seems that there are distinct brain areas involved specifically in mathematical thinking outside of those that are language-specific, which would mean that mathematical thinking is more than language [3]. [1] For a criticism regarding galileo's mediocrity and historically unfounded idealisation https://intellectualmathematics.com/blog/the-case-against-galileo-s01-overview/ https://intellectualmathematics.com/blog/the-case-against-ga... [2] Actually, a collection of languages, as different parts of mathematics form different languages or dialects that are not easy to communicate in between even if they refer to similar objects, eg category theory vs set theory, analysis vs probability theory. [3] https://anthonybonato.com/2017/09/19/this-is-your-brain-on-mathematics-2/ https://anthonybonato.com/2017/09/19/this-is-your-brain-on-m...

ramblenode 3y ago
> And, crucially, it is not supported by neuroscience [3]. It seems that there are distinct brain areas involved specifically in mathematical thinking outside of those that are language-specific, which would mean that mathematical thinking is more than language [3]. I would not construe a small study of 30 people (15 math experts) as having the backing of the field of neuroscience. The steel man position of mathematics-as-language would be something along the lines of "Mathematics is mostly a language + some other stuff". The paper in question only shows there is "some other stuff"---but that is also true of almost every non-contrived use of language. The paper shows that the other stuff for sentences about history and nature appear differently in the brain than the other stuff for sentences about math, but that is far from demonstrating that math on the whole is not mostly a linguistic phenomenon. Personally, I think case studies of famous mathematicians or physicists discussing their thinking process (e.g. Einstein) is more convincing evidence against the hypothesis than fMRI studies.

sunday_serif 3y ago

I am no philosopher, and only an amateur mathematician… But to me it seems that even when you consider math as just being language… you still run into the same philosophical problem when you consider what the semantics of that language are. For example, you may believe that the practice of algebra is simply operations on stings of a language. But when you start asking what objects those sentences refer to, you end up asking what math is again.

AndrewKemendo 3y ago
This is where I finally landed with this question - there’s too much vagueness and importantly no agreed definitions of what is and is not “reality” Math has no fundamental non-differentiable epistemology qua mathematics. As you said there is no meaningfully discrete, bounded “object” that can be used as a universal reference - so the fact of relativity means that even if maths were “real” I’m not even sure such a thing exists (who says the rules of the universe are uniform, or static as we’re only have a few centuries of poor observation) and if it does exist, if humans have the ability to identify it
ilyagr 3y ago
This is a good point, but natural language has this problem to an even greater degree. What object does the word "consciousness" refer to? "Unicorn"? What makes a ship a ship? What if we include toy ships and drawn ships? "What can math describe" is an easier question than "what is reality?" or "what is the nature of all concepts a human mind can conceive of or be interested in?". Not that it's an easy question either.
- mr_toad 3y ago
  But we don’t expect natural language to be as precise and consistent as Maths. You could argue that maths was invented because natural language was too vague.

namirez 3y ago

I don’t think Einstein could have come up with general relativity if he had to rely on any other language than math. So I suspect math is not “just” another language.

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