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It’s important to note that no model using math has accurately represented nature. Just because simple approximations line up in a beautiful way is not reason

It’s important to note that no model using math has accurately represented nature. Just because simple approximations line up in a beautiful way is not reason to think they are 1:1 the same. Math is not sacred or divine. It’s a tool. Like all tools: its shape matters. Math, as we know it, is invented. It is one set of rules that we like to use to describe things. It is not the only way. I can imagine an intelligent species that has a structured set of logical rules that looks very different from our math but still be able to make useful models.

senthil_rajasek 6y ago
"It’s important to note that no model using math has accurately represented nature" (Edit: I am getting downvoted, so let me expand my answer a bit. Nature is explained using Physics and Physics uses Mathematics for its models of Nature.) This is part of Godel's discovery. Especially, his first Incompleteness Theorem. https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_incompletenes...
- tsimionescu 6y ago
  The incompleteness theorems don't have anything to do with nature, they are about pure mathematics.
  - senthil_rajasek 6y ago
    Physics explains nature using mathematics.
    - protonfish 6y ago
      And writers explain nature using words.
      - senthil_rajasek 6y ago
        Not as precisely as mathematics can as shown by F = ma or E = mc^2.
      - 082349872349872 6y ago
        Poets, like writers, use words. They often say things in few words subject to many internal constraints, so one must pay closer attention than when reading prose. (Indeed, many people write prose about the poetry as a guide.) Mathematicians use even fewer symbols to say things subject to even stronger internal constraints, so one must pay even closer attention than when reading poetry. (Indeed, almost all people write prose around the symbols as a guide.) Some poets explore what happens with more, some with fewer, constraints. Some mathematicians explore what happens with more, some with fewer, constraints. (Viewed as a MMO game, mathematics introduced the parallel postulate in or before 300 BC and although many people said it ought to be nerfed, the nerfing wouldn't happen until 1830.)
    - tsimionescu 6y ago
      The important point is that physics will likely never find a perfect 1:1 correspondence between nature and math because of the properties of nature (such as being too complex for the human mind to fully model), not because of the properties of math (such as the incompleteness theorem). That is to say, even if the incompleteness theorem hadn't been true, math still wouldn't have described nature with perfect accuracy. Not to mention, the incompleteness theorem hasn't prevented any kind of modelling in physics that I have ever heard of. Indeed, physics is often not entirely constrained by formal methods, with ad-hoc mathematical models that can be shown to work even if they are not fully formalized sometimes being preferred (such as the Dirac delta "function").
protonfish 6y ago
Unfortunately Platonists don't believe in evidence so this won't convince them that their religious beliefs are wrong.
- tim333 6y ago
  That seems a little sweeping. I imagine many do.
  - protonfish 6y ago
    Then they wouldn't be Platonists, would they? Wait, do people here even understand what Platonism means? Rejection of evidence is a key tenet of that philosophy. This is why the other school of thought is called Empiricists - because they believe in empirical evidence. It's frustrating getting all these downvotes by people that don't understand their own words, but I suppose it's not surprising coming from Platonists. Folks who, when faced with a contradiction between reality and their cherished beliefs, conclude reality must be wrong.
    - tim333 6y ago
      Neither the Britannica article on Platonism https://www.britannica.com/topic/Platonism https://www.britannica.com/topic/Platonism or the Wikipedia one seem to mention rejection of evidence. I think maybe you have an unusual take on Platonism?
aliceryhl 6y ago
The symbols may be invented and the things we have investigated may be different, but the things we prove are true, even to the aliens. Consider modeling the game of tic tac toe. You can prove that certain strategies always win. If you then proceed to play it with that strategy against the aliens, then you will certainly win the game, regardless of how their logic looks. (this assumes you're starting of course)
- willis936 6y ago
  That just means math is a set of self consistent rules that you could teach to an alien, not that they are the only set of rules that are self consistent or the only rules that can be used to model empirical observations of nature.
  - aliceryhl 6y ago
    I consider any set of self-consistent rules to be mathematics, even if aliens invented them.
    - dfischer 6y ago
      Self consistent rules doesn’t imply communication. https://en.m.wikipedia.org/wiki/Chinese_room https://en.m.wikipedia.org/wiki/Chinese_room Said another way, if you have a dictionary translating, it doesn’t mean the definition is equivalent. In fact, most definitions aren’t, as each one has its nuance. If following a dictionary to arrive at a result after some steps, it doesn’t imply anything other than a set of rules being followed to arrive at an arbitrary value. Also interesting is the perspective of mind: https://en.wikipedia.org/wiki/China_brain https://en.wikipedia.org/wiki/China_brain Universe as one mind depending on zoomed out vs zoomed in. However this attribution of mind is simply anything that follows rules. So it’s better to call it “logics” and humans have a property of “bio-logic” abilities. But then what exists outside of logic that allows arbitrary definition of things to represent said logic? The programmer, the consciousness. In general I feel mind exists outside of logic but that’s perhaps a matter of perspective and definition. It gets hard to know where one has defined a conceptual space between continuum and borders and assigning it a range of values.
naringas 6y ago
> I can imagine an intelligent species that has a structured set of logical rules that looks very different from our math but still be able to make useful models. but can you imagine such a set of rules?
koheripbal 6y ago
1+1=2. That isn't an approximation. It is a core principle, not just of math, but of the laws of our universe. It is so core, that it's impossible to imagine a universe without it. or maybe it is an approximation. Maybe there is some essence of the universe that gives is additiveness - and maybe if you could see closer/bigger/faster, maybe that too would break down. We should take nothing for granted.
- codegladiator 6y ago
  1 drop of water + 1 drop of water = 1 drop of water 1 river + 1 river = 1 river 1 stick - 1/2 stick = 2 stick
  - TeMPOraL 6y ago
    Now you're cheating, because + and - no longer mean "addition" and "subtraction", but merging and... well, weird division - the last example would be better as 1 stick / 2 = 2 sticks. But try the first two with the following substitutions: "drop of water" => "volume of a drop of water" "river" => "flow rate of a river" And you're back to addition and correct answers.
    - codegladiator 6y ago
      Sure we can measure something else to keep a track of whats happening and fit our definitions, but the math remain true, isn't it? It is still 1 river and 1 drop and 2 sticks.
      - Tagbert 6y ago
        But 1 river is not a defined measurement. It is has a range of meanings whereas 2 is defined. 2 = 2 is true 2 ≠ 2 is false But 1 river = 1 river is true 1 river ≠ 1 river is also true under some circumstances
        dfischer 6y ago
dfischer 6y ago
Thinking No model can ever represent nature. A model helps prediction which helps convergence to certainty through consensus. Although this opens a can of worms of what “accurately represented” means. A threshold comes into account. Then it unpacks into what is existence. Fun stuff... I’ve been thinking a lot of “errors” in our models (language, ideas, etc) are emergent from not considering change. Nothing in the universe is still. Everything changes or vibrates. The only things that are still, don’t exist. It makes me imagine that we need an evolution in our language and core reasoning that intrinsically embeds change and or perspective (relativity?) I’ve had this thought recently again based on studying what numbers are. The definition of numerals vs numbers vs integers and number theory, etc... Even the “numbers” that we try to reference seems to exist as non fixed value of infinite change. Also makes me think of what is that core “thing” that allows a number to expose itself within our frame of reality? It seems to consistently show itself through mathematics and information theory and computer science. Perhaps all related to geometry in some way?
afpx 6y ago
Did you read the article?
- dang 6y ago
  "Please don't comment on whether someone read an article. "Did you even read the article? It mentions that" can be shortened to "The article mentions that."" https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html
amw-zero 6y ago
We really don’t know that to be certain (the fact that math is invented). This is a very longstanding philosophical debate with arguments on both sides. Now, of course the symbols of math are invented. But math is the study of mathematical _objects_, and it’s very unclear if those are invented or discovered. For example, the Curry-Howard isomorphism shows that math proofs and computer programs are the same mathematical object. Does this mean that they are a part of nature, and we discovered them because we invented a sufficiently powerful mathematical system? Or are they simply properties of the invented system? We do not (and most likely can not) know the answer to that question. If someone is extremely convicted when offering an answer to it, consider me completely suspicious.
- foxdev 6y ago
  This is something that hit me when I found out how to translate between decimal and binary. If an A on a screen can be a grid of numbers, and those numbers can be reduced to a series of 1s and 0s and stored in a bunch of fancy wires, then who knows what else it could be translated to or from. Look back far enough and the A is a series of photons clashing with orbiting subatomic particles at certain frequencies. We use math based on the status of a sea of transistors to store it in other atoms with forces we're slowly discovering how to communicate about. It's all layers of finding new ways to describe something that was already there, spinning and bumping photons long before any of us had photon detectors in our heads or a chemical machine capable of deciding to call one pattern A and all the ways to translate it into sound waves. One pattern is called "solid state drive," and we seem to have some say in what that pattern does from our frame of reference. Maybe it was always a solid state drive and we're the language the universe uses to describe its smaller parts. I promise I'm not high, and probably not a Boltzmann brain. Though I'm not sure about the latter.
  - dfischer 6y ago
    Signal theory is deeply interesting based on your line of thinking here. I’ve been postulating similar things. The universe gives a lot of potential for things to communicate. In the end what is recognized as signal and or information is a matter of imagination and consistency given the plethora of potentially unlimited signals available.
temac 6y ago
I would argue that math is both invented and discovered at the same time. Our expression math is obviously invented but that's not very important (but not totally unimportant either, because it drivers further discoveries). The set of axioms we chose is way more invented than discovered, although we won't like others, it is likewise we won't like e.g. a too random or too complicated a mechanical apparatus, or even it would be useless. Given a chosen set of axioms, the theorems we can infer are so constrained that the only reason we can find an invented component is that we chose the places to look for in an absurdly large research space. So like we can argue that we don't "discover" a novel in the large research space of random characters, we can in a way argue that we in part invent the theorems. In the end does it really matter though? Is it maybe not just that inventions and discoveries have more things in common than some intuitively think? Tons of people are not really found of patents, or at least of some class of patents, because the line is arguably blurry (so yeah there is a practical way in which case it do matter, but only because of a completely arbitrary social organisation)
roenxi 6y ago
> Math, as we know it, is invented. But if it is invented then could we not have avoided inventing some of the more gnarly bits? If math is invented then, say, irrational numbers were a terrible mistake. The building blocks of math are entirely abstract, but once we define a circle in any half-satisfying way then all of the results in algebra, calculus and geometry are largely set in stone. After the invention of the circle in pre-written-history, most of the math that followed was discovery of the implications.
- thenoblesunfish 6y ago
  The fun part is that what we invented is a set of rules that governs what else we may or may not invent! That is, once you agree on what logic is and what some axioms are, you can show that certain statements can't be added as axioms. Similarly, once you've decide on your axioms, and your set of logical tools, you will find some things (like irrational numbers), that you wouldn't have introduced directly. It's pretty cool!
- Tainnor 6y ago
  I mean, irrational numbers are maybe "ugly" from a certain perspective, but they are the price we pay for other things that make our life easier. Rational numbers are arguably nice and easy. But if all we have are rational numbers, 2 doesn't have a square root. More in general, our number line has holes, and one of the most fundamental properties of analysis - that every non-empty bounded set has a least upper bound - goes away. We could do away with that (and arguably, irrational numbers don't really exist "in the outside world") but it would make all the calculations so much more complicated.
- threatofrain 6y ago
  Irrationality is behavior which emerges from rational machinery. It is unavoidable.

senthil_rajasek 6y ago

"It’s important to note that no model using math has accurately represented nature" (Edit: I am getting downvoted, so let me expand my answer a bit. Nature is explained using Physics and Physics uses Mathematics for its models of Nature.) This is part of Godel's discovery. Especially, his first Incompleteness Theorem. https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_incompletenes...

tsimionescu 6y ago
The incompleteness theorems don't have anything to do with nature, they are about pure mathematics.
- senthil_rajasek 6y ago
  Physics explains nature using mathematics.
  - protonfish 6y ago
    And writers explain nature using words.
    - senthil_rajasek 6y ago
      Not as precisely as mathematics can as shown by F = ma or E = mc^2.
    - 082349872349872 6y ago
      Poets, like writers, use words. They often say things in few words subject to many internal constraints, so one must pay closer attention than when reading prose. (Indeed, many people write prose about the poetry as a guide.) Mathematicians use even fewer symbols to say things subject to even stronger internal constraints, so one must pay even closer attention than when reading poetry. (Indeed, almost all people write prose around the symbols as a guide.) Some poets explore what happens with more, some with fewer, constraints. Some mathematicians explore what happens with more, some with fewer, constraints. (Viewed as a MMO game, mathematics introduced the parallel postulate in or before 300 BC and although many people said it ought to be nerfed, the nerfing wouldn't happen until 1830.)
  - tsimionescu 6y ago
    The important point is that physics will likely never find a perfect 1:1 correspondence between nature and math because of the properties of nature (such as being too complex for the human mind to fully model), not because of the properties of math (such as the incompleteness theorem). That is to say, even if the incompleteness theorem hadn't been true, math still wouldn't have described nature with perfect accuracy. Not to mention, the incompleteness theorem hasn't prevented any kind of modelling in physics that I have ever heard of. Indeed, physics is often not entirely constrained by formal methods, with ad-hoc mathematical models that can be shown to work even if they are not fully formalized sometimes being preferred (such as the Dirac delta "function").

protonfish 6y ago

Unfortunately Platonists don't believe in evidence so this won't convince them that their religious beliefs are wrong.

tim333 6y ago
That seems a little sweeping. I imagine many do.
- protonfish 6y ago
  Then they wouldn't be Platonists, would they? Wait, do people here even understand what Platonism means? Rejection of evidence is a key tenet of that philosophy. This is why the other school of thought is called Empiricists - because they believe in empirical evidence. It's frustrating getting all these downvotes by people that don't understand their own words, but I suppose it's not surprising coming from Platonists. Folks who, when faced with a contradiction between reality and their cherished beliefs, conclude reality must be wrong.
  - tim333 6y ago
    Neither the Britannica article on Platonism https://www.britannica.com/topic/Platonism https://www.britannica.com/topic/Platonism or the Wikipedia one seem to mention rejection of evidence. I think maybe you have an unusual take on Platonism?

aliceryhl 6y ago

The symbols may be invented and the things we have investigated may be different, but the things we prove are true, even to the aliens. Consider modeling the game of tic tac toe. You can prove that certain strategies always win. If you then proceed to play it with that strategy against the aliens, then you will certainly win the game, regardless of how their logic looks. (this assumes you're starting of course)

willis936 6y ago
That just means math is a set of self consistent rules that you could teach to an alien, not that they are the only set of rules that are self consistent or the only rules that can be used to model empirical observations of nature.
- aliceryhl 6y ago
  I consider any set of self-consistent rules to be mathematics, even if aliens invented them.
  - dfischer 6y ago
    Self consistent rules doesn’t imply communication. https://en.m.wikipedia.org/wiki/Chinese_room https://en.m.wikipedia.org/wiki/Chinese_room Said another way, if you have a dictionary translating, it doesn’t mean the definition is equivalent. In fact, most definitions aren’t, as each one has its nuance. If following a dictionary to arrive at a result after some steps, it doesn’t imply anything other than a set of rules being followed to arrive at an arbitrary value. Also interesting is the perspective of mind: https://en.wikipedia.org/wiki/China_brain https://en.wikipedia.org/wiki/China_brain Universe as one mind depending on zoomed out vs zoomed in. However this attribution of mind is simply anything that follows rules. So it’s better to call it “logics” and humans have a property of “bio-logic” abilities. But then what exists outside of logic that allows arbitrary definition of things to represent said logic? The programmer, the consciousness. In general I feel mind exists outside of logic but that’s perhaps a matter of perspective and definition. It gets hard to know where one has defined a conceptual space between continuum and borders and assigning it a range of values.

naringas 6y ago

> I can imagine an intelligent species that has a structured set of logical rules that looks very different from our math but still be able to make useful models. but can you imagine such a set of rules?

koheripbal 6y ago

1+1=2. That isn't an approximation. It is a core principle, not just of math, but of the laws of our universe. It is so core, that it's impossible to imagine a universe without it. or maybe it is an approximation. Maybe there is some essence of the universe that gives is additiveness - and maybe if you could see closer/bigger/faster, maybe that too would break down. We should take nothing for granted.

codegladiator 6y ago
1 drop of water + 1 drop of water = 1 drop of water 1 river + 1 river = 1 river 1 stick - 1/2 stick = 2 stick
- TeMPOraL 6y ago
  Now you're cheating, because + and - no longer mean "addition" and "subtraction", but merging and... well, weird division - the last example would be better as 1 stick / 2 = 2 sticks. But try the first two with the following substitutions: "drop of water" => "volume of a drop of water" "river" => "flow rate of a river" And you're back to addition and correct answers.
  - codegladiator 6y ago
    Sure we can measure something else to keep a track of whats happening and fit our definitions, but the math remain true, isn't it? It is still 1 river and 1 drop and 2 sticks.
    - Tagbert 6y ago
      But 1 river is not a defined measurement. It is has a range of meanings whereas 2 is defined. 2 = 2 is true 2 ≠ 2 is false But 1 river = 1 river is true 1 river ≠ 1 river is also true under some circumstances
      - dfischer 6y ago

dfischer 6y ago

Thinking No model can ever represent nature. A model helps prediction which helps convergence to certainty through consensus. Although this opens a can of worms of what “accurately represented” means. A threshold comes into account. Then it unpacks into what is existence. Fun stuff... I’ve been thinking a lot of “errors” in our models (language, ideas, etc) are emergent from not considering change. Nothing in the universe is still. Everything changes or vibrates. The only things that are still, don’t exist. It makes me imagine that we need an evolution in our language and core reasoning that intrinsically embeds change and or perspective (relativity?) I’ve had this thought recently again based on studying what numbers are. The definition of numerals vs numbers vs integers and number theory, etc... Even the “numbers” that we try to reference seems to exist as non fixed value of infinite change. Also makes me think of what is that core “thing” that allows a number to expose itself within our frame of reality? It seems to consistently show itself through mathematics and information theory and computer science. Perhaps all related to geometry in some way?

afpx 6y ago

Did you read the article?

dang 6y ago
"Please don't comment on whether someone read an article. "Did you even read the article? It mentions that" can be shortened to "The article mentions that."" https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html

amw-zero 6y ago

We really don’t know that to be certain (the fact that math is invented). This is a very longstanding philosophical debate with arguments on both sides. Now, of course the symbols of math are invented. But math is the study of mathematical _objects_, and it’s very unclear if those are invented or discovered. For example, the Curry-Howard isomorphism shows that math proofs and computer programs are the same mathematical object. Does this mean that they are a part of nature, and we discovered them because we invented a sufficiently powerful mathematical system? Or are they simply properties of the invented system? We do not (and most likely can not) know the answer to that question. If someone is extremely convicted when offering an answer to it, consider me completely suspicious.

foxdev 6y ago
This is something that hit me when I found out how to translate between decimal and binary. If an A on a screen can be a grid of numbers, and those numbers can be reduced to a series of 1s and 0s and stored in a bunch of fancy wires, then who knows what else it could be translated to or from. Look back far enough and the A is a series of photons clashing with orbiting subatomic particles at certain frequencies. We use math based on the status of a sea of transistors to store it in other atoms with forces we're slowly discovering how to communicate about. It's all layers of finding new ways to describe something that was already there, spinning and bumping photons long before any of us had photon detectors in our heads or a chemical machine capable of deciding to call one pattern A and all the ways to translate it into sound waves. One pattern is called "solid state drive," and we seem to have some say in what that pattern does from our frame of reference. Maybe it was always a solid state drive and we're the language the universe uses to describe its smaller parts. I promise I'm not high, and probably not a Boltzmann brain. Though I'm not sure about the latter.
- dfischer 6y ago
  Signal theory is deeply interesting based on your line of thinking here. I’ve been postulating similar things. The universe gives a lot of potential for things to communicate. In the end what is recognized as signal and or information is a matter of imagination and consistency given the plethora of potentially unlimited signals available.

temac 6y ago

I would argue that math is both invented and discovered at the same time. Our expression math is obviously invented but that's not very important (but not totally unimportant either, because it drivers further discoveries). The set of axioms we chose is way more invented than discovered, although we won't like others, it is likewise we won't like e.g. a too random or too complicated a mechanical apparatus, or even it would be useless. Given a chosen set of axioms, the theorems we can infer are so constrained that the only reason we can find an invented component is that we chose the places to look for in an absurdly large research space. So like we can argue that we don't "discover" a novel in the large research space of random characters, we can in a way argue that we in part invent the theorems. In the end does it really matter though? Is it maybe not just that inventions and discoveries have more things in common than some intuitively think? Tons of people are not really found of patents, or at least of some class of patents, because the line is arguably blurry (so yeah there is a practical way in which case it do matter, but only because of a completely arbitrary social organisation)

roenxi 6y ago

> Math, as we know it, is invented. But if it is invented then could we not have avoided inventing some of the more gnarly bits? If math is invented then, say, irrational numbers were a terrible mistake. The building blocks of math are entirely abstract, but once we define a circle in any half-satisfying way then all of the results in algebra, calculus and geometry are largely set in stone. After the invention of the circle in pre-written-history, most of the math that followed was discovery of the implications.

thenoblesunfish 6y ago
The fun part is that what we invented is a set of rules that governs what else we may or may not invent! That is, once you agree on what logic is and what some axioms are, you can show that certain statements can't be added as axioms. Similarly, once you've decide on your axioms, and your set of logical tools, you will find some things (like irrational numbers), that you wouldn't have introduced directly. It's pretty cool!
Tainnor 6y ago
I mean, irrational numbers are maybe "ugly" from a certain perspective, but they are the price we pay for other things that make our life easier. Rational numbers are arguably nice and easy. But if all we have are rational numbers, 2 doesn't have a square root. More in general, our number line has holes, and one of the most fundamental properties of analysis - that every non-empty bounded set has a least upper bound - goes away. We could do away with that (and arguably, irrational numbers don't really exist "in the outside world") but it would make all the calculations so much more complicated.
threatofrain 6y ago
Irrationality is behavior which emerges from rational machinery. It is unavoidable.