9 ms·

About math limitations

41 points by alexmolas 3y ago

roenxi 3y ago
It is an interesting argument. However, it is possible to simulate continuous phenomenon on discrete systems (eg, we can simulate real numbers on computers, and certainly build computer systems that reason about real numbers). Similarly the human brain can contemplate infinity although it is obvious that we can't comprehend it in totality because brains are finite. So it isn't clear that maths is limited by physics. It might or might not be.
- arketyp 3y ago
  The simulation part I'd argue isn't as interesting. That other part is though, namely that you, or a computer, can reason symbolically. We can treat the irrational length of the hypotenuse as though it exists in some concrete sense and get sensible and even useful results out of it. The limits of this approach is indeed a question.
constantcrying 3y ago
>This first idea that math have some intrinsic limitations - eg: you can’t compute all the Busy Beaver Numbers But this is not a limitation of mathematics at least not any more so than there being no largest prime number is a limitation of mathematics. It is a limitation on abstract systems, such as Turing machines. The halting problem is not that you can not deduce whether any given Turing machine halts, but that there is no Turing machine that deducing this. Gödel's (ö and o are different letters, by the way) theorems are far more wide reaching, as they appy to large classes of axiomatic systems, but they are only making claims about the systems in relation to themselves.
- bigbillheck 3y ago
  > no smallest prime number Behold: 2.
  - olddustytrail 3y ago
    -2 is smaller... (I'll let someone better educated expand on prime ideals)
- kbrkbr 3y ago
  As the smallest prime number is 2 you probably meant "there being no largest prime number".
  - constantcrying 3y ago
    Indeed...
- tux3 3y ago
  The Busy Beaver numbers express a limitation of classes of axiomatic systems, not merely of Turing Machines. It's not merely that computers aren't guaranteed to be able to calculate these. BB(n) for some n <= 745 is independent of the ZFC axioms of set theory. Furthermore, Gödel's theorem and Turing's problem are related. It is undecidable whether a Turing machine halts, it's not just that you can't write a Turing machine that decides whether another Turing machine halts. Without Gödel's theorem, you could try to enumerate all possible proofs in your complete and consistent axiomatic system until you find one that proves whether an arbitrary Turing machine of your choosing halts, and you'd have conceptually solved the halting problem. (Then, you could encode that whole algorithm in a Turing machine (TM), and have that new TM be your decider that solves the halting problem, if you wanted). Because of Gödel you can't do this, whether on paper, with a TM, or even in principle. Your axiomatic system is not complete, so there are things it simply cannot prove. Like the halting problem, or in the case of ZF the value of BB(745).
zogomoox 3y ago
If you consider that mathematical ideas are discovered rather than invented, then the whole point becomes moot. Sure, we may not be smart enough to discover all of it, but we can be pretty certain there is still mathematics beyond our ability to understand it. As an analogy, we'll never be able to look at or visit beyond our cosmic horizon, but it's certain there are stars and galaxies out there.
- DecayingOrganic 3y ago
  Math as we know it, due to Gödel's Incompleteness Theorems, is not fully consistent - we can't even prove everything that is true with our current mathematical framework. This means that our understanding of math is indeed limited, not just by our intellectual capabilities, but by the very structure of the math we currently use also. This makes me wonder, will we be able to develop new mathematical frameworks that bypass these issues? And if so, what will they look like?
  - kbrkbr 3y ago
    > is not fully consistent Not being able to prove the consistency of a system within the system does not entail that the system is inconsistent (or not fully consistent).
    - pixl97 3y ago
      I think it means we could only ever prove it is inconsistent, and never prove that it is consistent.
      - tux3 3y ago
        If we can only prove that an inconsistent system is inconsistent, and we can never prove our (presumably) consistent system is consistent, then it is incomplete. We can't say it's inconsistent just because we can't prove otherwise. We can say it may be incomplete (or inconsistent, and we haven't noticed yet)
  - 3y ago
arketyp 3y ago
>There’s no pure knowledge, since the substract where the knowledge lies is material and thence not pure. I think you mean substrate.
- hgsgm 3y ago
  Thank you for divising a helpful addition.
  - arketyp 3y ago
    I think you mean devising. It's crass, but I know I appreciate such corrections in the end myself.
    - hgsgm 3y ago
      Look closer.
      - arketyp 3y ago
        I did, times a few. Went over my head. Cheers.
  - noSyncCloud 3y ago
    The grammar corrections are multiplying as we speak
- alexmolas 3y ago
  Probably yes, English is not my first language. Thanks for the help :)
bluefone 3y ago
And I don't think that Scott Aaronson's comment makes philosophical sense - Why do we need to extend our senses to know the truths beyond our world? An ant doesn't care to know whether there is a coast line or mountain ahead at 100km distance. It has no business with those distances. There is nothing wrong in imagining Jupiter as a humanified god and its moon as his wife. This is basically tying back things which are outside of human reach to things that make sense in our world. That's how it should be, instead of saying that Jupiter is a big ball of gas or rock. That doesn't matter to our life and evolution on Earth. Speed of light doesn't matter too. 99.99% of the Science and progress we made is unnecessary and childish work done by people who have luxury of not needing to bother about their food and family. And the catastrophic results of such childish "inventions" by these rich kids (scientists) was imposed on global populations. These so called inventions are Oil, Plastics, industrialization, racial mix and related conflicts, urbanization, weaponry, weird social trends, climate changes, general degradation of human biological abilities, natural quality of food, biological diversity and so on. And what did Science achieve? We proudly announce that we busted the dangerous myths enjoyed by our ancestors!! Oh, You discovered that Earth is not the center of the universe and Sun doesn't go around Earth? Please go a step further to realize that there is no absolute frame of reference that shows what goes around what. Earth going around the Sun is just as equally true as Sun going around Earth. And yes, there is nothing wrong in saying Earth is the center of universe as you can claim any place in the universe as "the center". And you spent countless amount human energy and money to establish the half-truths like this, and call it Science and progress?
- noSyncCloud 3y ago
  This is, frankly, a surreal and almost unbelievable opinion to read on the internet in 2023. On HN of all places: although (almost) nothing surprises me any more, this sure did.
  - bluefone 3y ago
    It surprises most of us because we have been brain-washed to think that science is good, myths are bad, to think that we can ignore our biology, gender, nature etc. How does relativity matter to human life? Why is the world filled with nukes, plastics, global-warming gases, GM foods, confusion about gender and biological roles? It's time to realize that the progress we made is actually degenerate and regressive from the view point of biological adaptation which is the only goal for humans.
strogonoff 3y ago
“Mathematics is limited by the physics because physics constrains our brains” only works if you hold the philosophical view of physicalism. People who hold that view believe that the current physical model of reality is all there is and the utmost absolute truth. Materialistic monism is a popular but not particularly robust view in my opinion, because those models can never be complete and always evolve in directions guided by the applications and predictions our minds seek. (Other views are available.) I would encourage more people to apply the same principle at the higher level, changing “our brains are limited by physics, hence our mathematics is limited by physics” to “our model of reality, including physics, is limited due to our minds operating within that reality”. In other words, the model that makes you think your consciousness comes from neurones and is thus limited by physics is itself a product of your consciousness, and conclusions you make based on that model (such as “mathematics is limited by physics”) should be taken with a grain of salt (not just as possibly wrong but, worse, not-even-wrong) because you can’t construct a complete model of a system from within said system. But the kicker is that this logic itself is rooted in mathematical reasoning (see Gödel’s incompleteness theorem), so take it for what you will.
- ezion 3y ago
  The whole idea of materialism is simply a bad and inconsistent worldview. If you start under the assumption that materialism is true you have to also eat the idea that free will doesn't exist, and that you can't trust any of your thoughts since they are: 1. Out of your control, since it's all physics. 2. Created by a brain that evolved to survive, not to create logic. Thus the idea of materialism itself would be just an illusion created by your brain's need to survive. This implies it is a self-defeating worldview from the get-go - because you've reached a contradiction: the brain generates the thoughts in a deterministic fashion and you cannot trust your brain, so you cannot trust the thoughts you have about the material reality including materialism. And the armchair philosophers misunderstand Godel literally all the time, including this time. All it says is that there are statements that are true that cannot be proven within the constraints of a finite set of axioms. It's not a limitation of math, it's a limitation of what can be proven true within the confines of a set of axioms. Meaning there could be another set of axioms under which the limitation to prove that such a statement is true disappears. But it doesn't make math any less true. This article was written by someone with very poor philosophical training.
jostylr 3y ago
I think having a language that helps understand those limitations is a useful achievement. Much of mathematics does have that. A notable exception is the definition of real numbers. They are usually presented as a string of infinite decimals, or a converging sequence, or a set of numbers less than something. All of those notions obscure the basic limitation of knowing the real number and give a veneer of similarity to rational number. Rational numbers are numbers that we can have in our hand while irrational numbers are ones which we can never have. It is important to have a setup that respects that difference. This is what motivated me to come up with a new definition of real numbers, namely, they are objects (I call them oracles) that answer Yes or No when asked if the number ought to be between two given rational numbers. Abstracting out what properties such an object should have, one can come up with a space of these oracles, define an arithmetic, and prove that they satisfy the axioms of real numbers. For details: https://github.com/jostylr/Reals-as-Oracles/ https://github.com/jostylr/Reals-as-Oracles/ In many ways, this is giving a definitional support to the use of interval analysis which is, of course, a very practical concern. It also brings our some cool stuff about mediants and continued fractions (nothing new about that, but nicely motivated). It also fits in with the adjacent post about busy beaver numbers and its conclusion about knowing a number is in an interval.
- auggierose 3y ago
  > It is important to have a setup that respects that difference. No, it is not. Or rather, which differences matter depends on your application. A circle of radius 1 is a very natural thing to me. I don't think 3/4 is more natural.
- ginnungagap 3y ago
  It's not clear to me what this approach offers over Dedekind cuts, you are specifying a real number by saying in which rational intervals it lies, a Dedekind cut specifies a real number by saying which rationals are below it and which are above, and translating back and forth between the two is immediate
  - deleted 3y ago
    [deleted]
cvoss 3y ago
There's another limitation not mentioned in the article: Certain true theorems may be unprovable in the mathematical system we have, not because of a fundamental theoretical limitation of the system, but simply because carrying out the proof would require more matter/energy than exists, or more time than the universe will exist for if the proof is carried out slowly enough to avoid forming a black hole because of too much matter in close proximity. If we lived in a bigger or more long lived universe, we could attain some of these inaccessible proofs.
- evanb 3y ago
  Along the same lines, the "Math is limited by math" really contains a hidden physical assumption, which is this: the (quantum, extended) Church-Turing thesis is an assumption about physics. The (classical version of the) argument goes like this: Suppose you want to solve the Halting problem for Turing problems that can execute one step per second. If I have a machine that takes 1 second for the zeroeth step step, and 0.5 seconds for the first step, and 0.25 seconds for the second step, and so on, I can just run the "normal" machines' entire infinite history in 2 seconds. If I want to check if the twin prime conjecture is true, I'll just check every integer. If I want to check if an arbitrary normal machine halts, I just run it on my unusual machine. As far as I can tell all of the arguments against the reasonableness of such an unusual machine rely on some physical argument. For example, computational speed requires energy, an increasing speed would require an unbounded amount of energy, which is both impractical and would also presumably collapse into a black hole. So, I guess I should be careful a bit. The Halting problem is a perfectly well defined mathematical question and it has a definite answer. But why is it an interesting question at all? That's a question about the nature of the world.
  - cvoss 3y ago
    Partially agree, partially disagree. Yes. Absolutely, Church-Turing is a claim not just about math and computation but about physics too, because any and all math "runs on" hardware or wetware or some other system that obeys the laws of physics. But I disagree with your assessment of the halting problem. It is not possible to have, even in theory, a machine that solves it. This is from the classic proof of undecidability of the problem, which involves, just as you have, supposing the existence of a machine to oracularly answer the question. You can then construct a logical paradox using a machine that enters an infinite loop if its input is a halting machine, and halts if its input is a non-halting machine, and simply run this machine on itself. It cannot both halt and not halt, so you have to back up and invalidate the assumption that the halting-problem machine could exist. No appeal to physics is required for this proof. Ok, so now the fun part: Let this mathematical impossibility speak into the domain physics by way of Church-Turing. Your accelerating machine is not only mathematically impossible, it _must_ also be physically impossible, else we have to throw out Church-Turing. We deduce that the laws of physics are somehow constrained so as to make it impossible.

41 points by alexmolas 3y ago

It is an interesting argument. However, it is possible to simulate continuous phenomenon on discrete systems (eg, we can simulate real numbers on computers, and certainly build computer systems that reason about real numbers). Similarly the human brain can contemplate infinity although it is obvious that we can't comprehend it in totality because brains are finite. So it isn't clear that maths is limited by physics. It might or might not be.

arketyp 3y ago
The simulation part I'd argue isn't as interesting. That other part is though, namely that you, or a computer, can reason symbolically. We can treat the irrational length of the hypotenuse as though it exists in some concrete sense and get sensible and even useful results out of it. The limits of this approach is indeed a question.

constantcrying 3y ago

>This first idea that math have some intrinsic limitations - eg: you can’t compute all the Busy Beaver Numbers But this is not a limitation of mathematics at least not any more so than there being no largest prime number is a limitation of mathematics. It is a limitation on abstract systems, such as Turing machines. The halting problem is not that you can not deduce whether any given Turing machine halts, but that there is no Turing machine that deducing this. Gödel's (ö and o are different letters, by the way) theorems are far more wide reaching, as they appy to large classes of axiomatic systems, but they are only making claims about the systems in relation to themselves.

bigbillheck 3y ago
> no smallest prime number Behold: 2.
- olddustytrail 3y ago
  -2 is smaller... (I'll let someone better educated expand on prime ideals)
kbrkbr 3y ago
As the smallest prime number is 2 you probably meant "there being no largest prime number".
- constantcrying 3y ago
  Indeed...
tux3 3y ago
The Busy Beaver numbers express a limitation of classes of axiomatic systems, not merely of Turing Machines. It's not merely that computers aren't guaranteed to be able to calculate these. BB(n) for some n <= 745 is independent of the ZFC axioms of set theory. Furthermore, Gödel's theorem and Turing's problem are related. It is undecidable whether a Turing machine halts, it's not just that you can't write a Turing machine that decides whether another Turing machine halts. Without Gödel's theorem, you could try to enumerate all possible proofs in your complete and consistent axiomatic system until you find one that proves whether an arbitrary Turing machine of your choosing halts, and you'd have conceptually solved the halting problem. (Then, you could encode that whole algorithm in a Turing machine (TM), and have that new TM be your decider that solves the halting problem, if you wanted). Because of Gödel you can't do this, whether on paper, with a TM, or even in principle. Your axiomatic system is not complete, so there are things it simply cannot prove. Like the halting problem, or in the case of ZF the value of BB(745).

zogomoox 3y ago

If you consider that mathematical ideas are discovered rather than invented, then the whole point becomes moot. Sure, we may not be smart enough to discover all of it, but we can be pretty certain there is still mathematics beyond our ability to understand it. As an analogy, we'll never be able to look at or visit beyond our cosmic horizon, but it's certain there are stars and galaxies out there.

DecayingOrganic 3y ago
Math as we know it, due to Gödel's Incompleteness Theorems, is not fully consistent - we can't even prove everything that is true with our current mathematical framework. This means that our understanding of math is indeed limited, not just by our intellectual capabilities, but by the very structure of the math we currently use also. This makes me wonder, will we be able to develop new mathematical frameworks that bypass these issues? And if so, what will they look like?
- kbrkbr 3y ago
  > is not fully consistent Not being able to prove the consistency of a system within the system does not entail that the system is inconsistent (or not fully consistent).
  - pixl97 3y ago
    I think it means we could only ever prove it is inconsistent, and never prove that it is consistent.
    - tux3 3y ago
      If we can only prove that an inconsistent system is inconsistent, and we can never prove our (presumably) consistent system is consistent, then it is incomplete. We can't say it's inconsistent just because we can't prove otherwise. We can say it may be incomplete (or inconsistent, and we haven't noticed yet)
- 3y ago

arketyp 3y ago

>There’s no pure knowledge, since the substract where the knowledge lies is material and thence not pure. I think you mean substrate.

hgsgm 3y ago
Thank you for divising a helpful addition.
- arketyp 3y ago
  I think you mean devising. It's crass, but I know I appreciate such corrections in the end myself.
  - hgsgm 3y ago
    Look closer.
    - arketyp 3y ago
      I did, times a few. Went over my head. Cheers.
- noSyncCloud 3y ago
  The grammar corrections are multiplying as we speak
alexmolas 3y ago
Probably yes, English is not my first language. Thanks for the help :)

bluefone 3y ago

And I don't think that Scott Aaronson's comment makes philosophical sense - Why do we need to extend our senses to know the truths beyond our world? An ant doesn't care to know whether there is a coast line or mountain ahead at 100km distance. It has no business with those distances. There is nothing wrong in imagining Jupiter as a humanified god and its moon as his wife. This is basically tying back things which are outside of human reach to things that make sense in our world. That's how it should be, instead of saying that Jupiter is a big ball of gas or rock. That doesn't matter to our life and evolution on Earth. Speed of light doesn't matter too. 99.99% of the Science and progress we made is unnecessary and childish work done by people who have luxury of not needing to bother about their food and family. And the catastrophic results of such childish "inventions" by these rich kids (scientists) was imposed on global populations. These so called inventions are Oil, Plastics, industrialization, racial mix and related conflicts, urbanization, weaponry, weird social trends, climate changes, general degradation of human biological abilities, natural quality of food, biological diversity and so on. And what did Science achieve? We proudly announce that we busted the dangerous myths enjoyed by our ancestors!! Oh, You discovered that Earth is not the center of the universe and Sun doesn't go around Earth? Please go a step further to realize that there is no absolute frame of reference that shows what goes around what. Earth going around the Sun is just as equally true as Sun going around Earth. And yes, there is nothing wrong in saying Earth is the center of universe as you can claim any place in the universe as "the center". And you spent countless amount human energy and money to establish the half-truths like this, and call it Science and progress?

noSyncCloud 3y ago
This is, frankly, a surreal and almost unbelievable opinion to read on the internet in 2023. On HN of all places: although (almost) nothing surprises me any more, this sure did.
- bluefone 3y ago
  It surprises most of us because we have been brain-washed to think that science is good, myths are bad, to think that we can ignore our biology, gender, nature etc. How does relativity matter to human life? Why is the world filled with nukes, plastics, global-warming gases, GM foods, confusion about gender and biological roles? It's time to realize that the progress we made is actually degenerate and regressive from the view point of biological adaptation which is the only goal for humans.

strogonoff 3y ago

“Mathematics is limited by the physics because physics constrains our brains” only works if you hold the philosophical view of physicalism. People who hold that view believe that the current physical model of reality is all there is and the utmost absolute truth. Materialistic monism is a popular but not particularly robust view in my opinion, because those models can never be complete and always evolve in directions guided by the applications and predictions our minds seek. (Other views are available.) I would encourage more people to apply the same principle at the higher level, changing “our brains are limited by physics, hence our mathematics is limited by physics” to “our model of reality, including physics, is limited due to our minds operating within that reality”. In other words, the model that makes you think your consciousness comes from neurones and is thus limited by physics is itself a product of your consciousness, and conclusions you make based on that model (such as “mathematics is limited by physics”) should be taken with a grain of salt (not just as possibly wrong but, worse, not-even-wrong) because you can’t construct a complete model of a system from within said system. But the kicker is that this logic itself is rooted in mathematical reasoning (see Gödel’s incompleteness theorem), so take it for what you will.

ezion 3y ago
The whole idea of materialism is simply a bad and inconsistent worldview. If you start under the assumption that materialism is true you have to also eat the idea that free will doesn't exist, and that you can't trust any of your thoughts since they are: 1. Out of your control, since it's all physics. 2. Created by a brain that evolved to survive, not to create logic. Thus the idea of materialism itself would be just an illusion created by your brain's need to survive. This implies it is a self-defeating worldview from the get-go - because you've reached a contradiction: the brain generates the thoughts in a deterministic fashion and you cannot trust your brain, so you cannot trust the thoughts you have about the material reality including materialism. And the armchair philosophers misunderstand Godel literally all the time, including this time. All it says is that there are statements that are true that cannot be proven within the constraints of a finite set of axioms. It's not a limitation of math, it's a limitation of what can be proven true within the confines of a set of axioms. Meaning there could be another set of axioms under which the limitation to prove that such a statement is true disappears. But it doesn't make math any less true. This article was written by someone with very poor philosophical training.

jostylr 3y ago

I think having a language that helps understand those limitations is a useful achievement. Much of mathematics does have that. A notable exception is the definition of real numbers. They are usually presented as a string of infinite decimals, or a converging sequence, or a set of numbers less than something. All of those notions obscure the basic limitation of knowing the real number and give a veneer of similarity to rational number. Rational numbers are numbers that we can have in our hand while irrational numbers are ones which we can never have. It is important to have a setup that respects that difference. This is what motivated me to come up with a new definition of real numbers, namely, they are objects (I call them oracles) that answer Yes or No when asked if the number ought to be between two given rational numbers. Abstracting out what properties such an object should have, one can come up with a space of these oracles, define an arithmetic, and prove that they satisfy the axioms of real numbers. For details: https://github.com/jostylr/Reals-as-Oracles/ https://github.com/jostylr/Reals-as-Oracles/ In many ways, this is giving a definitional support to the use of interval analysis which is, of course, a very practical concern. It also brings our some cool stuff about mediants and continued fractions (nothing new about that, but nicely motivated). It also fits in with the adjacent post about busy beaver numbers and its conclusion about knowing a number is in an interval.

auggierose 3y ago
> It is important to have a setup that respects that difference. No, it is not. Or rather, which differences matter depends on your application. A circle of radius 1 is a very natural thing to me. I don't think 3/4 is more natural.
ginnungagap 3y ago
It's not clear to me what this approach offers over Dedekind cuts, you are specifying a real number by saying in which rational intervals it lies, a Dedekind cut specifies a real number by saying which rationals are below it and which are above, and translating back and forth between the two is immediate
- deleted 3y ago
  [deleted]

cvoss 3y ago

There's another limitation not mentioned in the article: Certain true theorems may be unprovable in the mathematical system we have, not because of a fundamental theoretical limitation of the system, but simply because carrying out the proof would require more matter/energy than exists, or more time than the universe will exist for if the proof is carried out slowly enough to avoid forming a black hole because of too much matter in close proximity. If we lived in a bigger or more long lived universe, we could attain some of these inaccessible proofs.

evanb 3y ago
Along the same lines, the "Math is limited by math" really contains a hidden physical assumption, which is this: the (quantum, extended) Church-Turing thesis is an assumption about physics. The (classical version of the) argument goes like this: Suppose you want to solve the Halting problem for Turing problems that can execute one step per second. If I have a machine that takes 1 second for the zeroeth step step, and 0.5 seconds for the first step, and 0.25 seconds for the second step, and so on, I can just run the "normal" machines' entire infinite history in 2 seconds. If I want to check if the twin prime conjecture is true, I'll just check every integer. If I want to check if an arbitrary normal machine halts, I just run it on my unusual machine. As far as I can tell all of the arguments against the reasonableness of such an unusual machine rely on some physical argument. For example, computational speed requires energy, an increasing speed would require an unbounded amount of energy, which is both impractical and would also presumably collapse into a black hole. So, I guess I should be careful a bit. The Halting problem is a perfectly well defined mathematical question and it has a definite answer. But why is it an interesting question at all? That's a question about the nature of the world.
- cvoss 3y ago
  Partially agree, partially disagree. Yes. Absolutely, Church-Turing is a claim not just about math and computation but about physics too, because any and all math "runs on" hardware or wetware or some other system that obeys the laws of physics. But I disagree with your assessment of the halting problem. It is not possible to have, even in theory, a machine that solves it. This is from the classic proof of undecidability of the problem, which involves, just as you have, supposing the existence of a machine to oracularly answer the question. You can then construct a logical paradox using a machine that enters an infinite loop if its input is a halting machine, and halts if its input is a non-halting machine, and simply run this machine on itself. It cannot both halt and not halt, so you have to back up and invalidate the assumption that the halting-problem machine could exist. No appeal to physics is required for this proof. Ok, so now the fun part: Let this mathematical impossibility speak into the domain physics by way of Church-Turing. Your accelerating machine is not only mathematically impossible, it _must_ also be physically impossible, else we have to throw out Church-Turing. We deduce that the laws of physics are somehow constrained so as to make it impossible.