17 ms·

Intuitionism

179 points by rotartsi 3y ago

smokel 3y ago
I have liked intuitionism from the very moment I first heard about it. I often entertain the idea that all the patterns we observe are merely things that match our capability of understanding. This could explain the "unreasonable effectiveness of mathematics in the natural sciences". It may also help to guide us away from the "why is there something rather than nothing" problem. If existence is total chaos, then we as humans could be limited to the hyperplane of our own observable patterns, which fools us into thinking there is some inherent order -- which there isn't. This leaves us with "why is there chaos rather than nothing", so I doubt it's of any help :) Great ideas to ponder, but rather hard to reason about. (Edit: To avoid thinking that I'm a crackpot, with "capability of understanding", I am referring to the physical processes that lead to the existence and dynamics of neurons, not to the platonic world of ideas on top of that. If someone could point out how unoriginal or nonsensical this idea is, it would save me from writing a blog post about it.)
- barrenko 3y ago
  It's not chaos, it's infinitely ordered order most would rather die than accept.
  - kibwen 3y ago
    The system began at chaos and is seeking order, though the existence of inherently unpredictable quantum effects means that there's plenty of chaos to go around.
- danbruc 3y ago
  I often entertain the idea that all the patterns we observe are merely things that match our capability of understanding. This could explain the "unreasonable effectiveness of mathematics in the natural sciences". I have a similar but somewhat different take on this. The universe comes first, it behaves - for what ever reasons - in the way it does. Than we humans show up and invent logic, a set of rules that is useful to reason about our universe. It either rains or it doesn't makes sense in our universe. But the universe could potentially have been different, for example something like the many-worlds interpretation but where the inhabitants of that universe can experience all the branches. Their logic might say it rains and it doesn't. Other ideas like objects, properties, space, time, causation or countability might also be influenced by the way our universe works and how we perceive it and they might be far from universal or useful across different possible universes. On top of that we than construct mathematics and explore what all the things are that we can build from those ideas and our laws of logic. It should probably not be especially surprising that we can on the one hand construct things that are not realized in our universe and on the other hand find structures that are useful for describing and understanding our universe.
  - Nasrudith 3y ago
    I suspect something opposite, that math comes before the universe because the universe needs math but math doesn't need the universe. Math only calls for internal consistency which radically does not depend upon the universe. It describe perfectly self-consistent realities that do not exist. Math could model say for instance a world of continuous matter as opposed to our mostly empty matter for instance. We have calculated numbers larger than the universe itself could render after all.
    - smokel 3y ago
      I beg to differ. Math typically requires abstract thought, symbols, and humans to produce and enjoy it. Especially the latter is quite a dependency. We could reduce the requirements down to an implementation of a Turing machine, or something similar. (For the argument I simply ignore whether the machine is conscious -- that seems irrelevant in this context.) That still requires some kind of discrete switch, which may seem fairly minimal to a human observer, but in reality consists of tens of thousands of atoms to operate. Atoms used to be simple, but turn out to be quite complex as well. Representing, say, a circle in a fairly minimal system such as this would probably require the cooperation of millions of atoms.
- ttctciyf 3y ago
  > fools us into thinking there is some inherent order -- which there isn't. Bold claim :) Even while maintaining a willful agnosticism about Platonic realism, it seems clear that the business of doing mathematics - intuitionistic or otherwise - depends on the ability to state and follow unambiguous rules, or else how to establish a proof within some axiomatic system or other? But if mathematicians have this ability, even as an asymptotic ideal, and are able to make judgments as to when rules are not being followed correctly, mustn't this in turn depend on some pre-existing regularity (i.e. order) in the universe? Ineluctable physical law seems a natural candidate for this order.
  - smokel 3y ago
    Thank you for your reply. I'll try some more! :) I noticed that in nature (i.e. in the physical universe) there appears to be no logic. There are not even two things exactly the same, as far as I can tell. So that lead me to believe that "counting" (or abstraction) is not even a property of the universe, but possibly only a human (or animal, or Turing machine) construct. To me, logic only starts to occur when a very complex amalgam of matter comes together [1] to realize a discrete switch. Using these discrete switches, organisms build memory and abstraction mechanisms, start counting, and do mathematics. So my thesis is that things that are considered to be "basic" by most, such as logic or mathematics, are in fact quite specific systems built on top of chaos. From there, it remains to be proven that all physical laws that we observe are no more than projections of the chaos onto such a system. Perhaps a metaphor that helps to take on my perspective, is to look at a screen filled with random noise, and then observe some patterns in there. Now replace the screen with an infinite dimensional set of chaos, and then observe a pattern that is our universe. With the added twist that we are part of this chaos, and observing the pattern, possibly in the form of physical laws. Of course, there are many problems with this theory. What does the chaos reside in? How can there be discernible parts in the chaos? Is time an emergent property inside the chaos? How can it be that the patterns that we observe are so consistent? However, to me this theory seems more fruitful than merely accepting that we cannot say anything about things that science cannot observe, or that some deity created all this. [1] With "comes together" I do not refer to a dynamic process, but to the accidental occurrence of stuff in such a shape or form. Obviously, my ridiculous theory asserts a chaos chock full of dimensions, where time and space are but supporting actors.
- simonh 3y ago
  The physical world is a persistent system that exhibits highly consistent behaviour. Because the behaviour is consistent we can describe it in a highly consistent formal language, mathematics. What Plato called forms are just descriptions. We have a description of what a circle is, and anything that matches that description is a circle.
  - peteradio 3y ago
    Highly consistent!? Not really. Some things are, but like OP just said, those are the ones we glom onto. But don't mistake some things for every things, there's a whole big wide world out there. We can hardly describe a ripple in a stream let alone why I've had the 5th argument in five weeks about the order I have to fix the kitchen with me wife, yet my intuition told me it was a comin.
    - simonh 3y ago
      So you consistently perceive streams, the world, have a body, a life, a wife, a kitchen with a flaw that has persisted over time, an order in to fix it. Also the world is consistent enough that you could predict that argument in advance. That sounds like an awful lot of consistency :)
      - peteradio 3y ago
        Yes, I've already conceded that some things are consistent. Will you concede that we tend to glom onto them or will I interpret your reply where you once again glommed onto them as a concession? ;)
        archgoon 3y ago
        It's not glomming on when these things are basically the core aspects of reality. You haven't really come to grips with what a world that was mostly inconsistent would look like.
        3y ago
- GoblinSlayer 3y ago
  If universe was total chaos, then we would observe it, because absence of chaos isn't required by anthropic principle, but instead we observe universe to have at most pseudorandom processes, therefore universe isn't total chaos.
  - smokel 3y ago
    My point is that we as humans can observe only a (highly structured) part of the chaos. I am merely assuming a chaotic universe, because that seems more reasonable than an empty universe or an ordered universe. (Yes I know that one cannot use "reasonable" as an argument for a context in which human arguments do not apply. The entire thing is a thought experiment, nothing more.)
- timeagain 3y ago
  To paraphrase Wittgenstein, is it more likely that we “discovered” chess or that we invented it? He suggests all of mathematics should be thought of in this way, as more and more complex ways of stating tautologies.
- downsplat 3y ago
  I get the appeal, it's a minimalism thing. The thinking mind makes up patterns and checks them against each other, there is no ultimate reality, blah blah blah. Besides, a mathematical ground truth would be a kind of transcendence, in that it's prior to human thought, which makes it uncomfortably close to ideas of God. Still, I've come around basically to mathematical platonism. The structure is out there, we just happen to be smart enough to tease some of it out. I have two arguments for this. The first is the very existence of long-standing problems, and their eventual resolution either way. For centuries we were able to wonder whether Fermat's last "theorem" (which was really a conjecture at the time) was actually true, and eventually Wiles came around with an extremely complicated proof and it was settled. And we do believe that math/logic is consistent enough that someone else couldn't just have followed a different train of thought and come up with a proof of the opposite. How does a strict intuitionist account for this kind of situation? The second, and possibly deeper argument, has to do with structural equivalences. I've been out of the field for decades, but I know that a standard trick in academic math is to develop structural equivalences between disparate fields. You want to prove something in an area of math, but it's hard, so you prove that the whole structure of that subfield has a one-to-one correspondence with the structure of another subfield, and then prove the corresponding theorem in the other subfield, which happens to be easier (see: analytic number theory). Again, this sounds like exploring an existing territory, not like arbitrarily building thought-bridges here and there. The bridges are where they are, and if you try to build one where reality didn't put it, your proof will get nowhere. An even stronger form of this is that, in advanced mathematics, all kinds of notions of universality appear all the time. One of the most famous is probably computability theory. Just using a few basic symbols (say, integers, first order logic and some additive operations), you get theories of varying power. But as soon as you hit a certain level of richness, bang, all of a sudden, you've hit computability. Your theory is rich enough to embed a Turing machine, and therefore is exactly as rich and expressive as any other computable - even if one is based on numbers and multiplication, and the other on graphs or some such other weird thing. Universality shows up in lots of places. I'm too far out of the field to remember them, but it starts with the very integer numbers - there are plenty of ways to formalize their initial construction, but the eventual result is exactly the same. At this point my general thinking is that the bulk of the structure is pre-given. I have no special conjecture to make about how that comes to be - it's all a logical structure, prior to matter or thought, so unlike physics, it's not like there could be another universe out there with different basic mathematics.
- joe-collins 3y ago
  Isn't this just the anthropic principle from a different angle?
barrkel 3y ago
The article seems poorly written, or at least self-contradictory in a way that makes me uncertain what it means. The introduction talks about the intuitionism framing mathematics as a human construction, in opposition to an objective reality. But the subsequent paragraph talks about the truth of proofs themselves as being subjective. It seems to me that these are two different claims. I think mathematics is a human construction and doesn't have a real substance. Instead, mathematics is a system of assumptions and generative rules, and more generally a discipline around creating and operating such systems. But "truth" within a system of assumptions and generative rules is not subjective, it's mechanically provable. A possible confusion remains in what "true" means. Can something relating to an imaginary system can be true, or is it false because truth can only apply to objective reality? I think it's trivially true. Gollum was a hobbit, within the system of Middle Earth. If it's not true, then we need a different word for true that does mean this, because this is what most people mean when they use the word about all sorts of imaginary constructs, from institutions to cultural symbols.
- mjw1007 3y ago
  I think the Stanford Encyclopedia of Philosophy article is better. https://plato.stanford.edu/entries/intuitionism/ https://plato.stanford.edu/entries/intuitionism/
  - barrkel 3y ago
    Yeah, that's much better.
- chongli 3y ago
  But the subsequent paragraph talks about the truth of proofs themselves as being subjective. This reflects the development of intuitionism. The project was first started by L. E. J. Brouwer, who rejected formalism. It was then developed by his student Arend Heyting, who formalized it with the Heyting algebra, a restricted variant of Boolean algebra that lacks the double negation elimination (~~p => p) and law of the excluded middle (p OR ~p). Pretty much all work in intuitionistic mathematics continues the work of Heyting. Brouwer would have rejected the entire enterprise as subjective, so he is mainly of historical and philosophical interest.
RubyRidgeRandy 3y ago
Because humans never experience anything in and of itself, but only the output of the interaction between sensory data and a brain, literally everything is purely the result of human mental activity.
- bluetomcat 3y ago
  This is the transcendental idealism of Kant. He makes a distinction between the noumena (things in themselves beyond human cognition) and phenomena (things as they appear to us through our senses). Our mind constructs "transcendental objects" which are merely abstract ideas based on the appearances.
- simonh 3y ago
  I would agree that everything we experience is a model of the world that we construct from sense data, interpreted by our sensory systems and cognitive faculties. Donald Hoffman is good on this and worth looking up, although I disagree with some of his conclusions. That doesn't mean the external physical world doesn't exist, the information we use to construct that model must come from somewhere, and we can deduce that the source is a persistent and consistent one. The philosopher Husserl said: “The tree plain and simple, the thing of nature, is as different as it can be from this perceived tree as such, which as perceptual meaning belongs to the perception, and that inseparably.” He came up with the idea of the noema which is our experience of something, and noesis which is our conscious act of perception. For me, that's our act of interpretation of our sensory perceptions. Sometimes this all goes wrong and we construct a flawed model that does not correspond perfectly to actual external reality, such as when we are deceived by optical illusions, stage magic or just hallucinate. Fortunately we can test and correct our perceptions through action in the physical world. I'm an out-and-out physicalist but I think he is quite correct, we must distinguish between our internal perception of things and how things actually are. Fortunately science is extremely powerful in this regard. It has allowed us to decouple our model of the world from the limitations of our perceptual system, and come up with rigorous models of reality such as Relativity and Quantum Mechanics that are not tied to direct interpretation by our perceptive systems.
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simonh 3y ago
There are several things we could call mathematics. There is the abstract collection of all possible mathematical objects, statements, proofs, etc. There is the subset which is the actual body of knowledge we have explored. Within those parameters potentially there are those mathematical objects, statements, etc that are provable, and there are those that are not provable (or consistent, yes I know about Godel). The latter probably aren't mathematics in a strict sense, but there's also the act of doing mathematics, and even if we are calculating nonsense or exploring ideas that don't work out, it's still mathematics in the sense of the act. I prefer to think of mathematics in general as a language, we are constructing descriptions of relationships between concepts, and hopefully those descriptions turn out to be consistent ones. When a description is proved to be consistent, we say that it is true. I suppose that makes me an intuitionist perhaps? Everything I said in the first paragraph above about mathematics also applies to descriptions in any language. There is the abstract set of all possible statements in English. There is the set of statements that have been made, or at least that exist in writing, recordings and people's minds and therefore exist encoded physically. There are statements that are grammatically correct or accord with linguistic conventions, and also those not unlike what the appearance sensibly is or may not be. There are also statements that correspond with reality, such as a biography of a real person, and ones do not like The Lord of the Rings. I think of these things in terms of physically encoded information. There is the collection of hypothetical information that could exist such as plays Shakespeare never wrote, and the subset of information that does because it exists in a physically encoded form. I'm not a Platonist, what we call a circle is a description of a geometric form, and a real geometric form is a circle to the extent that it matches that description. There is no abstract form called circles that exists in any sense, or any world of forms for them to exist in. Actually I don't think Plato thought there was either but he didn't have a robust account of information to work with. Taking this to the relationship with science, there are many, many valid and consistent mathematical formulae, descriptions, theorems, etc that are proven consistent but have no correspondence with anything in physical reality. No process, no physical structure that they describe. These are like literary fictions describing a fantasy world, although they may be mathematically rigorous. However there are some mathematical descriptions that do accurately correspond to relationships and processes in the physical world, and we can use them to predict the behaviours of those physical processes. This is because, fortunately, the physical processes occurring in the world are highly consistent and persistent, and therefore can be described in a highly consistent formal language such as mathematics. We call those physical laws, though I hate the term laws. They are simply highly accurate and predictive descriptions of behaviour we have observed.
vlf99 3y ago
There's room for contemplation on the subjective nature of our perception and its influence on the understanding of reality
mg 3y ago
In the philosophy of mathematics, intuitionism is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality Wouldn't that mean that not only mathematics is pure mental activity, but every thought? When we say (or think) "Joe and Sue went to the grocery", it seems inherent to this statement that there are two distinct actors. Joe and Sue. But that is already math, isn't it? I have the feeling to avoid math, we would have to avoid "something" in the first place. As it already implies that "something" is in a category, consists of a collection of other things etc. So we could not talk about anything anymore.
- constantcrying 3y ago
  >Wouldn't that mean that not only mathematics is pure mental activity, but every thought? That statements seems very obviously true.
  - mg 3y ago
    "pure mental activity" as opposed to "mental activity related to a reality outside of the mental activity".
    - constantcrying 3y ago
      I don't see how you could distinguish those. Even the purest form mathematics involves things like rules and symbols and those have to exist in reality (certainly they are at least as real as any emotion).
  - BSEdlMMldESB 3y ago
    counterpoint: remove all blackboards and chalk, and paper and pencils. can you still do math?
    - constantcrying 3y ago
      Sure, it just takes a lot of remembering. But the point is that still that isn't "pure" mental activity. I am still imagining symbols and rules, those exist in reality at least as much as emotions do.
trabant00 3y ago
I think most here would know that math is not complete, consistent or decidable. (https://www.youtube.com/watch?v=HeQX2HjkcNo https://www.youtube.com/watch?v=HeQX2HjkcNo) But I'm going to leave that aside as it's pretty high level math for me and I never run into those problems in my life. My personal problem with math that prevents me from seeing it as "discovery of fundamental principles claimed to exist in an objective reality" is natural numbers. It's impossible for me to clearly find the number 1 (for example) anywhere. The boundaries between one unit and 0.X or 1.Y seem arbitrary and chosen to help us create models. Religion and philosophy deal with this, but it's also apparent in digital signal processing. Or medicine: are the numerous bacteria in our bodies us, or not? Is the heat radiation from my body me? What about the fact that 1 + 1 rarely if ever equals 2? Meaning that two things together physically interact to create more than the sum of both parts. From celestial bodies to a replicated database the complexity goes through the roof when you have more than one thing.
- constantcrying 3y ago
  >I think most here would know that math is not complete, consistent or decidable. There is zero evidence ZFC is inconsistent. Even if "1" does not exist in reality mathematics still describes fundamental universal principles. As long as you believe that these fundamental principles exist at all they exist as mathematical ones. Not even hardcore Platonists would claim that the number 1 exists in physical reality. But that does not mean it doesn't exist in some abtract sense. You can construct Models of reality using the natural numbers and these models about real objects are just imperfect descriptions of reality.
  - Tainnor 3y ago
    There's also zero conclusive evidence that ZFC is consistent. And even worse: if you found a proof (within ZFC or a weaker system) that ZFC was consistent, you would immediately know (by Gödel's second theorem) that it is actually inconsistent. The most we could hope for is that we couls prove its consistency in another system (one that hopefully convinces us more of its evident truth?). ZFC is weird (especially choice). It's not implausible, but there's little a priori reason to assume that it describes some phyiscal reality. It just happens to give a foundation to a lot of really useful mathematics. You could take a theory such as Peano Arithmetic and argue that that one is self-evident. But unfortunately, again by the second theorem, you can't use PA to prove ZFC consistent. That's, roughly, what Hilbert wanted to do in order to convince his critics, and he failed.
Der_Einzige 3y ago
The law of excluded middle always seemed like BS to me from the moment it was taught. It's wonderful that by removing it, proofs become "harder" to make, but consequently constructive and thus more rigoris. Too many people believe in the law of excluded middle, especially in their own lives, much to the folly of civilization itself.
- constantcrying 3y ago
  It directly leads to the continuum being inseperable. If you believe that "now" is real, you believe in LEM.
  - Der_Einzige 3y ago
    Well, I'm an eternalist who remembers the basics of the Relativity of simultaneity, so I reject that "now is real" as such.
    - constantcrying 3y ago
      Do you believe that any thing can be seperated into two part, which, if put together exactly the same as before become the initial part?
- ndriscoll 3y ago
  Even if you "believe" in LEM, I think it could be helpful to conceptualize proofs that use it as instead being proofs in the "Reader monad" over LEM. So instead of proving `A => B`, you prove `A => Reader[LEM, B] = A => (LEM => B)`. If you really examine the proof you're doing, probably you don't need LEM in its full power, but actually some specific `Not[Not[X]] => X` (or a handful of specific instances like that). The neat thing about this is that in principle it could be tracked in a proof assistant with type inference. The ZIO framework for Scala has a super slick system for dependencies where e.g. an `RIO[Foo,A]` (an IO that requires a Foo and returns an A) and `A => RIO[Bar,B]` can compose to form an RIO[Foo&Bar, B] with the types inferred. So you could in principle have a proof system that lets you infer types like `A => Reader[LEM[Foo] & LEM[Bar], B]` (where LEM[T] = Not[Not[T]] => T), i.e. you get explicit types that show all of the instances of an LEM function you need to make your proof constructive, and since you're thinking of things as living in the Reader monad, you have the ergonomics of a proof that just assumes LEM. Same idea could be used with AC or any other "controversial" axiom. So you don't need to "believe" them to use them, and if you do "believe" them, you can benefit from pretending you don't.
bowsamic 3y ago
I agree with this but I would also take it further. I think that all science is purely the mental activity of humans. What is really happening is that we are probing the nature of our own mind, even in fields like physics (and I say this as a physicist myself). My justification for this is Kantian. We can't get away from the fact that our brain structures reality so that we can understand it, for example by organising things via our notions of space and time. We can't access the "noumena", i.e. the things as they truly are, independent of our perception of them. Therefore the study of physics, or indeed literally any activity, can only ever be transcendental and reflexive, rather than actually reaching out into the "objective world".
kshahkshah 3y ago
The older I get the more I feel like pursuing these trains of thought will lead to mental instability and sadness. We model the world and the world follows the model and the exceptions add to our refinement of the model and that’s what makes it true. Perception and reality are yin and yang and that is about as deep as my philosophy needs to go here
- rambojohnson 3y ago
  I understand where you're coming from, and pragmatic perspectives certainly have their place. However, philosophy, including intuitionism, can offer us new ways to comprehend the world. It's not about destabilizing our understanding, but rather enriching it. Although it can be challenging, this exploration can also bring greater depth to our perception and reality, much like the yin and yang you mentioned. Even if we don't adopt a new philosophy wholeheartedly, engaging with it can open us up to valuable insights.
  - glcheetham 3y ago
    Sounds like GPT wrote this comment
    - rambojohnson 3y ago
      and AI thinks the Constitution was written by AI: https://arstechnica.com/information-technology/2023/07/why-ai-detectors-think-the-us-constitution-was-written-by-ai/ https://arstechnica.com/information-technology/2023/07/why-a...
    - staplers 3y ago
      An illustration of AI disrupting society in indirect ways. If this comment were in fact written by human hands, your perception of it is lessened by the mere possibility it wasn't.
- chongli 3y ago
  There’s a whole lot more stuff in math than models of the world. Arguably, the vast majority of math has as much to do with the world as chess.
bell-cot 3y ago
Paraphrasing a math professor I had [mumble] decades ago: "In theory, mathematics is all pure & abstract logic, with no connection whatsoever to the real world. "In practice, if you want funding for your mathematical research, or for more than a puny handful of people to ever look at what you did, then you had best pay plenty of attention to the real-world usefulness of it."
- mjh2539 3y ago
  > In theory, mathematics is all pure & abstract logic, with no connection whatsoever to the real world. I understand what your math professor was trying to communicate, but at the same time, I think framing things this way begs the question that mathematical entities are not "real". It's more accurate (or at least less question begging) to say something like "mathematics, or mathematical objects, doesn't seem to have any obvious or direct connection to the material or physical world". Putting things this way doesn't reify mathematical entities, but it also doesn't presume that they don't exist.
  - bell-cot 3y ago
    I imagine that my old prof. would say something like: "Mathematicians agreed long ago on a very short and exacting definition for 'real' numbers, and got on with doing useful work. Philosophers never agree on short nor exacting definitions for anything, and certainly don't want to do anything useful."
speak_plainly 3y ago
There's something truly evil about psychologizing math. If anyone feels compelled by this you should probably start by learning about the many rock-solid antipsychologistic arguments out there that are at least 130 years old: https://plato.stanford.edu/entries/psychologism/#FreAntArg https://plato.stanford.edu/entries/psychologism/#FreAntArg
- bmc7505 3y ago
  Not sure why you are getting downvoted, as I believe there is a kernel of truth to what you are saying. Although I am sympathetic to the idealist position, based on what we know today about computers, I have come to believe that the thing some would call mathematics today is not grounded in reality. Whenever mathematics departs from reality, it leads those who practice it down a path that ends with insanity. In the same way that psychology can be harmful to the pursuit of truth, there is an evil or deformity in what pure mathematics represents (or became), as Cantor and Gödel both discovered trying to eliminate its paradoxes. Mathematics as practiced by, e.g., Leibniz and Euler, was much more interested in calculation, i.e., mechanization. It is this form of "mathematics", as the thing has come to be called, that should have been developed, instead of trying to axiomatize infinite sets, explain the continuum, or fix impredicativity. But despite the many paradoxes of mathematics (e.g., AoC, LEM) and machines (i.e., bugs), computer science has happily chugged along and made many practical contributions which explain the universe and shed light into people's minds, fulfilling the role that pure mathematics once served. Today, we are very close to mechanizing both minds and mathematics, and can replicate many aspects of reality (i.e., mathematical or otherwise) inside computers. It is not unimaginable that one day, computer science as it is practiced today will be seen as a kind of telescope for revealing the nature of mathematics and/or reality, which the mind alone cannot fathom.
- naasking 3y ago
  Intuitionism has nothing to do with psychologizing math. The intuitionist is perfectly fine accepting that "the mind of the mathematician" is merely a metaphor for some system capable of mathematical constructions, and that the "mathematician" in this argument can be replaced by a computer performing the same steps. In fact, a lot of computer science, particularly programming language theory, depends on this very notion.
sunsetdive 3y ago
This is post-modernism applied to math. It concludes in solipsism. Since the mind and its mental constructs are a part of the objective reality, they will end up describing aspects of objective reality. If they don't, they break down, become chaotic and incomprehensible to those grounded in the objective reality.
- fnordsensei 3y ago
  They don’t need to describe aspects of reality, they just need to describe something analogous that’s useful enough when applied. I’m no physicist, but I understand that Newtonian physics aren’t strictly true as such, but they are a good enough analogy to put a person on a different planetary body. So I think it’s fine to be agnostic and practical about the outcomes without needing say much about the metaphysics either way. Anyway, both perspectives tickle my curiosity.
  - wpietri 3y ago
    For sure. I'm of the "all models are wrong, some models are useful" school of thought. My best guess is that the platonic-ideals-are-real folks are mistaking something in their head for something outside it. That's not to deny that there is an objective reality out there, just that I have no particular reason to think that it's perfectly representable in 3 pounds of primate headmeat and expressible by squirting air through our meat-flaps. [1] But ultimately, it doesn't matter too much to me, because the practical utility of both models is pretty high. It does make me wish to meet intelligent beings from different evolutionary backgrounds, though, as I think there would be a lot of "So you think what exactly?" that would be very revealing about which things are pan-human quirks and which are more universal. [1] Credit goes to Terry Bisson here for the last bit: https://www.mit.edu/people/dpolicar/writing/prose/text/thinkingMeat.html https://www.mit.edu/people/dpolicar/writing/prose/text/think...
- mistercheph 3y ago
  The offer of 'objective reality' as the antithesis to solipsistic mental constructs is exactly the naivety that give both of these impotent families of epistemology any continued sway. Both bad, both wrong, and the crowd swings from one to the other, and at each arrival, anew recognizes the flaws of the mode and turns back.
dwheeler 3y ago
Intuitionism is very interesting, but those of us who "grew up" on classical logic can easily accidentally depend on the law of excluded middle applying in all cases. The Metamath system lets you specify the axioms you want to use, and then can verify that your proofs only use those axioms (directly or indirectly). There's a Metamath database specifically for intuitionistic logic: https://us.metamath.org/ileuni/mmil.html https://us.metamath.org/ileuni/mmil.html More things have been proven using intuitionistic logic over time.
mrwnmonm 3y ago
So why every human would agree that there are no squared circles?
mensetmanusman 3y ago
"My intuition has 10 significant digits of accuracy when it comes to calculating physical constants with quantumchromodynamics"
wanda 3y ago
The reality is that we can only have a sensible conversation about, or write sensibly about, concepts and facts within the context of our epistemic limit as a species. You may have noticed that a lot of the cliché philosophical questions haven't really shifted for as long as people have been asking them. It isn't because there is no answer, but rather the question itself is null and void. It's answer is beyond our ability to find or compute because you would have remove yourself from the human frame of reference to find it. The meaning of life? Doesn't make sense outside of the context of human life. In that context, it's whatever you want it to be or decide it is to you. Outside, well, you'd have to die to see if there's anything beyond, and as Spock points out in ST4, it would be impossible to discuss without a common frame of reference with someone who hasn't died. ~~~ To claim that math is a human invention or a noumenal language of nature is futile, the answer is not within our ability to determine. It is, for all intents and purposes, undefined. Unknowable. That's not to say that the conversation is meaningless, as the logical positivists would claim, as this is also a leap too far. Early Wittgenstein, big influence on the logical positivists, didn't try to claim that philosophy was meaningless, just that if you try to talk beyond the facts, you won't get anywhere. Bradley aways summed it up nicely for me: "anyone ready to dispense with metaphysics is a brother metaphysician with a rival theory of his own" In other words, to rule out metaphysics (defined as concepts and ideas that lack empirical data or basis in fact) is to make a metaphysical proposition, since by definition, there is no data to disprove the metaphysical propositions, just as much as there is no data to validate them. Most metaphysical philosophies caught on because they can be presented rationally as a chain of propositions and conclusions, but they only add up within their own framework that is upheld by an unprovable assumption or set of assumptions. Example, cogito ergo sum makes perfect sense, provided that you accept that the speaker is I, that the speaker thinks rather than simply utters, and that to be means anything at all. In the end, cogito ergo sum can be accepted intuitively within the framework of "let's not be sceptical of every thing" but what can you conclude from it? "I think therefore I am" well, I is a thing that thinks -> "I thinks, therefore it exists" I think presupposes I's existence -> "I thinks" That is a definition of I I = thinks === I is a consciousness which is a tautology. And that's why is makes sense, because it doesn't go anywhere than where it started, it's algebra. It starts off as a = b where b = a and can be reduced to simply a Math is privileged in that its algebraic statements can be used to model things in the world because it's numbers and functions of numbers, but ultimately the usefulness emerges from proving that a equals a very complicated statement that isn't obviously tautologically equal, or not equal to, a. Math is a tool. It's a way of reasoning that comes bundled with reasonably standardised notation that enables boosted productivity compared to reasoning about the same problems in regular languages. ~~~ It's the same reason why I find "simulated reality" questions rather dull. If the world is a perfect simulation, we have no way to distinguish it from the real thing, and so to our frame of reference, there is no difference about which we can have a discussion that makes any sense. If there are cracks in the simulation, sure, then you have a fact to talk about. But then the discussion isn't philosophical, it's practical: We've been brain-hacked, what do we do? and the conversation ends rather abruptly.
ineedasername 3y ago
>mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality In the context of the Godel Incompleteness Theorem I have always found it difficult to reconcile mathematics w/ an objective reality. A mathematical system would simply be incapable of completely encompassing objective reality.
- mensetmanusman 3y ago
  What is the opposite of an objective reality?
  - ineedasername 3y ago
    I don’t know, but an objective reality could exist without it being fully accessible to us or directly corresponding to our mathematical systems. I guess the opposite would be something like solipsism though.
    - mensetmanusman 3y ago
      Solipsism assumes the objective reality of the self.
      - ineedasername 3y ago
        That would fit though, wouldn’t it? Objective reality == exists independently of the self. Solipsism == only the self is a given. At the very least it’s opposite-ish. And if not, then I simply don’t have a better answer, but it wasn’t my question to begin with.
        mensetmanusman 3y ago
        It seems that it builds towards your questions though, and we would be better off if we could come to some agreement on objective reality. Although I understand it’s unlikely philosophers will ever agree on what reality is.
kazinator 3y ago
> To an intuitionist, the claim that an object with certain properties exists is a claim that an object with those properties can be constructed But, like, just in someone's mind ... to keep the truth of that existence purely subjective.

179 points by rotartsi 3y ago

I have liked intuitionism from the very moment I first heard about it. I often entertain the idea that all the patterns we observe are merely things that match our capability of understanding. This could explain the "unreasonable effectiveness of mathematics in the natural sciences". It may also help to guide us away from the "why is there something rather than nothing" problem. If existence is total chaos, then we as humans could be limited to the hyperplane of our own observable patterns, which fools us into thinking there is some inherent order -- which there isn't. This leaves us with "why is there chaos rather than nothing", so I doubt it's of any help :) Great ideas to ponder, but rather hard to reason about. (Edit: To avoid thinking that I'm a crackpot, with "capability of understanding", I am referring to the physical processes that lead to the existence and dynamics of neurons, not to the platonic world of ideas on top of that. If someone could point out how unoriginal or nonsensical this idea is, it would save me from writing a blog post about it.)

barrenko 3y ago
It's not chaos, it's infinitely ordered order most would rather die than accept.
- kibwen 3y ago
  The system began at chaos and is seeking order, though the existence of inherently unpredictable quantum effects means that there's plenty of chaos to go around.
danbruc 3y ago
I often entertain the idea that all the patterns we observe are merely things that match our capability of understanding. This could explain the "unreasonable effectiveness of mathematics in the natural sciences". I have a similar but somewhat different take on this. The universe comes first, it behaves - for what ever reasons - in the way it does. Than we humans show up and invent logic, a set of rules that is useful to reason about our universe. It either rains or it doesn't makes sense in our universe. But the universe could potentially have been different, for example something like the many-worlds interpretation but where the inhabitants of that universe can experience all the branches. Their logic might say it rains and it doesn't. Other ideas like objects, properties, space, time, causation or countability might also be influenced by the way our universe works and how we perceive it and they might be far from universal or useful across different possible universes. On top of that we than construct mathematics and explore what all the things are that we can build from those ideas and our laws of logic. It should probably not be especially surprising that we can on the one hand construct things that are not realized in our universe and on the other hand find structures that are useful for describing and understanding our universe.
- Nasrudith 3y ago
  I suspect something opposite, that math comes before the universe because the universe needs math but math doesn't need the universe. Math only calls for internal consistency which radically does not depend upon the universe. It describe perfectly self-consistent realities that do not exist. Math could model say for instance a world of continuous matter as opposed to our mostly empty matter for instance. We have calculated numbers larger than the universe itself could render after all.
  - smokel 3y ago
    I beg to differ. Math typically requires abstract thought, symbols, and humans to produce and enjoy it. Especially the latter is quite a dependency. We could reduce the requirements down to an implementation of a Turing machine, or something similar. (For the argument I simply ignore whether the machine is conscious -- that seems irrelevant in this context.) That still requires some kind of discrete switch, which may seem fairly minimal to a human observer, but in reality consists of tens of thousands of atoms to operate. Atoms used to be simple, but turn out to be quite complex as well. Representing, say, a circle in a fairly minimal system such as this would probably require the cooperation of millions of atoms.
ttctciyf 3y ago
> fools us into thinking there is some inherent order -- which there isn't. Bold claim :) Even while maintaining a willful agnosticism about Platonic realism, it seems clear that the business of doing mathematics - intuitionistic or otherwise - depends on the ability to state and follow unambiguous rules, or else how to establish a proof within some axiomatic system or other? But if mathematicians have this ability, even as an asymptotic ideal, and are able to make judgments as to when rules are not being followed correctly, mustn't this in turn depend on some pre-existing regularity (i.e. order) in the universe? Ineluctable physical law seems a natural candidate for this order.
- smokel 3y ago
  Thank you for your reply. I'll try some more! :) I noticed that in nature (i.e. in the physical universe) there appears to be no logic. There are not even two things exactly the same, as far as I can tell. So that lead me to believe that "counting" (or abstraction) is not even a property of the universe, but possibly only a human (or animal, or Turing machine) construct. To me, logic only starts to occur when a very complex amalgam of matter comes together [1] to realize a discrete switch. Using these discrete switches, organisms build memory and abstraction mechanisms, start counting, and do mathematics. So my thesis is that things that are considered to be "basic" by most, such as logic or mathematics, are in fact quite specific systems built on top of chaos. From there, it remains to be proven that all physical laws that we observe are no more than projections of the chaos onto such a system. Perhaps a metaphor that helps to take on my perspective, is to look at a screen filled with random noise, and then observe some patterns in there. Now replace the screen with an infinite dimensional set of chaos, and then observe a pattern that is our universe. With the added twist that we are part of this chaos, and observing the pattern, possibly in the form of physical laws. Of course, there are many problems with this theory. What does the chaos reside in? How can there be discernible parts in the chaos? Is time an emergent property inside the chaos? How can it be that the patterns that we observe are so consistent? However, to me this theory seems more fruitful than merely accepting that we cannot say anything about things that science cannot observe, or that some deity created all this. [1] With "comes together" I do not refer to a dynamic process, but to the accidental occurrence of stuff in such a shape or form. Obviously, my ridiculous theory asserts a chaos chock full of dimensions, where time and space are but supporting actors.
simonh 3y ago
The physical world is a persistent system that exhibits highly consistent behaviour. Because the behaviour is consistent we can describe it in a highly consistent formal language, mathematics. What Plato called forms are just descriptions. We have a description of what a circle is, and anything that matches that description is a circle.
- peteradio 3y ago
  Highly consistent!? Not really. Some things are, but like OP just said, those are the ones we glom onto. But don't mistake some things for every things, there's a whole big wide world out there. We can hardly describe a ripple in a stream let alone why I've had the 5th argument in five weeks about the order I have to fix the kitchen with me wife, yet my intuition told me it was a comin.
  - simonh 3y ago
    So you consistently perceive streams, the world, have a body, a life, a wife, a kitchen with a flaw that has persisted over time, an order in to fix it. Also the world is consistent enough that you could predict that argument in advance. That sounds like an awful lot of consistency :)
    - peteradio 3y ago
      Yes, I've already conceded that some things are consistent. Will you concede that we tend to glom onto them or will I interpret your reply where you once again glommed onto them as a concession? ;)
      - archgoon 3y ago
        It's not glomming on when these things are basically the core aspects of reality. You haven't really come to grips with what a world that was mostly inconsistent would look like.
        3y ago
GoblinSlayer 3y ago
If universe was total chaos, then we would observe it, because absence of chaos isn't required by anthropic principle, but instead we observe universe to have at most pseudorandom processes, therefore universe isn't total chaos.
- smokel 3y ago
  My point is that we as humans can observe only a (highly structured) part of the chaos. I am merely assuming a chaotic universe, because that seems more reasonable than an empty universe or an ordered universe. (Yes I know that one cannot use "reasonable" as an argument for a context in which human arguments do not apply. The entire thing is a thought experiment, nothing more.)
timeagain 3y ago
To paraphrase Wittgenstein, is it more likely that we “discovered” chess or that we invented it? He suggests all of mathematics should be thought of in this way, as more and more complex ways of stating tautologies.
downsplat 3y ago
I get the appeal, it's a minimalism thing. The thinking mind makes up patterns and checks them against each other, there is no ultimate reality, blah blah blah. Besides, a mathematical ground truth would be a kind of transcendence, in that it's prior to human thought, which makes it uncomfortably close to ideas of God. Still, I've come around basically to mathematical platonism. The structure is out there, we just happen to be smart enough to tease some of it out. I have two arguments for this. The first is the very existence of long-standing problems, and their eventual resolution either way. For centuries we were able to wonder whether Fermat's last "theorem" (which was really a conjecture at the time) was actually true, and eventually Wiles came around with an extremely complicated proof and it was settled. And we do believe that math/logic is consistent enough that someone else couldn't just have followed a different train of thought and come up with a proof of the opposite. How does a strict intuitionist account for this kind of situation? The second, and possibly deeper argument, has to do with structural equivalences. I've been out of the field for decades, but I know that a standard trick in academic math is to develop structural equivalences between disparate fields. You want to prove something in an area of math, but it's hard, so you prove that the whole structure of that subfield has a one-to-one correspondence with the structure of another subfield, and then prove the corresponding theorem in the other subfield, which happens to be easier (see: analytic number theory). Again, this sounds like exploring an existing territory, not like arbitrarily building thought-bridges here and there. The bridges are where they are, and if you try to build one where reality didn't put it, your proof will get nowhere. An even stronger form of this is that, in advanced mathematics, all kinds of notions of universality appear all the time. One of the most famous is probably computability theory. Just using a few basic symbols (say, integers, first order logic and some additive operations), you get theories of varying power. But as soon as you hit a certain level of richness, bang, all of a sudden, you've hit computability. Your theory is rich enough to embed a Turing machine, and therefore is exactly as rich and expressive as any other computable - even if one is based on numbers and multiplication, and the other on graphs or some such other weird thing. Universality shows up in lots of places. I'm too far out of the field to remember them, but it starts with the very integer numbers - there are plenty of ways to formalize their initial construction, but the eventual result is exactly the same. At this point my general thinking is that the bulk of the structure is pre-given. I have no special conjecture to make about how that comes to be - it's all a logical structure, prior to matter or thought, so unlike physics, it's not like there could be another universe out there with different basic mathematics.
joe-collins 3y ago
Isn't this just the anthropic principle from a different angle?

barrkel 3y ago

The article seems poorly written, or at least self-contradictory in a way that makes me uncertain what it means. The introduction talks about the intuitionism framing mathematics as a human construction, in opposition to an objective reality. But the subsequent paragraph talks about the truth of proofs themselves as being subjective. It seems to me that these are two different claims. I think mathematics is a human construction and doesn't have a real substance. Instead, mathematics is a system of assumptions and generative rules, and more generally a discipline around creating and operating such systems. But "truth" within a system of assumptions and generative rules is not subjective, it's mechanically provable. A possible confusion remains in what "true" means. Can something relating to an imaginary system can be true, or is it false because truth can only apply to objective reality? I think it's trivially true. Gollum was a hobbit, within the system of Middle Earth. If it's not true, then we need a different word for true that does mean this, because this is what most people mean when they use the word about all sorts of imaginary constructs, from institutions to cultural symbols.

mjw1007 3y ago
I think the Stanford Encyclopedia of Philosophy article is better. https://plato.stanford.edu/entries/intuitionism/ https://plato.stanford.edu/entries/intuitionism/
- barrkel 3y ago
  Yeah, that's much better.
chongli 3y ago
But the subsequent paragraph talks about the truth of proofs themselves as being subjective. This reflects the development of intuitionism. The project was first started by L. E. J. Brouwer, who rejected formalism. It was then developed by his student Arend Heyting, who formalized it with the Heyting algebra, a restricted variant of Boolean algebra that lacks the double negation elimination (~~p => p) and law of the excluded middle (p OR ~p). Pretty much all work in intuitionistic mathematics continues the work of Heyting. Brouwer would have rejected the entire enterprise as subjective, so he is mainly of historical and philosophical interest.

RubyRidgeRandy 3y ago

Because humans never experience anything in and of itself, but only the output of the interaction between sensory data and a brain, literally everything is purely the result of human mental activity.

bluetomcat 3y ago
This is the transcendental idealism of Kant. He makes a distinction between the noumena (things in themselves beyond human cognition) and phenomena (things as they appear to us through our senses). Our mind constructs "transcendental objects" which are merely abstract ideas based on the appearances.
simonh 3y ago
I would agree that everything we experience is a model of the world that we construct from sense data, interpreted by our sensory systems and cognitive faculties. Donald Hoffman is good on this and worth looking up, although I disagree with some of his conclusions. That doesn't mean the external physical world doesn't exist, the information we use to construct that model must come from somewhere, and we can deduce that the source is a persistent and consistent one. The philosopher Husserl said: “The tree plain and simple, the thing of nature, is as different as it can be from this perceived tree as such, which as perceptual meaning belongs to the perception, and that inseparably.” He came up with the idea of the noema which is our experience of something, and noesis which is our conscious act of perception. For me, that's our act of interpretation of our sensory perceptions. Sometimes this all goes wrong and we construct a flawed model that does not correspond perfectly to actual external reality, such as when we are deceived by optical illusions, stage magic or just hallucinate. Fortunately we can test and correct our perceptions through action in the physical world. I'm an out-and-out physicalist but I think he is quite correct, we must distinguish between our internal perception of things and how things actually are. Fortunately science is extremely powerful in this regard. It has allowed us to decouple our model of the world from the limitations of our perceptual system, and come up with rigorous models of reality such as Relativity and Quantum Mechanics that are not tied to direct interpretation by our perceptive systems.
- 3y ago

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simonh 3y ago

There are several things we could call mathematics. There is the abstract collection of all possible mathematical objects, statements, proofs, etc. There is the subset which is the actual body of knowledge we have explored. Within those parameters potentially there are those mathematical objects, statements, etc that are provable, and there are those that are not provable (or consistent, yes I know about Godel). The latter probably aren't mathematics in a strict sense, but there's also the act of doing mathematics, and even if we are calculating nonsense or exploring ideas that don't work out, it's still mathematics in the sense of the act. I prefer to think of mathematics in general as a language, we are constructing descriptions of relationships between concepts, and hopefully those descriptions turn out to be consistent ones. When a description is proved to be consistent, we say that it is true. I suppose that makes me an intuitionist perhaps? Everything I said in the first paragraph above about mathematics also applies to descriptions in any language. There is the abstract set of all possible statements in English. There is the set of statements that have been made, or at least that exist in writing, recordings and people's minds and therefore exist encoded physically. There are statements that are grammatically correct or accord with linguistic conventions, and also those not unlike what the appearance sensibly is or may not be. There are also statements that correspond with reality, such as a biography of a real person, and ones do not like The Lord of the Rings. I think of these things in terms of physically encoded information. There is the collection of hypothetical information that could exist such as plays Shakespeare never wrote, and the subset of information that does because it exists in a physically encoded form. I'm not a Platonist, what we call a circle is a description of a geometric form, and a real geometric form is a circle to the extent that it matches that description. There is no abstract form called circles that exists in any sense, or any world of forms for them to exist in. Actually I don't think Plato thought there was either but he didn't have a robust account of information to work with. Taking this to the relationship with science, there are many, many valid and consistent mathematical formulae, descriptions, theorems, etc that are proven consistent but have no correspondence with anything in physical reality. No process, no physical structure that they describe. These are like literary fictions describing a fantasy world, although they may be mathematically rigorous. However there are some mathematical descriptions that do accurately correspond to relationships and processes in the physical world, and we can use them to predict the behaviours of those physical processes. This is because, fortunately, the physical processes occurring in the world are highly consistent and persistent, and therefore can be described in a highly consistent formal language such as mathematics. We call those physical laws, though I hate the term laws. They are simply highly accurate and predictive descriptions of behaviour we have observed.

vlf99 3y ago

There's room for contemplation on the subjective nature of our perception and its influence on the understanding of reality

mg 3y ago

In the philosophy of mathematics, intuitionism is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality Wouldn't that mean that not only mathematics is pure mental activity, but every thought? When we say (or think) "Joe and Sue went to the grocery", it seems inherent to this statement that there are two distinct actors. Joe and Sue. But that is already math, isn't it? I have the feeling to avoid math, we would have to avoid "something" in the first place. As it already implies that "something" is in a category, consists of a collection of other things etc. So we could not talk about anything anymore.

constantcrying 3y ago
>Wouldn't that mean that not only mathematics is pure mental activity, but every thought? That statements seems very obviously true.
- mg 3y ago
  "pure mental activity" as opposed to "mental activity related to a reality outside of the mental activity".
  - constantcrying 3y ago
    I don't see how you could distinguish those. Even the purest form mathematics involves things like rules and symbols and those have to exist in reality (certainly they are at least as real as any emotion).
- BSEdlMMldESB 3y ago
  counterpoint: remove all blackboards and chalk, and paper and pencils. can you still do math?
  - constantcrying 3y ago
    Sure, it just takes a lot of remembering. But the point is that still that isn't "pure" mental activity. I am still imagining symbols and rules, those exist in reality at least as much as emotions do.

trabant00 3y ago

I think most here would know that math is not complete, consistent or decidable. (https://www.youtube.com/watch?v=HeQX2HjkcNo https://www.youtube.com/watch?v=HeQX2HjkcNo) But I'm going to leave that aside as it's pretty high level math for me and I never run into those problems in my life. My personal problem with math that prevents me from seeing it as "discovery of fundamental principles claimed to exist in an objective reality" is natural numbers. It's impossible for me to clearly find the number 1 (for example) anywhere. The boundaries between one unit and 0.X or 1.Y seem arbitrary and chosen to help us create models. Religion and philosophy deal with this, but it's also apparent in digital signal processing. Or medicine: are the numerous bacteria in our bodies us, or not? Is the heat radiation from my body me? What about the fact that 1 + 1 rarely if ever equals 2? Meaning that two things together physically interact to create more than the sum of both parts. From celestial bodies to a replicated database the complexity goes through the roof when you have more than one thing.

constantcrying 3y ago
>I think most here would know that math is not complete, consistent or decidable. There is zero evidence ZFC is inconsistent. Even if "1" does not exist in reality mathematics still describes fundamental universal principles. As long as you believe that these fundamental principles exist at all they exist as mathematical ones. Not even hardcore Platonists would claim that the number 1 exists in physical reality. But that does not mean it doesn't exist in some abtract sense. You can construct Models of reality using the natural numbers and these models about real objects are just imperfect descriptions of reality.
- Tainnor 3y ago
  There's also zero conclusive evidence that ZFC is consistent. And even worse: if you found a proof (within ZFC or a weaker system) that ZFC was consistent, you would immediately know (by Gödel's second theorem) that it is actually inconsistent. The most we could hope for is that we couls prove its consistency in another system (one that hopefully convinces us more of its evident truth?). ZFC is weird (especially choice). It's not implausible, but there's little a priori reason to assume that it describes some phyiscal reality. It just happens to give a foundation to a lot of really useful mathematics. You could take a theory such as Peano Arithmetic and argue that that one is self-evident. But unfortunately, again by the second theorem, you can't use PA to prove ZFC consistent. That's, roughly, what Hilbert wanted to do in order to convince his critics, and he failed.

Der_Einzige 3y ago

The law of excluded middle always seemed like BS to me from the moment it was taught. It's wonderful that by removing it, proofs become "harder" to make, but consequently constructive and thus more rigoris. Too many people believe in the law of excluded middle, especially in their own lives, much to the folly of civilization itself.

constantcrying 3y ago
It directly leads to the continuum being inseperable. If you believe that "now" is real, you believe in LEM.
- Der_Einzige 3y ago
  Well, I'm an eternalist who remembers the basics of the Relativity of simultaneity, so I reject that "now is real" as such.
  - constantcrying 3y ago
    Do you believe that any thing can be seperated into two part, which, if put together exactly the same as before become the initial part?
ndriscoll 3y ago
Even if you "believe" in LEM, I think it could be helpful to conceptualize proofs that use it as instead being proofs in the "Reader monad" over LEM. So instead of proving `A => B`, you prove `A => Reader[LEM, B] = A => (LEM => B)`. If you really examine the proof you're doing, probably you don't need LEM in its full power, but actually some specific `Not[Not[X]] => X` (or a handful of specific instances like that). The neat thing about this is that in principle it could be tracked in a proof assistant with type inference. The ZIO framework for Scala has a super slick system for dependencies where e.g. an `RIO[Foo,A]` (an IO that requires a Foo and returns an A) and `A => RIO[Bar,B]` can compose to form an RIO[Foo&Bar, B] with the types inferred. So you could in principle have a proof system that lets you infer types like `A => Reader[LEM[Foo] & LEM[Bar], B]` (where LEM[T] = Not[Not[T]] => T), i.e. you get explicit types that show all of the instances of an LEM function you need to make your proof constructive, and since you're thinking of things as living in the Reader monad, you have the ergonomics of a proof that just assumes LEM. Same idea could be used with AC or any other "controversial" axiom. So you don't need to "believe" them to use them, and if you do "believe" them, you can benefit from pretending you don't.

bowsamic 3y ago

I agree with this but I would also take it further. I think that all science is purely the mental activity of humans. What is really happening is that we are probing the nature of our own mind, even in fields like physics (and I say this as a physicist myself). My justification for this is Kantian. We can't get away from the fact that our brain structures reality so that we can understand it, for example by organising things via our notions of space and time. We can't access the "noumena", i.e. the things as they truly are, independent of our perception of them. Therefore the study of physics, or indeed literally any activity, can only ever be transcendental and reflexive, rather than actually reaching out into the "objective world".

kshahkshah 3y ago

The older I get the more I feel like pursuing these trains of thought will lead to mental instability and sadness. We model the world and the world follows the model and the exceptions add to our refinement of the model and that’s what makes it true. Perception and reality are yin and yang and that is about as deep as my philosophy needs to go here

rambojohnson 3y ago
I understand where you're coming from, and pragmatic perspectives certainly have their place. However, philosophy, including intuitionism, can offer us new ways to comprehend the world. It's not about destabilizing our understanding, but rather enriching it. Although it can be challenging, this exploration can also bring greater depth to our perception and reality, much like the yin and yang you mentioned. Even if we don't adopt a new philosophy wholeheartedly, engaging with it can open us up to valuable insights.
- glcheetham 3y ago
  Sounds like GPT wrote this comment
  - rambojohnson 3y ago
    and AI thinks the Constitution was written by AI: https://arstechnica.com/information-technology/2023/07/why-ai-detectors-think-the-us-constitution-was-written-by-ai/ https://arstechnica.com/information-technology/2023/07/why-a...
  - staplers 3y ago
    An illustration of AI disrupting society in indirect ways. If this comment were in fact written by human hands, your perception of it is lessened by the mere possibility it wasn't.
chongli 3y ago
There’s a whole lot more stuff in math than models of the world. Arguably, the vast majority of math has as much to do with the world as chess.

bell-cot 3y ago

Paraphrasing a math professor I had [mumble] decades ago: "In theory, mathematics is all pure & abstract logic, with no connection whatsoever to the real world. "In practice, if you want funding for your mathematical research, or for more than a puny handful of people to ever look at what you did, then you had best pay plenty of attention to the real-world usefulness of it."

mjh2539 3y ago
> In theory, mathematics is all pure & abstract logic, with no connection whatsoever to the real world. I understand what your math professor was trying to communicate, but at the same time, I think framing things this way begs the question that mathematical entities are not "real". It's more accurate (or at least less question begging) to say something like "mathematics, or mathematical objects, doesn't seem to have any obvious or direct connection to the material or physical world". Putting things this way doesn't reify mathematical entities, but it also doesn't presume that they don't exist.
- bell-cot 3y ago
  I imagine that my old prof. would say something like: "Mathematicians agreed long ago on a very short and exacting definition for 'real' numbers, and got on with doing useful work. Philosophers never agree on short nor exacting definitions for anything, and certainly don't want to do anything useful."

speak_plainly 3y ago

There's something truly evil about psychologizing math. If anyone feels compelled by this you should probably start by learning about the many rock-solid antipsychologistic arguments out there that are at least 130 years old: https://plato.stanford.edu/entries/psychologism/#FreAntArg https://plato.stanford.edu/entries/psychologism/#FreAntArg

bmc7505 3y ago
Not sure why you are getting downvoted, as I believe there is a kernel of truth to what you are saying. Although I am sympathetic to the idealist position, based on what we know today about computers, I have come to believe that the thing some would call mathematics today is not grounded in reality. Whenever mathematics departs from reality, it leads those who practice it down a path that ends with insanity. In the same way that psychology can be harmful to the pursuit of truth, there is an evil or deformity in what pure mathematics represents (or became), as Cantor and Gödel both discovered trying to eliminate its paradoxes. Mathematics as practiced by, e.g., Leibniz and Euler, was much more interested in calculation, i.e., mechanization. It is this form of "mathematics", as the thing has come to be called, that should have been developed, instead of trying to axiomatize infinite sets, explain the continuum, or fix impredicativity. But despite the many paradoxes of mathematics (e.g., AoC, LEM) and machines (i.e., bugs), computer science has happily chugged along and made many practical contributions which explain the universe and shed light into people's minds, fulfilling the role that pure mathematics once served. Today, we are very close to mechanizing both minds and mathematics, and can replicate many aspects of reality (i.e., mathematical or otherwise) inside computers. It is not unimaginable that one day, computer science as it is practiced today will be seen as a kind of telescope for revealing the nature of mathematics and/or reality, which the mind alone cannot fathom.
naasking 3y ago
Intuitionism has nothing to do with psychologizing math. The intuitionist is perfectly fine accepting that "the mind of the mathematician" is merely a metaphor for some system capable of mathematical constructions, and that the "mathematician" in this argument can be replaced by a computer performing the same steps. In fact, a lot of computer science, particularly programming language theory, depends on this very notion.

sunsetdive 3y ago

This is post-modernism applied to math. It concludes in solipsism. Since the mind and its mental constructs are a part of the objective reality, they will end up describing aspects of objective reality. If they don't, they break down, become chaotic and incomprehensible to those grounded in the objective reality.

fnordsensei 3y ago
They don’t need to describe aspects of reality, they just need to describe something analogous that’s useful enough when applied. I’m no physicist, but I understand that Newtonian physics aren’t strictly true as such, but they are a good enough analogy to put a person on a different planetary body. So I think it’s fine to be agnostic and practical about the outcomes without needing say much about the metaphysics either way. Anyway, both perspectives tickle my curiosity.
- wpietri 3y ago
  For sure. I'm of the "all models are wrong, some models are useful" school of thought. My best guess is that the platonic-ideals-are-real folks are mistaking something in their head for something outside it. That's not to deny that there is an objective reality out there, just that I have no particular reason to think that it's perfectly representable in 3 pounds of primate headmeat and expressible by squirting air through our meat-flaps. [1] But ultimately, it doesn't matter too much to me, because the practical utility of both models is pretty high. It does make me wish to meet intelligent beings from different evolutionary backgrounds, though, as I think there would be a lot of "So you think what exactly?" that would be very revealing about which things are pan-human quirks and which are more universal. [1] Credit goes to Terry Bisson here for the last bit: https://www.mit.edu/people/dpolicar/writing/prose/text/thinkingMeat.html https://www.mit.edu/people/dpolicar/writing/prose/text/think...
mistercheph 3y ago
The offer of 'objective reality' as the antithesis to solipsistic mental constructs is exactly the naivety that give both of these impotent families of epistemology any continued sway. Both bad, both wrong, and the crowd swings from one to the other, and at each arrival, anew recognizes the flaws of the mode and turns back.

dwheeler 3y ago

Intuitionism is very interesting, but those of us who "grew up" on classical logic can easily accidentally depend on the law of excluded middle applying in all cases. The Metamath system lets you specify the axioms you want to use, and then can verify that your proofs only use those axioms (directly or indirectly). There's a Metamath database specifically for intuitionistic logic: https://us.metamath.org/ileuni/mmil.html https://us.metamath.org/ileuni/mmil.html More things have been proven using intuitionistic logic over time.

mrwnmonm 3y ago

So why every human would agree that there are no squared circles?

mensetmanusman 3y ago

"My intuition has 10 significant digits of accuracy when it comes to calculating physical constants with quantumchromodynamics"

wanda 3y ago

The reality is that we can only have a sensible conversation about, or write sensibly about, concepts and facts within the context of our epistemic limit as a species. You may have noticed that a lot of the cliché philosophical questions haven't really shifted for as long as people have been asking them. It isn't because there is no answer, but rather the question itself is null and void. It's answer is beyond our ability to find or compute because you would have remove yourself from the human frame of reference to find it. The meaning of life? Doesn't make sense outside of the context of human life. In that context, it's whatever you want it to be or decide it is to you. Outside, well, you'd have to die to see if there's anything beyond, and as Spock points out in ST4, it would be impossible to discuss without a common frame of reference with someone who hasn't died. ~~~ To claim that math is a human invention or a noumenal language of nature is futile, the answer is not within our ability to determine. It is, for all intents and purposes, undefined. Unknowable. That's not to say that the conversation is meaningless, as the logical positivists would claim, as this is also a leap too far. Early Wittgenstein, big influence on the logical positivists, didn't try to claim that philosophy was meaningless, just that if you try to talk beyond the facts, you won't get anywhere. Bradley aways summed it up nicely for me: "anyone ready to dispense with metaphysics is a brother metaphysician with a rival theory of his own" In other words, to rule out metaphysics (defined as concepts and ideas that lack empirical data or basis in fact) is to make a metaphysical proposition, since by definition, there is no data to disprove the metaphysical propositions, just as much as there is no data to validate them. Most metaphysical philosophies caught on because they can be presented rationally as a chain of propositions and conclusions, but they only add up within their own framework that is upheld by an unprovable assumption or set of assumptions. Example, cogito ergo sum makes perfect sense, provided that you accept that the speaker is I, that the speaker thinks rather than simply utters, and that to be means anything at all. In the end, cogito ergo sum can be accepted intuitively within the framework of "let's not be sceptical of every thing" but what can you conclude from it? "I think therefore I am" well, I is a thing that thinks -> "I thinks, therefore it exists" I think presupposes I's existence -> "I thinks" That is a definition of I I = thinks === I is a consciousness which is a tautology. And that's why is makes sense, because it doesn't go anywhere than where it started, it's algebra. It starts off as a = b where b = a and can be reduced to simply a Math is privileged in that its algebraic statements can be used to model things in the world because it's numbers and functions of numbers, but ultimately the usefulness emerges from proving that a equals a very complicated statement that isn't obviously tautologically equal, or not equal to, a. Math is a tool. It's a way of reasoning that comes bundled with reasonably standardised notation that enables boosted productivity compared to reasoning about the same problems in regular languages. ~~~ It's the same reason why I find "simulated reality" questions rather dull. If the world is a perfect simulation, we have no way to distinguish it from the real thing, and so to our frame of reference, there is no difference about which we can have a discussion that makes any sense. If there are cracks in the simulation, sure, then you have a fact to talk about. But then the discussion isn't philosophical, it's practical: We've been brain-hacked, what do we do? and the conversation ends rather abruptly.

ineedasername 3y ago

>mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality In the context of the Godel Incompleteness Theorem I have always found it difficult to reconcile mathematics w/ an objective reality. A mathematical system would simply be incapable of completely encompassing objective reality.

mensetmanusman 3y ago
What is the opposite of an objective reality?
- ineedasername 3y ago
  I don’t know, but an objective reality could exist without it being fully accessible to us or directly corresponding to our mathematical systems. I guess the opposite would be something like solipsism though.
  - mensetmanusman 3y ago
    Solipsism assumes the objective reality of the self.
    - ineedasername 3y ago
      That would fit though, wouldn’t it? Objective reality == exists independently of the self. Solipsism == only the self is a given. At the very least it’s opposite-ish. And if not, then I simply don’t have a better answer, but it wasn’t my question to begin with.
      - mensetmanusman 3y ago
        It seems that it builds towards your questions though, and we would be better off if we could come to some agreement on objective reality. Although I understand it’s unlikely philosophers will ever agree on what reality is.

kazinator 3y ago

> To an intuitionist, the claim that an object with certain properties exists is a claim that an object with those properties can be constructed But, like, just in someone's mind ... to keep the truth of that existence purely subjective.