[HN Gopher] Intuitionism
___________________________________________________________________
Intuitionism
Author : rotartsi
Score : 151 points
Date : 2023-07-14 05:59 UTC (17 hours ago)
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| bell-cot wrote:
| 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 wrote:
| > 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 wrote:
| 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."
| sunsetdive wrote:
| 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 wrote:
| 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 wrote:
| 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/think...
| mistercheph wrote:
| 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.
| wpietri wrote:
| Beautifully put.
| mecsred wrote:
| Thank god a philosopher has arrived to tell us we're all
| wrong. Being so wise, you must have the correct answer for
| us. What's it going to be today "you're not smart enough to
| understand my genius solution" or "enlightenment can't be
| taught, only achieved"?
| cubefox wrote:
| Yeah. There are some approaches in philosophy of
| mathematics which try to avoid platonism (the view that
| mathematical objects have a mind independent existence) but
| while also retaining classical logic. That's not easily
| done though. (Currently popular is "structuralism", but
| this theory has its own problems.)
| Schiphol wrote:
| Intuitionism is not at all chaotic, though. It's a fully cogent
| way of doing math, it's just not a perfect overlap with
| traditional mathematics: some things you can prove
| intuitionistically that you cannot prove otherwise, and vice
| versa.
| naasking wrote:
| No, you're reading too much into it. The origin of the idea was
| a skepticism around seemingly paradoxical mathematical
| constructions, like uncountable infinities. Intuitionism
| eliminates some methods of proving that such things exist
| _without needing to construct a proof of their existence_.
|
| And via Curry-Howard, any intuitionistic proof is also a
| computer program. Intuitionism thus unifies computation and
| mathematics in a very direct way, which has been extremely
| useful.
| tunesmith wrote:
| I was initially surprised to read this because when I hear
| Intuitionism, I hear Intuitionist Logic. But IL doesn't have
| anything to do with denying objective reality; it can use facts
| on the way to proof. So I don't really know why Intuitionism is
| so much more adamant about denying constructive reality, or why
| it's thought to "give rise" to Intuitionist Logic, which at
| this point seems like a totally different thing. In other
| words, it's true that truth != proof, but that doesn't mean
| truth doesn't exist.
| cubefox wrote:
| If there is an independent source of truth (external
| reality), then classical logic makes sense and intuitionistic
| logic doesn't. But intuitionists say mathematics, unlike the
| physical would, doesn't have such an independent reality.
| There is no platonic mathematical reality apart from explicit
| mathematical construction. Then classical logic is
| inappropriate and intuitionistic logic has to be used for
| mathematics.
| smokel wrote:
| 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.)
| joe-collins wrote:
| Isn't this just the anthropic principle from a different angle?
| ttctciyf wrote:
| > 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 wrote:
| 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.
| danbruc wrote:
| _There are not even two things exactly the same, as far as
| I can tell._
|
| That is a common definition of identity, two things are the
| same if all their properties are the same. So by definition
| there can not be two different things with all their
| properties the same as this would make them
| indistinguishable and therefore the same thing. But if you
| relax this a bit, then for example elementary particles
| like electrons are - as far as we know - all completely
| identical up to their position, momentum and spin.
| pixl97 wrote:
| >There are not even two things exactly the same, as far as
| I can tell.
|
| https://medium.com/physics-as-a-foreign-language/how-do-
| we-k...
|
| https://www.popularmechanics.com/science/news/a27731/what-
| if...
| smokel wrote:
| Your url-based reply is quite ironic.
|
| The first link is an article that explains that all
| electrons are exactly identical, suggesting that there
| are indeed things in the universe that are exactly the
| same. However, the second link discusses the one-electron
| idea by Wheeler, which suggests the exact opposite :)
| ttctciyf wrote:
| > built on top of chaos.
|
| I don't think you've taken on the full force of the
| argument that regularity in human activity (such as
| _building systems_ ) requires a source of order for it not
| to simply dissolve into chaos itself.
|
| > How can it be that the patterns that we observe are so
| consistent?
|
| How can we claim to discern consistency (or inconsistency)
| without the ability to follow a rule correctly? And how can
| we follow a rule without a source of order or regularity in
| the cosmos? Wouldn't it be like trying to build a the
| Eiffel tower out of live slugs?
|
| If you insist on an absence of order in the physical
| universe, the onus is on you to explain how regularity in
| human activity (required for mathematics of any kind) can
| be achieved without it.
|
| This is not an argument for Platonic realism, BTW, or
| against intuitionism, roughly construed as the view that
| "mathematics is a creation of the mind" as per [1], or in
| your formulation that 'mathematical abstractions are not
| part of the physical universe' (if I understand what you're
| saying). You can perfectly well believe that mathematics is
| a mental construct and at the same time acknowledge that
| it's possible to observe regularities and order in the
| cosmos. If you want to insist that the regularity doesn't
| come from physical law, then I find it hard to see how
| you'll escape from some kind of Platonic belief in a non-
| physical realm that serves as the source of order :)
|
| In your TV screen dots analogy, isn't it usually thought
| that patterns appear only because of the structured,
| generative activity of law-observing physical components,
| specifically the neurons comprising your grey matter?
|
| 1: https://plato.stanford.edu/entries/intuitionism/
| klik99 wrote:
| This argument completely ignores the observer which is
| bound by the same limits of our processing - indeed they
| are paired together.
|
| Entering purely theoretical space here: On the timescale
| of eternity this might be a local pocket of some logical
| organization but there is no fundamental logic governing
| everything. Our observation is limited so we can't
| perceive chaos, instead evolved to only recognize
| patterns. Over infinity, pure chaos does not preclude
| long pockets of what looks like order. What we consider
| fundamental rules could very well be local phenomenon,
| which we are a product of.
|
| Of course this purely theoretical - all I'm saying is
| that intuitism could be true while also math being useful
| to predict things right now. We could also only exist for
| an instant and all our memories just construct, but
| that's not very useful. It's more useful to believe in
| scientific method because what's repeatable is provable,
| whereas chaos is by its nature unprovable - which doesn't
| make it impossible.
| ttctciyf wrote:
| Well, not a possibility I considered, but nothing in the
| argument depends on the regularity being a permanent
| feature of the cosmos, just that systematic human
| endeavours, such as mathematics or indeed meaningful
| debate, depend on it, so when it goes they go. In that
| sense, if you wish to consider this conversation
| meaningful you are kind of ceding the point that _for
| now_ chaos doesn 't, in fact, reign. If you don't
| consider it meaningful, then why are you having it? :)
| klik99 wrote:
| Not arguing anything - it's more an interesting thought
| experiment. This view doesn't change much other than
| never finding the "true unified theory of everything",
| which I'm not sure how many people think is truly
| possible (at least anytime soon) anyway.
|
| It is useful to focus on repeating things, and useless to
| focus on randomness. But I don't think it's necessarily
| true that randomness (probably a better word than chaotic
| since chaotic systems are complex mathematical
| interactions) doesn't reign. We evolved to take advantage
| of repeatable things, our sense organs and perception are
| all focused on things that are repeatable. Our definition
| of usefulness (what is useful/what isn't) depends on
| repeatable things. I believe there's a pretty high
| likelihood that we are blind to anything outside of that,
| such as true randomness. IE, we literally cannot conceive
| of true randomness since we are products of an
| environment that rewarded it.
|
| To your point, I guess this isn't exactly Intuitionism,
| since Intuitionism says it's a totally human construct,
| and mathematics has provenly predicted things in nature
| from purely theoretical models, whereas I just find the
| part that supposes mathematics isn't a fundamental part
| of objective reality possible.
|
| Either way, I don't think it changes anything about how
| we do math or science or anything - how could you even
| study this? By definition understanding and using things
| depends on repeatability. If there truly were cracks in
| it, they by definition couldn't be repeatable. It
| certainly won't help us get food.
|
| _EDIT_ > If you don't consider it meaningful, then why
| are you having it?
|
| I just find it interesting since I've had this thought
| before and seeing what other people think of it.
| smokel wrote:
| Something to consider in the context of "repetition", is
| that it requires abstraction, and possibly memory. As
| noted before, I do not see any kind of repetition
| (identical things, counting) in nature. I think
| abstraction and memory are both emergent properties from
| human brains (or machines, brains in other mammals,
| octopuses, etc.) My pet theory also initially discards
| "things", because that again requires abstraction.
|
| For reference, my views are somewhat related to
| "emergentism", "connectionism", and "realism", but I
| haven't found a school of philosophy that I feel
| comfortable with.
|
| > how could you even study this?
|
| This is indeed the biggest challenge. I am currently
| studying this from a conceptual art perspective, because
| philosophy and science do not seem adequately equipped
| for this kind of problem.
| smokel wrote:
| Thanks again for taking the time for a thoughtful reply.
| I am aware that I'm using terminology very loosely, and I
| omit many details that may be required for a full
| understanding.
|
| With respect to the full force of the argument: I assume
| that the "regularity" stems from the physical systems
| that make up our brain. Just as replication through DNA
| offers some stability in life, the shape of our neurons
| (and perhaps the dynamics of space-time, and the laws of
| physics) offer some regularity in the chaotic universe.
|
| In my view, this regularity is but an accidental blip in
| the totality of existence, but to us, who cannot observe
| the rest of the chaos, it seems fundamental -- which from
| the universal context, it isn't.
|
| The biggest problem that I cannot get around is that I
| somehow assume this chaos exists, and allows for things
| to exist inside of it. I do not know how to provide
| arguments for that, other than the negative one that it
| seems highly unlikely that "there is something rather
| than nothing". Likeliness, and the fact that I can define
| these abstract concepts, only make sense in the realm of
| human thought, so I am sort of stuck in a recursive loop
| there.
|
| With regards to the second part of your reply, again it
| is us humans who do the discerning. It is an emergent
| property of our brain (or possibly of slightly simpler,
| but still rather complex "discrete switches") that we can
| discern things. In the underlying universe of total
| chaos, there is no context, no logic, no measure to
| discern things.
|
| So, the source of order does arise through physical
| constructs, that happen to have a certain structure that
| allows observation. It is humans, mammals, octopuses,
| computers, that can use this universal form of
| observation to process input, and then do observation as
| we know it. So I suppose my idea is some kind of realism,
| but my reality is nothing more than pure and utter,
| unbounded chaos. And we live in some corner of that.
|
| The grand claim is that mathematics is nothing more than
| the result of some self-observing shapes in the chaos
| that is existence.
|
| Again, I feel sorry for all the readers who try to make
| sense of all my overloaded concepts. I wish I had the
| skills to write down my thoughts more rigorously. Or
| perhaps someone can save me a lot of time [1].
|
| [1] https://xkcd.com/386/
| dr_dshiv wrote:
| > There are not even two things exactly the same, as far as
| I can tell
|
| Aren't all basic particles defined by the fact that they
| are exactly the same? And countable things rely on some
| difference, such as different spatial locations -- or else
| they wouldn't be countable, they'd just be the same.
|
| While two neutron stars are distinguishable they are also
| classifiable as a real type of star in a manner that seems
| to go beyond human perceptual idiosyncrasy.
|
| But I'm a deep Platonist/Pythagorean -- so my bias is that
| "all is number" and the world is made of math. Math is real
| :)
| antonvs wrote:
| If the world is made of math, do new physical objects pop
| into being whenever a mathematician writes down or thinks
| of a structure or a proof? Or does only some math get to
| become physical?
|
| In general, your view seems like a definitional issue to
| me. If you want to call what the world is made of "math",
| then what you and I mean by math are two different
| things, and using the same word to describe them only
| leads to confusion.
| dr_dshiv wrote:
| Well, we try to discover math. That's straight forward?
| antonvs wrote:
| Again, definitional issues are critical here.
|
| I can make up all sorts of math that isn't really
| "discovered", it's more "invented". Most of it won't
| correspond usefully to the physical universe. See e.g.:
| https://plato.stanford.edu/entries/formalism-mathematics/
| :
|
| > "mathematics is not a body of propositions representing
| an abstract sector of reality but is much more akin to a
| game, bringing with it no more commitment to an ontology
| of objects or properties than ludo or chess."
|
| I take the brevity and lack of commitment to a position
| in your reply as an indication that you're not really
| willing or able to defend your position. That's fine,
| it's a complex topic, as the many articles on the SEP
| linked above attest. But if you want to claim "the world
| is made of math", then the onus is on you to define what
| you mean by that. To me, it looks a little incoherent.
| dr_dshiv wrote:
| Take the brevity as a lack of hubris. The notion that
| "the world is made of math" is one of the oldest and most
| influential ideas of all time. If you find Pythagoras,
| Plato and Newton a little incoherent, that's not unusual.
| But the onus doesn't lie with them (or me). In any case,
| I remain interested in your ideas!
| antonvs wrote:
| None of Pythagoras, Plato, or Newton claimed that the
| "world is made of math". Also, Aristotle's philosophy of
| mathematics is considered an alternative to Plato's, so
| trying to seek solace in both at the same time seems
| inconsistent.
|
| Plato described math as a realm distinct from both the
| physical world and the world of consciousness. This
| doesn't support the idea the "the world is made of math".
|
| Newton described the world as operating in accordance
| with the rules of math, but that's not the same as being
| "made of math." Plato's view is compatible with this:
| what Frege described as the "third realm", the realm of
| abstract objects, can have a relationship with the
| physical realm without requiring that the latter be "made
| of" the former.
|
| Aristotle explicitly distinguished between physics and
| mathematics, saying in his Metaphysics that physics is
| concerned with things that change, whereas mathematics
| encompasses things that are eternal, do not change, and
| are not substances. So Aristotle seems to explicitly
| reject your view.
|
| As such, I don't accept your claim that your position is
| "one of the oldest and most influential ideas of all
| time."
|
| > But the onus doesn't lie with them (or me).
|
| If you make a claim, the onus certainly lies on you to
| support that claim.
| narag wrote:
| I've seen statistics proposed as the force that makes
| reality, that would be fundamentally random, coherent. But
| statistics laws are themselves very strong when numbers get
| big.
| derefr wrote:
| In other words, the Cartesian principle extended to two
| parties?
|
| _Cogito sicut tu cogitas, ergo sum sicut structus es_ : I
| think as you think (at least for a moment); therefore, I am,
| at some level, structured as you are (at least for a moment);
| and therefore, there is order in the universe, at least in
| the temporary alignment of our structures that enables us to
| think along these same lines?
|
| (Saying nothing, of course, of whether you "exist." I may be
| a brain in a vat, and your thoughts may be the emission of an
| evil demon running a simulation -- but that still means that
| my thoughts and the evil demon's simulation share structure
| that implies an underlying shared set of computational
| axioms!)
| ttctciyf wrote:
| Yes, thank you!
| pilgrim0 wrote:
| > mustn't this in turn depend on some pre-existing regularity
| (i.e. order) in the universe?
|
| Yes. And such order exists within the observer, the
| mediators. We are measuring instruments. Scales are bound to
| the ruler and not to what's being ruled. So mathematics
| represents order only in as much as there are people to
| validate it. Should all rulers be broken and forgotten, then
| there's no measure at all. This is the same for those orders
| not yet established, that is, the future of science.
| Nevertheless, I do believe there's a principle which allows
| order, but it's not order itself, but a foundation of order,
| which I believe to be Unit. Unit is not mere duality, because
| duality implies two, Unit would be more like Cause-Effect,
| wherein one is the same as the other (either Cause-Cause, or
| Effect-Effect, doesn't really matter). This is also different
| than yin-yang, since each is discrete, and discreetness
| itself cannot exist prior to Unit. I like to think that all
| numbers are qualities of Unit, and the whole of mathematical
| theories are different theories of Unit, so they will be
| consistent every time Unit is maintained consistently, when
| something "follows" from what has already "followed",
| following some definition of "following", whatever it may be.
| It would explain the effectiveness of mathematics in the
| sense that the whole Universe is Unit out of self-similar
| Unit. The fact that all of information can be encoded in 0s
| and 1s is a great illustration of the power of Unit, and the
| fact that binary streams only makes "sense" upon
| interpretation is a great illustration of consciousness,
| which is an expression of Bias. Even if the Universe would
| change so all of the physical "laws" would mutate, Unit would
| still persist unchanged. New things would still "follow". I
| haven't come across anyone realizing that a theory of
| "everything" can exist but be useless, just like the concept
| of "everything" is useless as a particularity, and theories
| aim to be particularly applicable, so a theory which really
| applies to every thing applies to no thing. This would be the
| utmost conclusion of Godel's incompleteness. How would a
| theory of absolutely everything be different than an
| infinitely long ruler without any subdivisions, or even with
| infinitely many zero-spaced subdivisions? One wouldn't be
| able to measure any particular thing with such universal
| ruler.
| smokel wrote:
| Does your "principle which allows order" presume the
| existence of space and time, and more specifically the
| ordering of time? I would say that in a universe without
| time, "cause and effect" have little meaning.
|
| I think I lost your train of thought when you say that
| numbers are qualities of Unit. Does your universe involve
| only "Unit", or are there other principles at play as well?
| pilgrim0 wrote:
| The Peano arithmetic hints at the proposition that
| numbers are gradations within a unified principle, in
| this case the principle of "succession". If you start
| with nothing (zero) and recursively apply the same
| quality (succession) you get all integers. So we can
| rationally assume that all numbers are different
| qualifications of some primitive. As for time and space,
| they do not fundamentally exist and they do not configure
| a necessity within the universe. It takes a being able to
| record and internally persist events for time to appear.
| Space is similar, being also a referentiality. In
| essence, time and space are different framings of the
| same phenomenon. The length distance between two points
| is also a temporal ratio between the points. It is just
| two perceptions, two expediences, not grounded in
| singular reality. I believe "beings" are Unit juxtaposed
| over Unit, in the spirit of Wheeler ideas [1]. I don't
| think there are fundamental necessities other than Unit.
| Unit is not causality, I have just used the terms for
| illustration. If there are other fundamental necessities
| other than Unit then one would need to explain how they
| came into being, and it would eventually recurse into
| Unit. From nucleosynthesis to procreation, it's Unit all
| over. The way I grasp it, mathematics is a coloring of
| Unit, just like for seeing wind one need to sprinkle
| something over it. It boils down to the ascription of
| Parts within Wholes. Parts are mere subjections.
|
| [1] https://www.reddit.com/r/holofractal/comments/6wnekw/
| this_be...
| zoogeny wrote:
| > mustn't this in turn depend on some pre-existing regularity
|
| Consider the set of all things, including the ordered and
| disordered. Consider the operation of taking subsets of that
| set. Consider that conscious entities such as us only arise
| in ordered subsets (for some definition of ordered).
|
| Those conscious entities would see their proximal environment
| as ordered. They might assume that the only things in the set
| of all things are those things that are ordered in the same
| way as the proximal environment from which they arose.
|
| Our universe may be just a subset of a subset. That is, our
| universe might be one kind of universe that has some apparent
| quantum-field or relativity-geometry based regularity. There
| might be some number of universes that share that kind of
| ordering and perhaps not in all of them conscious entities
| arise.
|
| It is still an interesting question to ask: why would
| conscious entities arise within this particular kind of
| ordered universe? But it no longer remains the question of
| whether or not regularity is a fundamental property of all
| possible states of existence. And in fact it leaves open the
| question as to whether or not some kind of consciousness
| (perhaps very different from our own) could arise in what
| would appear to us as chaotic universes.
| smokel wrote:
| I spent quite some time thinking along these lines, but
| then I realized that it is very unlikely to be the case in
| our universe.
|
| If our world would be an "accidental" subset of the
| universe, with ordering, then how can it be that this
| ordering is so consistent? Would it not be more likely that
| we'd live in a world that has only _some_ order, or varying
| order depending on one 's position in space and time? For
| example, I would expect a lot more miracles to happen, but
| every physics experiment turns out to be extremely
| consistent.
|
| This led me to believe that there is another process at
| play. My thought experiment now assumes that observation
| enforces a certain kind of consistency in the laws of
| physics. That is, the systems that we use to observe
| something, must by their very existence result in very
| consistent patterns in the chaos. What this precisely looks
| like and how it operates is left as an exercise to the
| reader.
| zoogeny wrote:
| > observation enforces a certain kind of consistency in
| the laws of physics.
|
| That sounds like a typical chicken-and-egg problem.
|
| > how can it be that this ordering is so consistent?
|
| One thing that I didn't explicitly state is that the
| universe only needs to _seem_ ordered. You are making an
| assumption that isn 't necessarily founded. 99.9% of the
| time we aren't checking on the ordered state of the
| universe. It is the exceedingly rare case that we measure
| with sufficient granularity to expect to see quantum
| effect, for example.
|
| > I would expect a lot more miracles to happen
|
| I totally understand this. You might expect a table upon
| which your keyboard rests to change color as we pass
| through a chaotic portion of some multi-verse. But that
| assumes order can't exist at the time-scales of
| universes.
|
| Imagine it like a picture. The set of all possible
| pictures at 640x480 is massive but finite. The vast
| majority of images in that space are like white noise.
| But you've probably seen millions of images at that
| resolution that are totally coherent. You don't expect
| the middle block of some specific image to randomly be a
| different color. Same with a movie. The set of all
| sequences of 640x480 images at 30 fps with 1 minute
| duration is finite. It is full of total chaos. But when
| you watch a movie you don't expect chaos to ensue at any
| moment.
|
| If you consider our entire universe like a single image
| from the set of all images or a movie from the set of all
| movies, it isn't surprising that it is entirely ordered
| from beginning to end. It just feels weird because the
| scale in both time and space is so large in the case of
| the universe. But given the scale of the entire set, such
| a large subset having internal consistency isn't totally
| unreasonable.
| californical wrote:
| I don't have anything to add, but I just genuinely love
| your analogy. Totally helped me see the idea in a
| different way
| danbruc wrote:
| _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 wrote:
| 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 wrote:
| 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.
| barrenko wrote:
| It's not chaos, it's infinitely ordered order most would rather
| die than accept.
| kibwen wrote:
| 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.
| timeagain wrote:
| 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.
| simonh wrote:
| 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.
| darkclouds wrote:
| > The physical world is a persistent system that exhibits
| highly consistent behaviour.
|
| Thats because chemicals shape/influence our personalities and
| emotions. You'll see the same consistent behaviours in
| animals as you do in humans, when given the same chemicals.
| darkclouds wrote:
| Dont know why this got down voted.
|
| https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/RS0
| 1....
|
| Your mobile phone can be used as a weapon against you,
| triggering calcium waves in the body which could be harmful
| to your liberty and your life.
|
| I know because I've had it done to me!
| peteradio wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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.
| peteradio wrote:
| > what a world that was mostly inconsistent would look
| like
|
| Probably one that had trouble evolving high-order
| lifeforms, aka anthropic principle.
| downsplat wrote:
| 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.
| GoblinSlayer wrote:
| 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 wrote:
| 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.)
| [deleted]
| speak_plainly wrote:
| 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
| bmc7505 wrote:
| 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 Godel 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 wrote:
| 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.
| kshahkshah wrote:
| 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 wrote:
| 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 wrote:
| Sounds like GPT wrote this comment
| staplers wrote:
| 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 wrote:
| 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.
| theamk wrote:
| I started being much happier about philosophy in general once I
| realized a very important truth: "unlike hard sciences,
| philosophy only affects philosophers"
|
| For example the toplevel article says things like "independent
| existence in an objective reality".. this sounds important,
| right? Perhaps depending on the truth of the intuitionism, we
| might discover some facts about real world?
|
| Nope. The whole "is intuitionism true" question only affects
| the person thinking about it. The whole books are written that
| basically quibble about dictionary definitions of "fundamental
| principles", "objective reality", "mental activity" and other
| complex words.
|
| (This is especially visible in discussion of "consciousness":
| there are tons of texts about it, and yet none of them matter
| in any practical way. The practical applications -- generative
| text models, NLP, neuroscience, etc.. -- just ignore all the
| philosophic cruft)
| nextaccountic wrote:
| Intuitionism has had a great impact on computer science and
| the design of many programming languages.
|
| It's now widely known that an intuitionistic proof has
| computational content (through the curry-howard
| correspondence)
|
| Those things have a real world impact
| wanda wrote:
| 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 cliche 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.
| barrkel wrote:
| 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.
| bowsamic wrote:
| I agree and this is actually a common problem in philosophical
| discussions. People struggle to differentiate between two kinds
| of objectivity: the "total" objectivity of knowledge that is
| completely context-free and unstructured, and the objective
| _modulo_ a given set of assumptions (e.g., "the human brain",
| or at least, the human way of structuring and understanding
| reality). Mathematical truths are objective with respect to the
| latter kind of objectivity, but not the former.
| chongli wrote:
| _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.
| [deleted]
| mjw1007 wrote:
| I think the Stanford Encyclopedia of Philosophy article is
| better.
|
| https://plato.stanford.edu/entries/intuitionism/
| barrkel wrote:
| Yeah, that's much better.
| constantcrying wrote:
| >mathematics is a system of assumptions and generative rules,
| and more generally a discipline around creating and operating
| such systems.
|
| This is essentially the standard mathematical approach
| developed during the early last century. From a few basic
| axioms (which are not really justifiable) new statements are
| proven and structures are built up. All notions of truth and
| provability are relative to that system. In standard ZFC (those
| are the standard axioms) mathematics "1+1=2" is, like all other
| statements, a statement about sets. The statement is true by
| the definitions of 1,2, "=" and "+". In an alternative system
| with different axioms or definitions the statement is false.
|
| This is not the view of intuitionists though. For them the
| symbols 1,2 or + aren't formalized objects (e.g. sets in this
| case). They are just symbols, which transfer some (hopefully)
| shared meaning to another person, who then (potentially with
| some addition arguments) might also accept the truth of that
| statement.
|
| In the former the question of "truth" is fully formal there can
| be no "interpretation" in any meaningful sense. Intuitionism
| places mathematics fully inside the mind of the mathematician
| and "truth" can only be found there.
|
| >Can something relating to an imaginary system can be true, or
| is it false because truth can only apply to objective reality?
|
| In formalized mathematics "truth" is fully formal. There can
| not be any "external" truth derived from it, as any such
| statement is non-sensical.
| practal wrote:
| > 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.
|
| Mathematics is a human construction, but it certainly has a
| real substance. What does "mechanically provable" even mean, if
| there is no absolute truth? Do you believe in the definition of
| a proof or not? Do you believe that whenever the assumptions of
| your theorem are true, and you have a proof of a conclusion,
| that then the conclusion is also true? If you do believe that,
| that's your absolute truth, then. If you don't believe that, a
| proof is meaningless, isn't it?
| constantcrying wrote:
| >What does "mechanically provable" even mean, if there is no
| absolute truth? Do you believe in the definition of a proof
| or not?
|
| Different Axioms lead to different provable statements.
|
| Believing in standard mathematics basically means that you
| can not believe in absolute truth. Unless you also believe
| that some guys a hundred years ago figured the sole and
| completely perfect rules which totally correspond to reality.
| denotational wrote:
| > Believing in standard mathematics basically means that
| you can not believe in absolute truth.
|
| I agree that if one follows an axiomatic approach strictly
| and consider "truth" to be a shorthand for "provable from
| in some logic from some set of non-logical axioms" [1] then
| one is rejecting any notion absolute truth, since
| everything is relative to some set of axioms, but I don't
| agree with the charactedisation of this as "standard"; it
| seems to me to be a very Formalist stance.
|
| I'd argue that most mathenaticians consider themselves
| Platonists, and believe that the mathematical objects they
| are describing are real enough to form some kind of
| metamathematical "standard model", and "absolute truth" can
| be defined in the model-theoretic sense relative to this
| standard model, even if this is somewhat unavoidably
| handwavy.
|
| [1] : Even if you do think this, "truth" is generally used
| by logicians in the model-theoretic sense of "truth in some
| _specific_ model /structure compatible with the language".
| markisus wrote:
| Yet still professional mathematicians have an underlying
| notion of truth outside of any axiom systems. I forgot who
| said it but if we were to find a contradiction using
| Peano's axioms, we would say that the axioms were wrong,
| rather than arithmetic itself.
|
| Even your comment references "perfect rules which totally
| correspond to reality" which seems to be another way to say
| "absolute truth".
| barrkel wrote:
| I think it's a system of symbols and rules. By mechanically
| provable, I mean that given axioms (assumptions) and rules,
| you can devise a machine (i.e. something which follows rules,
| with no independent thinking or homunculus) which generates
| statements which follow from the axioms and rules, and this
| is what "true" means in the system.
| practal wrote:
| So would you say that your system of symbols and rules is
| real? Could it be that we both use the same system of
| symbols and rules, with the same assumptions, but derive
| different conclusions? If not, why not?
| barrkel wrote:
| It has no substance other than its representations;
| there's nothing of it you can touch which is physical. At
| best, there is a correspondence between the physical and
| the system.
| practal wrote:
| I agree with you here, at least in the sense that the
| system is definitely not physical. You didn't answer my
| question, though. Is it real?
| simonh wrote:
| All information that exists is physical, encoded in a
| physical substrate. Beads in an abacus, holes in a
| punched card, distributions of charge in a computer
| memory. Hypothetical information that is not encoded
| physically cannot be causal. A book or computer program
| that have not been written can have no effects. Only
| information that exists in a physical encoding can be
| causal, by virtue of it's physicality.
| practal wrote:
| Not sure where you are going with this. What is
| causality? Is there a non-mathematical way of making it
| precise? And if I have found some way to make it precise,
| does it matter if I write it down here in this HN
| comment, or on a piece of paper, or just think it? Is the
| mathematical content different depending on how it is
| expressed in physical reality?
| BSEdlMMldESB wrote:
| if the system is sound, then (I think) by definition you
| cannot prove different (wrong) conclusions.
|
| if you derive a different result, by soundness those
| would be equivalent ???
| practal wrote:
| It does not really matter if the system is sound or not,
| right? Although of course a sound one is far more
| interesting. Anyway, any way of justifying this is
| mathematical (and so would be the definition of
| soundness, if it was relevant here). If math is not real,
| then there is no justification.
| BSEdlMMldESB wrote:
| math describes (fragments) of reality;
|
| therefore it is of no consequence if math as itself is
| "real" or not. it is intended to model whatever "real"
| even is.
|
| somewhat similarly: in modern logical theories whatever
| "true" (and/including "false") even mean doesn't matter.
| is left out of the logical theory and it is effectively a
| mere parameter.
|
| all the subject does is gurantee "truth in, truth out"
| (and complementarily "false in, false out")
|
| the precise details of true "and/including" false, seems
| to me, are somewhere in the boundary between "classical"
| and "intuitionism" (or "constructivism")
|
| the subtle distinction between intuitionism" and
| "constructivism" is above my pay grade (and seemingly
| above the paygrade of everybody I've had the chance of
| discussing this with)
| practal wrote:
| > math describes (fragments) of reality;
|
| This is only possible if math itself is real. Note that I
| am not saying that a particular axiom system like
| Euclidean geometry has some sort of "real physical
| manifestation". No, what I am saying is that logical
| reasoning itself is real. And our reasoning about logical
| reasoning is certainly real as well, even if logical
| reasoning itself happens in very abstract form. Math
| itself might be viewed by some as just a game of symbols.
| But that doesn't change the fact that _the game itself is
| real_. Would it be otherwise, then math would be about as
| important as chess.
| BSEdlMMldESB wrote:
| I like to draw a distinction between real and ideal.
|
| I insist that math is ideal. it models reality ideally.
|
| this distinction is important because otherwise we mix
| together something, and the ideas and concepts (e.g.
| symbols and rules) we use to describe and model said
| something.
|
| the game is not real. people playing the game are real,
| the game getting played is real. the game on its own as
| may be described in symbols is ideal.
|
| i suppose what it all is all about is the intersection
| between this reality and this _ideality_.
| practal wrote:
| You can say that a certain axiom system models a certain
| part of reality in an ideal way. But whatever is ideal,
| is also real, because otherwise there is nothing that
| could model anything. So your intersection of reality and
| ideality is just ideality itself.
| 112233 wrote:
| Wow, you excavated an ocean to cover a puddle. how this:
| "something which follows rules, with no independent
| thinking or homunculus" can be easier to prove and reason
| than the initial rules?
|
| For example, what is simpler to reason out: does a chess
| move violate chess rules, or, there exist a method to
| construct an electomechanical device, that will Correctly
| determine whether this chess move is legal?
| barrkel wrote:
| The reason I brought up a machine explain something as
| being mechanical is to clarify that it doesn't require
| intuition.
|
| We can make machines that count. A trivial example:
| pebbles in a bucket. Neither the pebbles nor the bucket
| need intelligence to act as (have a correspondence with)
| a counter.
| 112233 wrote:
| Completely agree about the machines. Many mathematical
| results become more rigorous as a result of "can be done
| on this kind of machine" type of proofs. However, if the
| goal is getting rid of intuition, machines don't help!
| Because no matter what the machine (bucket of pebbles or
| a Buchholz hydra), it takes a lot of intuition to prove,
| that particular machine correctly enforces intended
| rules. Usually more intuition, than the original problem.
|
| Without having proof for the machine itself that machine
| is declared axiomatic. It is a valid way to go about
| things of course, but I would hesitate calling it "not
| requiring intuition".
| trabant00 wrote:
| I think most here would know that math is not complete,
| consistent or decidable.
| (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.
| BSEdlMMldESB wrote:
| > It's impossible for me to clearly find the number 1 (for
| example) anywhere.
|
| what about that there is one of you?
| AnimalMuppet wrote:
| "1" works just fine for things that are discrete. Take eggs,
| for example. I have one egg. If it's smaller than other eggs, I
| don't have 0.9 eggs; no, I have exactly one egg or, if you
| prefer, I have exactly one _small_ egg, but still exactly one.
|
| I have exactly one wife. If she gained weight, I would not then
| have 1.01 wives.
|
| I have exactly one cat. If she had N kittens, I would then have
| exactly N+1 cats, not (1 * N/10) cats.
|
| And so on.
| athrowaway3z wrote:
| My looking at natural numbers gave me the exact opposite
| conclusion.
|
| Once the grey blob of infinite nothingness becomes distinct
| enough that there is a difference between one moment/point and
| another moment/point, you have enough to start counting
| different states (or do binary arithmetic). Its high school
| math to find out some numbers are different then others and
| eventually you'll find the primes.
|
| I'll bet there is no god, power, or alternative rules in any
| possible universe, fictional or real, that could not find the
| primes.
| constantcrying wrote:
| >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 wrote:
| 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 Godel'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.
| skissane wrote:
| Human thought is paraconsistent - relevance/relevant logic
| best models how implication works in natural language, and
| that is paraconsistent; I think the intelligibility of
| inconsistent fiction such as Graham Priest's _Sylvan's Box_
| [0] is also evidence of that. If one believes mathematics
| is ultimately grounded in human thought, and if human
| thought is ultimately paraconsistent, that suggests
| paraconsistent logic may be a better foundation for
| mathematics than classical logic. It also suggests that
| maybe we should seriously consider taking the inconsistency
| horn of Godel's trilemma (incomplete or inconsistent or
| weak), given the paraconsistent rejection of the principle
| of explosion means that doing so is non-trivial.
| Inconsistent theories can be strong, complete and non-
| trivial.
|
| [0] https://projecteuclid.org/journals/notre-dame-journal-
| of-for...
| roywiggins wrote:
| I would bet that, even if ZFC is not consistent, there's
| another set of axioms which is, and in which all the stuff
| we've proven in ZFC still holds. That is, ZFC just happens
| to be a useful framework for the mathematics we're
| interested in. Even if ZFC collapses it seems very unlikely
| that all the stuff we've proven within it will; instead,
| we'll fix ZFC, like ZFC "fixed" naive set theory.
| simonh wrote:
| That's not quite correct. Systems of mathematics cannot be both
| complete and consistent, but incomplete systems of mathematics
| can be consistent. For example Presburger arithmetic is
| provably consistent. There are limits to consistency for sure,
| but that doesn't mean there's no such thing as mathematical
| consistency.
| codeflo wrote:
| I'm not sure what you mean. Presburger arithmetic is famously
| complete. What a system can't be is consistent, complete,
| _and_ strong enough to perform a Godel encoding (which
| requires something multiplication-like). Drop any of the
| three requirements and it 's possible.
|
| Inconsistent: trivial, from falsehood follows anything.
|
| Incomplete: Peano.
|
| Weak: Presburger.
| skissane wrote:
| > Inconsistent: trivial, from falsehood follows anything.
|
| Only trivial if you accept the principle of explosion ( _ex
| falso quodlibet_ or _ex contradictione quodlibet_ ). If you
| reject it, you end up with paraconsistent logic, from which
| one can develop nontrivial inconsistent mathematics see
| https://plato.stanford.edu/entries/mathematics-
| inconsistent/ and https://ir.canterbury.ac.nz/bitstream/han
| dle/10092/5626/1263...
| simonh wrote:
| The comment could be interpreted as meaning that such
| systems cannot be complete or consistent, I'm just pointing
| out they can be one or the other. As I understand it, it is
| possible to consistently prove and decide things in
| mathematics, just not everything. Godel proved limits to
| mathematics, not that mathematics doesn't work. That's all.
| trabant00 wrote:
| > Godel proved limits to mathematics, not that
| mathematics doesn't work
|
| Nobody claimed it doesn't work. It clearly does. The
| question is if it's a fundamental property of the
| universe or just an useful but flowed human mental model.
| simonh wrote:
| Sure, personally I see it as a language for expressing
| relationships and processes. I expounded on this in
| detail in another comment.
| loicd wrote:
| > Systems of mathematics cannot be both complete and
| consistent
|
| No. They can't be at the same times complete, consistent,
| _decidable_ and powerful enough to express arithmetic. You
| can do complete, consistent and decidable though.
| mg wrote:
| 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.
| simonh wrote:
| I think both are descriptive. Mathematics is a descriptive
| language, and thoughts are descriptions of the world,
| ourselves, our intentions, etc or maybe the process of forming
| those descriptions.
| constantcrying wrote:
| >Wouldn't that mean that not only mathematics is pure mental
| activity, but every thought?
|
| That statements seems very obviously true.
| BSEdlMMldESB wrote:
| counterpoint:
|
| remove all blackboards and chalk, and paper and pencils.
|
| can you still do math?
| constantcrying wrote:
| 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.
| BSEdlMMldESB wrote:
| hmm, but imagining rules and symbols IS a pure mental
| activity, isn't it???
| haswell wrote:
| My first foray into math involved counting with my fingers.
| mg wrote:
| "pure mental activity" as opposed to "mental activity related
| to a reality outside of the mental activity".
| constantcrying wrote:
| 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).
| RubyRidgeRandy wrote:
| 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.
| cubefox wrote:
| Presumably the mental activity is itself explained by
| independent external reality. Intuitionists say there is no
| such independent reality for mathematics.
| simonh wrote:
| 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.
| RubyRidgeRandy wrote:
| I think there is a hard limit to what we know and what we can
| assume to know based of this point and in logic by the
| Munchhausen trilemma. It's interesting to think of the source
| of sense data as persistent or consistent when it could just
| be that our sense organs reduce varied data into persistent
| experience.
|
| When we look at a tree, it could very well be that the source
| of the tree is very much like the tree we experience, but it
| could also be wildly different. When we see a tree in a video
| game, we know there is no real source tree just like it, just
| ones and zeroes. I disagree that science fixes this problem.
| Tools are still just measuring the physical world. For
| example, if you used a tool to measure some aspect of the
| tree, you are still measuring the representation of the tree
| in this world. If I use the video game analogy again, my
| point is that you wouldn't be able to see true underlying
| 'source code' of the game tree by looking at it in the game.
| simonh wrote:
| I agree certain knowledge may be unattainable, but I don't
| care. Useful effective knowledge that helps me achieve my
| goals in life will do just fine. As long as my mental model
| of the tree is accurate and useful enough for me to chop it
| down and make a table out of it, I'm good.
|
| I don't expect any description to accord perfectly with the
| reality of the object it refers to. It's just a
| description, which may be more or less accurate or useful.
| Science, and investigation in general, is a way to test and
| improve such descriptions.
| bluetomcat wrote:
| 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.
| circlefavshape wrote:
| I've been thinking about this recently, and realised that your
| framing here casts humans as separate from the rest of reality.
| Your sense organs and your brain are _part_ of things-in-and-
| and-of-themselves.
| bowsamic wrote:
| Yeah, this was the big step from Kant to Hegel, the
| realisation that the object is actually totally inside the
| subject and vice versa. Unfortunately, when the subject and
| object get totally mixed up in that way, the philosophy seems
| to become much more difficult and complicated. Kant's
| Transcendental Idealism is really useful and easy to
| understand, but if you want to go a step further into what
| you describe then it's like moving from Newtonian gravity to
| General Relativity. Literally everything becomes way more
| difficult.
| RubyRidgeRandy wrote:
| I don't think it does. Humans are agents within reality and
| have perceptions of reality. Your brain having a
| representation in this reality that might be different from
| 'true reality' doesn't change the argument at all.
| circlefavshape wrote:
| I don't see how your perceptions can be anything other than
| a direct experience of reality interacting with itself
| _unless_ you imagine that your mind is separated from
| reality somehow
| pphysch wrote:
| We do have a direct experience with reality, but we are
| only capable of processing an approximate & infinitely
| simplified model of it.
|
| When you take a picture of an apple, you have a picture
| of an apple, not the apple itself. Both are real and
| related, but not the same thing.
| thriftwy wrote:
| I do not agree, humans experience some interactions with Turing
| machines "in and of itself", because sensory data becomes
| irrelevant. A bit is always a bit.
|
| This gives an argument about intuitionism: If you say that math
| is a byproduct of our wetware and nothing else, how come we can
| successfully teach it to turing machines, and have that process
| fill us on some holes we had in our understanding of maths, but
| not terribly large holes?
| tgv wrote:
| That doesn't really hold when you start measuring, unless you
| believe that your eyes can change the size of a measure tape
| depending on the subject.
| bowsamic wrote:
| > unless you believe that your eyes can change the size of a
| measure tape depending on the subject
|
| Well, of course, they kind of can. There are drugs that make
| the world look like it's squashed, such as ketamine. The way
| that we perceive reality really is totally dependent on the
| properties of the observer. Of course we all, with our sober
| minds, assert that we are perceiving the ruler the "right
| way", but all this means it that we perceive the ruler in a
| way that most humans agree with. Jumping from that to "this
| is the way that the ruler looks for all possible subjects" is
| a leap of faith.
| ineedasername wrote:
| >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.
| cubefox wrote:
| Godel was a platonist btw.
| mensetmanusman wrote:
| What is the opposite of an objective reality?
| ineedasername wrote:
| 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 wrote:
| "My intuition has 10 significant digits of accuracy when it comes
| to calculating physical constants with quantumchromodynamics"
| Der_Einzige wrote:
| 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.
| ndriscoll wrote:
| 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.
| constantcrying wrote:
| It directly leads to the continuum being inseperable.
|
| If you believe that "now" is real, you believe in LEM.
| Der_Einzige wrote:
| Well, I'm an eternalist who remembers the basics of the
| Relativity of simultaneity, so I reject that "now is real" as
| such.
| constantcrying wrote:
| 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?
| mrwnmonm wrote:
| So why every human would agree that there are no squared circles?
| simonh wrote:
| 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.
| bowsamic wrote:
| 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".
| dwheeler wrote:
| 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
|
| More things have been proven using intuitionistic logic over
| time.
| vlf99 wrote:
| There's room for contemplation on the subjective nature of our
| perception and its influence on the understanding of reality
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