[HN Gopher] About math limitations
___________________________________________________________________
About math limitations
Author : alexmolas
Score : 33 points
Date : 2023-07-20 06:45 UTC (2 days ago)
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| arketyp wrote:
| >There's no pure knowledge, since the substract where the
| knowledge lies is material and thence not pure.
|
| I think you mean substrate.
| hgsgm wrote:
| Thank you for divising a helpful addition.
| arketyp wrote:
| I think you mean devising. It's crass, but I know I
| appreciate such corrections in the end myself.
| hgsgm wrote:
| Look closer.
| arketyp wrote:
| I did, times a few. Went over my head. Cheers.
| noSyncCloud wrote:
| The grammar corrections are multiplying as we speak
| alexmolas wrote:
| Probably yes, English is not my first language. Thanks for the
| help :)
| roenxi wrote:
| It is an interesting argument. However, it is possible to
| simulate continuous phenomenon on discrete systems (eg, we can
| simulate real numbers on computers, and certainly build computer
| systems that reason about real numbers). Similarly the human
| brain can contemplate infinity although it is obvious that we
| can't comprehend it in totality because brains are finite.
|
| So it isn't clear that maths is limited by physics. It might or
| might not be.
| arketyp wrote:
| The simulation part I'd argue isn't as interesting. That other
| part is though, namely that you, or a computer, can reason
| symbolically. We can treat the irrational length of the
| hypotenuse as though it exists in some concrete sense and get
| sensible and even useful results out of it. The limits of this
| approach is indeed a question.
| cvoss wrote:
| There's another limitation not mentioned in the article: Certain
| true theorems may be unprovable in the mathematical system we
| have, not because of a fundamental theoretical limitation of the
| system, but simply because carrying out the proof would require
| more matter/energy than exists, or more time than the universe
| will exist for if the proof is carried out slowly enough to avoid
| forming a black hole because of too much matter in close
| proximity. If we lived in a bigger or more long lived universe,
| we could attain some of these inaccessible proofs.
| evanb wrote:
| Along the same lines, the "Math is limited by math" really
| contains a hidden physical assumption, which is this: the
| (quantum, extended) Church-Turing thesis is an assumption about
| physics. The (classical version of the) argument goes like
| this:
|
| Suppose you want to solve the Halting problem for Turing
| problems that can execute one step per second. If I have a
| machine that takes 1 second for the zeroeth step step, and 0.5
| seconds for the first step, and 0.25 seconds for the second
| step, and so on, I can just run the "normal" machines' entire
| infinite history in 2 seconds. If I want to check if the twin
| prime conjecture is true, I'll just check every integer. If I
| want to check if an arbitrary normal machine halts, I just run
| it on my unusual machine.
|
| As far as I can tell all of the arguments against the
| reasonableness of such an unusual machine rely on some physical
| argument. For example, computational speed requires energy, an
| increasing speed would require an unbounded amount of energy,
| which is both impractical and would also presumably collapse
| into a black hole.
|
| So, I guess I should be careful a bit. The Halting problem is a
| perfectly well defined mathematical question and it has a
| definite answer. But why is it an interesting question at all?
| That's a question about the nature of the world.
| constantcrying wrote:
| >This first idea that math have some intrinsic limitations - eg:
| you can't compute all the Busy Beaver Numbers
|
| But this is not a limitation of _mathematics_ at least not any
| more so than there being no largest prime number is a limitation
| of mathematics. It is a limitation on _abstract systems_ , such
| as Turing machines.
|
| The halting problem _is not_ that you can not deduce whether any
| given Turing machine halts, but that there is no _Turing machine_
| that deducing this.
|
| Godel's (o and o are different letters, by the way) theorems are
| far more wide reaching, as they appy to large classes of
| axiomatic systems, but they are only making claims about the
| systems _in relation to themselves_.
| kbrkbr wrote:
| As the smallest prime number is 2 you probably meant "there
| being no largest prime number".
| constantcrying wrote:
| Indeed...
| tux3 wrote:
| The Busy Beaver numbers express a limitation of classes of
| axiomatic systems, not merely of Turing Machines.
|
| It's not merely that computers aren't guaranteed to be able to
| calculate these. BB(n) for some n <= 745 is independent of the
| ZFC axioms of set theory.
|
| Furthermore, Godel's theorem and Turing's problem are related.
| It is undecidable whether a Turing machine halts, it's not just
| that you can't write a Turing machine that decides whether
| another Turing machine halts.
|
| Without Godel's theorem, you could try to enumerate all
| possible proofs in your complete and consistent axiomatic
| system until you find one that proves whether an arbitrary
| Turing machine of your choosing halts, and you'd have
| conceptually solved the halting problem. (Then, you could
| encode that whole algorithm in a Turing machine (TM), and have
| that new TM be your decider that solves the halting problem, if
| you wanted).
|
| Because of Godel you can't do this, whether on paper, with a
| TM, or even in principle. Your axiomatic system is not
| complete, so there are things it simply cannot prove. Like the
| halting problem, or in the case of ZF the value of BB(745).
| bigbillheck wrote:
| > no smallest prime number
|
| Behold: 2.
| olddustytrail wrote:
| -2 is smaller...
|
| (I'll let someone better educated expand on prime ideals)
| bluefone wrote:
| And I don't think that Scott Aaronson's comment makes
| philosophical sense - Why do we need to extend our senses to know
| the truths beyond our world? An ant doesn't care to know whether
| there is a coast line or mountain ahead at 100km distance. It has
| no business with those distances. There is nothing wrong in
| imagining Jupiter as a humanified god and its moon as his wife.
| This is basically tying back things which are outside of human
| reach to things that make sense in our world. That's how it
| should be, instead of saying that Jupiter is a big ball of gas or
| rock. That doesn't matter to our life and evolution on Earth.
| Speed of light doesn't matter too. 99.99% of the Science and
| progress we made is unnecessary and childish work done by people
| who have luxury of not needing to bother about their food and
| family. And the catastrophic results of such childish
| "inventions" by these rich kids (scientists) was imposed on
| global populations. These so called inventions are Oil, Plastics,
| industrialization, racial mix and related conflicts,
| urbanization, weaponry, weird social trends, climate changes,
| general degradation of human biological abilities, natural
| quality of food, biological diversity and so on. And what did
| Science achieve? We proudly announce that we busted the dangerous
| myths enjoyed by our ancestors!!
|
| Oh, You discovered that Earth is not the center of the universe
| and Sun doesn't go around Earth? Please go a step further to
| realize that there is no absolute frame of reference that shows
| what goes around what. Earth going around the Sun is just as
| equally true as Sun going around Earth. And yes, there is nothing
| wrong in saying Earth is the center of universe as you can claim
| any place in the universe as "the center". And you spent
| countless amount human energy and money to establish the half-
| truths like this, and call it Science and progress?
| silentwanderer wrote:
| It's human nature to make inferences about the world - to
| observe effect and infer cause. No matter whether the cause you
| land upon is Jupiter sending a storm to kill the crops, or
| global weather patterns, you're still using data to make
| guesses about how the world works. "Science" (capital s) isn't
| limited to discovering oil or landing men on the Moon, it's
| _also_ about feeding your family, or guessing what the best
| birthday present for your kid is. It's an inextricable part of
| us.
| noSyncCloud wrote:
| This is, frankly, a surreal and almost unbelievable opinion to
| read on the internet in 2023. On HN of all places: although
| (almost) nothing surprises me any more, this sure did.
| ponderings wrote:
| That is because you are not suppose to talk about it either
| because people get mad or under the assumption they won't be
| able to understand. Without asserting truth, where the topic
| goes wrong often Ill attempt to do a simple example, fail and
| prove my point (haha)
|
| We have a law here that roughly says for all government
| construction projects there must be 1% spend on art. The most
| convenient way to deal with this "problem" it to take the 1%
| and have some artist create one of those unsightly giant
| blobs of metal. The size justifies the price. No one needs
| this, few can appreciate it and with a lot of effort you
| could create something or many things that more people would
| enjoy.
|
| We also have science budgets. The most convenient way to deal
| with that problem is to have as large as possible experiments
| where size also justifies the cost. If one was to make a lot
| of effort one could create many experiments each with a
| weight in practicality. The very idea might be offensive for
| pure science. But say, improve wind, wave, solar, improve
| batteries, etc or even a big project like an attempt to fix
| scientific publishing, the study replication problem, the
| archiving of research data, etc Or you can spend everything
| on one giant particle collider. It would be interesting but
| when we sort the set by usefulness it wont be in the top
| 100k.
|
| One might look at great scientists and see what impact their
| work had on the real world. Faraday pops to mind. You could
| ignore everything Tesla did after his AC motor and it would
| still be stunning.
|
| And then Relativity Theory. A pile of free floating nonsense
| with no connection to anything. I could be accused of not
| understanding the elegance of it but I don't have to as
| nothing practical followed. There was no real world
| revolution of the kind that makes great scientists. Just
| saying it doesn't make it true.
|
| Pure science is to slow to meaningfully fill the void of
| possible questions. We have by comparison only a hand full of
| real problems to solve. We might actually make some progress
| there if we care for it. Those answers will trigger an
| infinite chain of new questions - practical questions.
| bluefone wrote:
| It surprises most of us because we have been brain-washed to
| think that science is good, myths are bad, to think that we
| can ignore our biology, gender, nature etc. How does
| relativity matter to human life? Why is the world filled with
| nukes, plastics, global-warming gases, GM foods, confusion
| about gender and biological roles? It's time to realize that
| the progress we made is actually degenerate and regressive
| from the view point of biological adaptation which is the
| only goal for humans.
| nonameiguess wrote:
| Why are you even on this site? This is a fully general
| argument against intellectual curiosity, and the very
| mission of this site is to submit and discuss "anything
| that gratifies one's intellectual curiosity."
| tux3 wrote:
| I could hardly disagree more, in light of all the goodness,
| the lives saved, the artifice that we have gained through
| hard work and choose to not throw away every day.
|
| The worries that we have today are, to me, preferable and
| much less violent, bloody, or tragic than the worries of
| yesterday.
|
| I think diversity of opinion can contribute useful
| perspective, but I'd encourage not calling most people
| brainwashed who disagree. Even if that were true, it's
| hardly conductive to changing anyone's mind.
|
| Sometimes people have different values and cannot agree.
| Assuming good faith can help cross that bridge.
| ponderings wrote:
| That is some great advice. For me the brainwashing
| diagnosis works like this: If you have control over
| someones mind the first thought to program into it is to
| have them think it isn't so. The false thoughts are
| locked into place with anger towards anyone who dares to
| question it. If there is a man behind a control panel or
| if the brainwashing evolved naturally doesn't really
| matter to the result but it does ask for different less
| crude terminology.
|
| Here are some triggers: People are going to die anyway,
| people want to make an infinite amount of babies at all
| cost, we really want others to have less if as a result
| of it we can have more for ourselves. We love animals,
| then we eat them. We love nature, then we burn and cut it
| all down for personal gain. etc
|
| I really believe eugenics is the dirtiest word in the
| dictionary. If you try talk about it Ill probably get
| angry much sooner than reasonable.
| strogonoff wrote:
| "Mathematics is limited by the physics because physics constrains
| our brains" only works if you hold the philosophical view of
| physicalism. People who hold that view believe that the current
| physical model of reality is all there is and the utmost absolute
| truth.
|
| Materialistic monism is a popular but not particularly robust
| view in my opinion, because those models can never be complete
| and always evolve in directions guided by the applications and
| predictions our minds seek. (Other views are available.)
|
| I would encourage more people to apply the same principle at the
| higher level, changing "our brains are limited by physics, hence
| our mathematics is limited by physics" to "our model of reality,
| _including physics,_ is limited due to our minds operating within
| that reality". In other words, the model that makes you think
| your consciousness comes from neurones and is thus limited by
| physics is itself a product of your consciousness, and
| conclusions you make based on that model (such as "mathematics is
| limited by physics") should be taken with a grain of salt (not
| just as possibly wrong but, worse, not-even-wrong) because you
| can't construct a complete model of a system from within said
| system.
|
| But the kicker is that this logic itself is rooted in
| mathematical reasoning (see Godel's incompleteness theorem), so
| take it for what you will.
| ezion wrote:
| The whole idea of materialism is simply a bad and inconsistent
| worldview.
|
| If you start under the assumption that materialism is true you
| have to also eat the idea that free will doesn't exist, and
| that you can't trust any of your thoughts since they are: 1.
| Out of your control, since it's all physics. 2. Created by a
| brain that evolved to survive, not to create logic.
|
| Thus the idea of materialism itself would be just an illusion
| created by your brain's need to survive.
|
| This implies it is a self-defeating worldview from the get-go -
| because you've reached a contradiction: the brain generates the
| thoughts in a deterministic fashion and you cannot trust your
| brain, so you cannot trust the thoughts you have about the
| material reality including materialism.
|
| And the armchair philosophers misunderstand Godel literally all
| the time, including this time. All it says is that there are
| statements that are true that cannot be proven within the
| constraints of a finite set of axioms. It's not a limitation of
| math, it's a limitation of what can be proven true within the
| confines of a set of axioms.
|
| Meaning there could be another set of axioms under which the
| limitation to prove that such a statement is true disappears.
| But it doesn't make math any less true.
|
| This article was written by someone with very poor
| philosophical training.
| smcameron wrote:
| > The whole idea of materialism is simply a bad and
| inconsistent worldview.
|
| Isn't your argument an example of Appeal to Consequences?[1]
| It appears that you don't accept materialism not because you
| can show that it's incorrect, but because you don't like the
| implications?
|
| [1] https://en.wikipedia.org/wiki/Appeal_to_consequences
| ezion wrote:
| Uhh... I proved that it's false because it leads to
| contradictions. That is literally a reductio ad absurdum -
| a proper proof. It's a transcendental argument against
| materialism.
|
| In addition to it, I personally reject materialism because
| it fails to describe the universe properly, in particular
| the Big Bang.
|
| The Big Bang is the beginning of all material universe -
| creatio ex nihilo. However, if there was a mathematical
| structure that sets the framework for the physical laws,
| there's a reality that transcends the material universe and
| exists prior to it. It's the only logically consistent
| worldview in my opinion.
|
| This has even further implications but, you know, you can
| figure it out yourself.
| strogonoff wrote:
| From my vantage point, if we take "as little magic as
| possible" to be a quality of an elegant model, then naive
| materialistic monism, which requires granting magical
| existence to _really many_ things (you know, matter with
| all those atoms, neurones, etc.) and arbitrary rules for
| their behavior, while often denying existence to
| consciousness ( _the only thing we can directly access is
| an illusion_ type of argument), certainly doesn't strike
| me as particularly elegant.
|
| Why we assume that an elegant model has a higher chance
| of being correct, or is otherwise somehow better, in the
| first place--it seems to be pretty much implied that
| beauty and elegance are desirable traits of a model--is a
| good and on topic question, but perhaps too much for this
| particular thread.
| chongli wrote:
| _creatio ex nihilo_
|
| "At first there was nothing, then out of nothing the Big
| Bang happened and the universe began" is a popular
| misconception of the Big Bang theory. It's completely
| wrong.
|
| The Big Bang theory does not attempt to explain what
| things were like "before" the Big Bang. It's not a theory
| of creation at all! It only offers a model starting with
| the inflationary epoch, when the universe was already
| expanding rapidly.
|
| Prior to that, our existing theories break down and we
| need a different physical theory (commonly called the
| Grand Unified Theory or GUT). We don't have it so we can
| go no further back.
|
| The point is: back in the inflationary epoch the universe
| was all there, it was just much smaller. The point of the
| theory is to try to explain the extreme uniformity
| (isotropy) in the distribution of galaxies, gas, dust,
| and light we observe in the universe today (among other
| things, such as dark matter). It's a much better model in
| terms of its simplicity and predictive power than any
| steady-state model of the universe that has been
| proposed.
|
| But it's got nothing to do with creation!
| ezion wrote:
| I don't claim to know anything before it either.
|
| All I'm claiming is that before whatever happened, there
| must have been a mathematical structure that allowed it
| to happen following certain physical laws. That's just a
| fact. So these mathematical laws exist _before_ anything.
| This implies a realm that 's not physical. Anything that
| is contingent such as all material things (why is there
| anything at all?) must have a prior cause - and that's an
| assumption of science.
|
| All theories presuppose these mathematical structures.
|
| That is, even if the current big bang is just one in a
| series of several, or however else you try to explain
| existence, you have to presuppose the mathematical
| structure that forces it to follow certain physical laws.
|
| I also don't think you fully understand inflation and why
| it's been proposed but I'll ignore it since it's not
| useful for the conversation. Also read up on the Borde-
| Guth-Vilenkin theorem as it basically proves that a
| beginning is necessary.
|
| In addition, note that right now it seems almost certain
| that the universe had a beginning and any other
| explanation is not evidence based.
| chongli wrote:
| _these mathematical laws exist before anything_
|
| You might want to study a bit of philosophy of
| mathematics, particularly the bit on the realism vs anti-
| realism debate. A great deal of people fall into the
| anti-realism camp, and they would disagree rather
| vociferously with your statement about "mathematical
| laws."
| ezion wrote:
| Can you explain why it is not the case everything happens
| at random? Why these particles seem to follow physical
| laws?
|
| Realism/Anti-Realism has no bearing to the matter. I
| personally think mathematics is a priori and non-causal,
| and not necessarily correspondent to the real world. This
| would imply I'm technically an anti-realist. Certain
| mathematics correlates to the real world, others do not
| and are simply (interesting) mathematical structures.
|
| Neither viewpoints can explain why certain mathematical
| statements actually correlate to reality. Mathematical
| entities are not causal - they do not have any causal
| connection to the real world since they are (obviously)
| abstract.
|
| Maybe a good place to start thinking about this problem
| would be to read Wigner's work "The Unreasonable
| Effectiveness of Mathematics in the Natural Sciences".
| chongli wrote:
| _Can you explain why it is not the case everything
| happens at random? Why these particles seem to follow
| physical laws?_
|
| What does random mean? Are you asking why the universe
| appears to be consistent and intelligible at all? Where
| would we be if it weren't?
|
| The weak anthropic principle covers this nicely. Perhaps
| it is an unsatisfying answer, but scientists and
| mathematicians aren't usually in the business of
| answering ontological questions.
|
| _Maybe a good place to start thinking about this problem
| would be to read Wigner 's work "The Unreasonable
| Effectiveness of Mathematics in the Natural Sciences"._
|
| I have studied all of this. Wigner's account is very
| important for the debates it inspired but it's far from
| the last word on the topic. There have been numerous
| replies addressing all of Wigner's claims. Try reading
| Hamming's replies to start.
| ezion wrote:
| The anthropic principle has nothing to do with why
| complex numbers are necessary for doing quantum
| mechanics. Complex numbers are purely abstract objects
| and yet are necessary for giving a proper account to the
| real world.
|
| "From all of this I am forced to conclude both that
| mathematics is unreasonably effective and that all of the
| explanations I have given when added together simply are
| not enough to explain what I set out to account for." -
| Hamming
| chongli wrote:
| Most of mathematics has no use whatsoever in physics.
| Physicists select the mathematical tools that best enable
| them to build their models. In other cases, the necessary
| tools do not exist and they are built for purpose.
|
| In many cases, physicists are not so successful. Look at
| the current debates in particle physics and the grand
| unified theory. Progress has largely stalled.
|
| In other cases, models are revised continually as new
| information comes to light. For example, the age of the
| universe as we know it may double [1] from 13.79 billion
| years to 26.7 billion years.
|
| You might call this "unreasonable effectiveness," I call
| it a process of messy refinement and rethinking over
| thousands of years, much of it later shown to be invalid,
| despite mathematical correctness.
|
| [1] https://phys.org/news/2023-07-age-universe-billion-
| years-pre...
| Koshkin wrote:
| > _That 's just a fact._
|
| No,
|
| > _That 's, like, just your opinion man ;)_
| ezion wrote:
| If the mathematics is not a priori, one must explain why
| it is the case particles follow particular mathematical
| laws instead of moving about randomly without any
| particular logic to them. How do these elementary
| particles know what to do and have always known?
| chongli wrote:
| _particles follow particular mathematical laws_
|
| We don't know that they do. We _assume_ they do, but our
| laws are not their laws.
|
| You're like the person who sees a birdwatcher recording
| their observations and asking "why do birds exist?"
| You're asking the wrong question to the wrong person and
| demanding satisfaction.
| nickelbob wrote:
| Where's the contradiction? Are you saying that "not being
| able to trust your thoughts" implies that materialism is
| false?
|
| > This article was written by someone with very poor
| philosophical training.
|
| Strange to throw shade on the author when your position is
| that an idea that's been taken seriously by philosophers for
| 2000+ years is obviously contradictory. Maybe some humility
| is in order?
| ezion wrote:
| Materialism is a self-defeating worldview because its
| implications are that you cannot trust it.
|
| The whole purpose of a worldview is to be able to trust it
| as one trust one's own eyes - because a good mental model
| of reality affects how one acts and moves around the world.
|
| If the worldview implies a contradiction it is not a
| worthwhile worldview to hold, because you know it to be
| logically false like any other set of statements that lead
| to a contradiction.
| rssoconnor wrote:
| > All [Goedel] says is that there are statements that are
| true that cannot be proven within the constraints of a finite
| set of axioms. It's not a limitation of math, it's a
| limitation of what can be proven true within the confines of
| a set of axioms.
|
| Minor correction. Goedel's theorem applies to decidable
| (a.k.a. computable) sets of axioms. e.g. Peano's axiom, when
| written in first order logic, contain an infinite set of
| induction axioms that follow a specific and recognizable
| (i.e. decidable) pattern.
|
| In fact, Goedel's theorem applies even beyond that including
| any set of axioms whose syntax can be specified by some
| (first order) mathematical predicate, even if that predicate
| is not computable in any way.
|
| > Meaning there could be another set of axioms under which
| the limitation to prove that such a statement is true
| disappears. But it doesn't make math any less true.
|
| Indeed, when studying logic it is common to consider the set
| of all true first-order (arithmetic) statements, which
| necessarily forms a complete and consistent set of axioms. Of
| course, given Godel's incompleteness theorem it must be the
| case that such a set of axioms isn't computable. Not only
| that, but it cannot be even defined in a first-order
| (arithmetic) way, which leads the the corollary of Tarski's
| undefinability theorem[0].
|
| [0] https://en.wikipedia.org/wiki/Tarski%27s_undefinability_t
| heo...
|
| P.S. There is no free will and one's thoughts are indeed out
| of one's control. I'm not certain that implies one should
| never trust one's thoughts. Though it does make sense to take
| them with a grain of salt. See for example, "The past is not
| true" (https://news.ycombinator.com/item?id=36798854).
|
| P.P.S I'm not saying your argument is invalid. It can both be
| the case that materialism is contradictory and there is no
| free will.
| strogonoff wrote:
| That your thoughts are not fully under what you believe
| would be your direct control, and supposed absence of free
| will, do not strike me as particularly following from one
| another (at least as long as you allow that as a conscious
| agent you are more than just what you identify as your
| thoughts).
| rssoconnor wrote:
| For me, the lack of free will follows from my study of
| physics, which seems to allow no room for free will.
|
| That my thoughts are not under my control is the counter
| argument to the claim "But then how is it that you feel
| like you do have free will?" The answer to this is that,
| upon more careful introspection, it seems I don't even
| really feel like I have free will either.
| ezion wrote:
| I disagree with your claim there's no free will and one's
| thoughts are out of one's control. I do not believe all of
| our actions and behaviors are determined. That's, like,
| just your opinion man ;)
|
| Where determinism truly fails is at the transcendental
| argument level. The act of affirming determinism is self-
| defeating like affirming materialism: to rationally accept
| something as true you must have the freedom to weigh the
| evidence, but if determinism is true there's no true
| deliberation as everything is the result of prior causes,
| including its acceptance.
|
| So you cannot trust your own belief system and thus your
| worldview is inconsistent at its core.
|
| It also fails to account for our mental lives, given the
| experience we all have.
| rssoconnor wrote:
| While it is true that one cannot trust one's belief
| system, I don't think that makes it inconsistent. I
| cannot know with complete certainty that the world is
| without free will, but that doesn't prevent me from
| concluding and thus believing the in the lack of free
| will, in as much as I believe anything else. It is, like,
| my opinion man, although I claim it is a reasonably
| informed opinion.
|
| Like, maybe the world is inhabited by leprechaun's
| holding up cardboard facades of the city when I'm walking
| through it always hiding themselves. Or maybe my brain is
| in vat, hooked up to a computer that is making me believe
| that I'm experiencing typing out this comment. Or maybe I
| have no brain at all and I'm just an improbably quantum
| fluctuation in unfathomably large dead universe that just
| for a fraction of a second creates an experience of
| typing on a keyboard and the sensation of having a
| history that never was, a Boltzmann brain.
|
| I accept that all the above are possible, but I don't
| "believe in them". I do believe in determinism (plus
| randomness) with about the same conviction that I believe
| the keyboard that I am tapping on exists.
|
| > It also fails to account for our mental lives, given
| the experience we all have.
|
| So this is what got me quite recently.
|
| I remember a while back Henk Barendregt remarked
| something to the effect that everything that happens
| either follows deterministically from past event or
| happens randomly, and in neither case is there any free
| will.
|
| I remember accepting that from what I know of physics
| this does seem to be the case. I'm not aware of any
| violations of the Borne rule that would allow for any
| event to be neither random nor deterministic. But, that
| the time, I remember thinking that it did seem strange
| that I feel like I have free will.
|
| But recently, I've been listening to Sam Harris's Waking
| Up app, and, in addition to meditation practices, he has
| a section where he argues, convincingly in my opinion,
| that if you pay attention, there isn't actually any
| feeling of free will either.
|
| For instance, if I am sitting and meditating and I am
| trying to pay attention to my breathing, or trying to pay
| attention the the bird chirping outside my window, then
| often, usually only after a few seconds, I suddenly find
| myself thinking about what I'm going to make for dinner,
| or remembering that time I was talking with my aunt about
| her birdwatching hobby, or any number of a million
| possible intrusive thoughts.
|
| I certainly do not freely will these thoughts into
| existence. My will is supposedly to be paying attention
| to what I am hearing, or the experience of breathing. Yet
| inevitability I eventually end up distracted by whatever
| pops into my head at any given moment.
|
| And after spending some time paying attention to my
| experience, I cannot help but see that Harris is right.
| He suggests taking the example of picking a movie, any
| movie. Just pick one and pay attention to the experience
| of freely choosing a movie title. For me, just now, I
| picked "Ghostbusters". But it was simply the first title
| that popped into my head.
|
| Is this free will? The title popped into my head the same
| way that thoughts intrude into my meditation, just
| appearing in consciousness with no source in sight. No
| particular explanation as to why one title appears and
| not another.
|
| Try again and think up three songs and choose one of
| them. Take as much time as you want. Make this choice as
| freely as you like. But also watch as the titles just
| appear in your consciousness from nowhere to be seen.
| Watch as you /deliberate/ about which one to pick.
| Reasons, like all thoughts, just start popping into one's
| head from nowhere to be seen. "I really enjoy X song" or
| "X song is quite popular" or "I haven't listened to X in
| a while, maybe I'll pick it then go listen to it". All
| these reasons for freely picking a title are just popping
| into consciousness in just the same way that the titles
| appeared in the first place. Maybe the thought pops into
| your head that you should just roll a die to choose one
| of the three.
|
| Upon examination, my experience of deliberation is just
| the same as my experiences of other kinds of thoughts. I
| simply experience reasons popping into my head relating
| to the choice at hand, but I have no real control over
| which reasons will appear. They all just start to show
| up. Maybe eventually a thought appears, "Ah, that is a
| good reason to do X". Did I just freely come to a choice?
| Am I in control of that thought? That last thought just
| popped into my head in the same way that every other
| thought did, and I don't recall choosing to come to a
| final conclusion.
|
| Anyhow upon this recent introspection on my own mental
| life, it now feels completely compatible with the lack of
| free will. While it doesn't conclude that I for sure have
| no free will, it doesn't actually look like I have it.
| The thoughts that enter my consciousness don't directly
| appear to have any origin. They just appear in
| consciousness in a way not so different from the
| appearance of the sound of a bird chirping outside my
| window does. I don't have any direct control over what
| thoughts will appear in my consciousness. And what type
| of thoughts that do appear in my head seem to be at least
| somewhat influenced from past experiences. i.e. they do
| kinda look like something that is offered up based on
| some combination of determinism and randomness.
| ezion wrote:
| If you cannot trust your belief system then maybe you
| should change it, instead of calling yourself an NPC.
|
| It's clear circumstance and your memory will impact your
| present - that doesn't mean you did not choose to reply
| to my comment with your thoughts.
|
| The whole point of a belief system is for it to correlate
| with reality and truth, and if there's an obvious
| contradiction in it, it clearly doesn't.
| rssoconnor wrote:
| I'd certainly prefer it to be the case that I have free
| will. However me having free will is neither not
| compatible with my current understanding of physics, nor
| is free will something that I've noticed having when I
| carefully consider my experiences.
|
| Free will or not, I don't see how I can change my belief
| system in such a way to exclude the possibility that I'm
| a brain in a vat, so in that sense I don't see how I can
| ever 100% believe my experiences match reality short of
| attempting to tautologically define my experiences as
| what reality is.
|
| Since you claim have free will, I'd be interested in how
| you experience the sensation of freely choosing a movie
| or song title and where exactly your free will comes into
| play within those sensations, and of how your sensations
| of the task differ from mine.
| plorkyeran wrote:
| Declaring something to be a contradiction because you
| don't like the conclusion they came to is not the most
| compelling argument.
| ezion wrote:
| If your set of beliefs implies that your set of beliefs
| are not true, that's a contradiction.
|
| In this case, thinking all of reality is material implies
| that you cannot believe your thoughts, including the
| thought that all of reality is material. But that was
| your original belief, meaning through your own worldview
| you've reached a contradiction.
| Zababa wrote:
| There is a meme that carries more or less the same idea:
| https://i.ibb.co/3CNSCq2/103.jpg
| strogonoff wrote:
| Love it. Some might take it as undermining knowledge or [some
| branches of] philosophy, but to me rather the opposite!
| oszai wrote:
| A Big Curveball Here Mathematics has been Likened to a Language
| and Through the Likes of Galileo the Language of the Universe.
|
| But Language Came First, in Order to have a Permutation of the
| Languages an Individual Could Speak.
|
| Would it not Follow that On the Hunt one Could Described a
| Series of Logical Steps that Much up to Logic, without Being
| Called Logic. To Predictably Bring down the Bison?
|
| Or Know Logically that they Could Frighten the Bison to Run off
| a Cliff and save themselves the Trouble of the Hunt in the
| First Place.
|
| That it would be Counter Intuitive to Describe the Hunt
| Mathematically, but the Hunt also had to Be Something that
| could be Repeated. So What other form of Language Prose exists
| regardless of Written Language that Allows for Something to Be
| Repeated?
|
| A Story, but how is the Story Composed to Pass down the Lesson
| to Frighten the Bison to Save the Trouble of the Hunt? But So
| too what Mechanism in those People would Recognize that Running
| the Entire Heard off the Cliff would Lead to Waste and
| Depreciate the Yearly Yield of this Strange Tradition? Logic
| without a Name.
|
| So Mathematics is a Late Understanding of Language that
| Provides a Condensed Symbol System to Perform Operations based
| on Some Mathematical Logic. Proven by Some Test we Call
| Theorems. To Inform that these Symbols are Not just Random
| Symbols, but Correspond to Reality via Some linking from the
| Individual doing the Maths to the Process where the Maths is
| used to Inform what Was Once the Hunt, but is now used in the
| Creation of a Product instead. Just a Slice on a Document
| Filled with Other Language to Inform the Creation of the
| Product where Math is Just an Additional Quality that informs
| that Processes Outcome.
|
| Interesting to Note Math was a Late Addition to Computer
| Science.
| webnrrd2k wrote:
| I voted this post down because it reads like an incoherent
| LLM creation.
| Chinjut wrote:
| What is this capitalization?
| zogomoox wrote:
| If you consider that mathematical ideas are discovered rather
| than invented, then the whole point becomes moot. Sure, we may
| not be smart enough to discover all of it, but we can be pretty
| certain there is still mathematics beyond our ability to
| understand it. As an analogy, we'll never be able to look at or
| visit beyond our cosmic horizon, but it's certain there are stars
| and galaxies out there.
| DecayingOrganic wrote:
| Math as we know it, due to Godel's Incompleteness Theorems, is
| not fully consistent - we can't even prove everything that is
| true with our current mathematical framework. This means that
| our understanding of math is indeed limited, not just by our
| intellectual capabilities, but by the very structure of the
| math we currently use also.
|
| This makes me wonder, will we be able to develop new
| mathematical frameworks that bypass these issues? And if so,
| what will they look like?
| zmgsabst wrote:
| The insight is that mathematics is not complete -- which is
| the property that a system can prove every true theorem.
|
| Consistency is there as a technical detail: an inconsistent
| system can prove every true theorem, by virtue of being able
| to prove every theorem.
| kbrkbr wrote:
| > is not fully consistent
|
| Not being able to prove the consistency of a system within
| the system does not entail that the system is inconsistent
| (or not fully consistent).
| pixl97 wrote:
| I think it means we could only ever prove it is
| inconsistent, and never prove that it is consistent.
| tux3 wrote:
| If we can only prove that an inconsistent system is
| inconsistent, and we can never prove our (presumably)
| consistent system is consistent, then it is incomplete.
|
| We can't say it's inconsistent just because we can't
| prove otherwise. We can say it may be incomplete (or
| inconsistent, and we haven't noticed yet)
| arketyp wrote:
| But it does become relevant if you consider the mathematics we
| can understand expanded to some ontological boundedness, for
| instance perhaps something that relates to the validity of the
| Church-Turing thesis. Then talking about mathematics beyond
| such a scope becomes moot too. With this I'm not saying we
| can't reason about Godel's results by the way.
| jostylr wrote:
| I think having a language that helps understand those limitations
| is a useful achievement. Much of mathematics does have that. A
| notable exception is the definition of real numbers. They are
| usually presented as a string of infinite decimals, or a
| converging sequence, or a set of numbers less than something. All
| of those notions obscure the basic limitation of knowing the real
| number and give a veneer of similarity to rational number.
| Rational numbers are numbers that we can have in our hand while
| irrational numbers are ones which we can never have. It is
| important to have a setup that respects that difference.
|
| This is what motivated me to come up with a new definition of
| real numbers, namely, they are objects (I call them oracles) that
| answer Yes or No when asked if the number ought to be between two
| given rational numbers. Abstracting out what properties such an
| object should have, one can come up with a space of these
| oracles, define an arithmetic, and prove that they satisfy the
| axioms of real numbers.
|
| For details: https://github.com/jostylr/Reals-as-Oracles/
|
| In many ways, this is giving a definitional support to the use of
| interval analysis which is, of course, a very practical concern.
| It also brings our some cool stuff about mediants and continued
| fractions (nothing new about that, but nicely motivated).
|
| It also fits in with the adjacent post about busy beaver numbers
| and its conclusion about knowing a number is in an interval.
| gunnihinn wrote:
| > they are objects (I call them oracles) that answer Yes or No
| when asked if the number ought to be between two given rational
| numbers.
|
| So a real number is a function from pairs of rationals to a
| two-element set (plus some sanity conditions)? Why is that
| better than the other constructions?
| jostylr wrote:
| Yes. The basic reason is lazy evaluation and answering the
| primary question most people care about when using an actual
| real number in a computation.
|
| The lazy evaluation is that defining the real number is about
| having a way of answering the question when presented with
| the rational interval, but one does not actually need to have
| the answers until asked.
|
| From this perspective, a program that computes Yes or No when
| given a rational interval via the rule "a<b Yes iff a^2< 2 <
| b^2" can be said to be the square root of 2. For the usual
| presentations of the other definitions, they cannot be
| embodied in a computer. One cannot literally have all the
| infinite elements of a Cauchy sequence, or the infinite
| sequence of digits in a decimal representation, or the
| uncountably many elements of a Dedekind cut, represented in a
| computer memory. One can have a function in memory.
|
| The other definitions can also be presented, in various ways,
| as functions as well, but I think, fundamentally, what we
| want in practice from a real number is an interval of small
| enough size to be of use in whatever we are doing with the
| real number. That is what the oracles facilitate.
| ginnungagap wrote:
| One can have a function in memory when the real number is
| nice in some sense, your example is algebraic.
|
| But what if I want to represent an uncomputable number?
|
| Or regardless of that, under any reasonable encoding of
| programs that can be held in memory by a computer, there
| are only countably many programs.
| jostylr wrote:
| We also cannot represent all natural numbers or rational
| numbers in a computer. But the ones we care about, we
| generally can. I guess one question is whether there are
| uncomputable numbers that we need to compute for some
| purpose other than just computing it as a challenge? And
| if there are such things, how is it usually done? The
| theoretical definition of an oracle is not problematized
| by being uncomputable by Turing machines, but it is in
| uncomfortable tension with the driving purpose of the
| definition. I think that is a bonus. I think we should
| pause to consider the relevance of numbers that are
| uncomputable.
|
| There are a number of real numbers that one can define
| which can depend on whether we can prove something or
| not. It may turn out that we can never prove it and the
| number is never resolved.
|
| My attitude, which may not be satisfactory, is that we do
| what we can and we should have a language/framework for
| facilitating that. I think the oracle approach highlights
| what we know and marks what we can't compute clearly. I
| call it the resolution of the oracle. I don't want known
| precision to be lacking just because of a poor definition
| of what a real number is.
|
| As for the example being algebraic, it is a particularly
| nice example and it is the same example in every example
| of a Dedekind cut. Another example of such a rule, one
| which is not algebraic, would be whether pi is in an
| interval or not. Given a<b, one rule could be that it is
| No if b < 3 or a > 4 or sin(a) _sin(b) > 0.
| Alternatively, it would say Yes if a >= 3 and b<= 4 and
| sin(a)_sin(b) <= 0. To compute this, one needs to compute
| sine of a and b sufficiently precisely to determine their
| sign.
|
| The flavor I am trying to convey is having a definition
| convey a useful goal. The interval approach says that we
| are trying to generate precision about our inaccuracy. I
| think this is something which would greatly benefit those
| learning about real numbers. Most of the time, the error
| is presented as secondary and an annoyance, the concerns
| of the error propagation in further computations is
| pushed to the side, and it is all relegated to experts or
| computers. The expansionary nature of arithmetic in error
| propagation is pushed to a usually unsatisfactory
| discussion about significant digits.
|
| My goal here is to change the mental framework so that
| these concerns come to the forefront. Ideally, they also
| come with useful tools to handle the uncertainties such
| as ways to compute how narrow the input intervals need to
| be when doing a computation. And maybe, just maybe,
| students could become more comfortable with fractions.
| ginnungagap wrote:
| It's not clear to me what this approach offers over Dedekind
| cuts, you are specifying a real number by saying in which
| rational intervals it lies, a Dedekind cut specifies a real
| number by saying which rationals are below it and which are
| above, and translating back and forth between the two is
| immediate
| jostylr wrote:
| In part, it is about framing it. The oracle approach is not
| the set of all rational intervals containing a real number,
| but rather it a rule that acts on a given rational number.
| You cannot literally hand over a set of all rational
| intervals that contain the square root of 2. But you can
| specify the rule that tells you whether the rational interval
| contains it, namely, if a<b is the rational interval to test,
| then the rule sees if a^2<2<b^2.
|
| Dedekind cuts can similarly by recast as a function that
| gives -1,0, or 1 if a number is below, the same, or above the
| real number. This would be a great improvement over the usual
| presentation.
|
| But beyond that difference, consider the solution to f(x) = 0
| in terms of the intermediate value theorem for an increasing
| function. The Dedekind cut solution is A = {x | f(x) < 0}.
| That is the answer. From the perspective of Dedekind cuts,
| there is nothing further to compute and it has provided
| nothing to compute with.
|
| The oracle answer would be the rule that says yes to a<b if
| f(a)*f(b) <=0. This gives a direct mechanism for testing
| intervals and the idea of narrowing intervals by picking a
| point inside a Yes interval is very natural from this
| perspective. I do not see that suggestion popping out of the
| Dedekind cut approach.
|
| The rational interval approach specifically is targeting
| narrowing solutions to the answer with error bounds built
| into it. If I tell you that 3 is in the set A, you have no
| intrinsic sense of how good of an approximation that is. But
| if I tell you that the target is in the interval 3 to 3.1,
| then you have a sense of the precision. Oracles promote
| having accuracy as foundational.
| [deleted]
| monkeyjoe wrote:
| > Rational numbers are numbers that we can have in our hand
| while irrational numbers are ones which we can never have. It
| is important to have a setup that respects that difference.
|
| Do you mean physically? Basic shapes like circles, squares and
| triangles allow us to hold irrational numbers in our hands as
| distances. Children playing with blocks can sense that root 2
| does not conform nicely with other (rational) distances.
| jostylr wrote:
| I did not mean physically. I meant, having it explicitly
| written out in a way that rational numbers can be. If I write
| 3/4 + 1/2, I can compute out 5/4 and that is infinitely
| accurate.
|
| If I want to compute pi + e, an infinitely accurate version
| is pi + e. That's about it. So what we are actually looking
| for in this computation is an estimation algorithm, one which
| can be made as accurate as we wish, but finitely so. The
| natural way to express this is with rational intervals as
| rationals are precise and intervals give a containment of the
| numbers.
|
| For arithmetic, we can have a mechanism for figuring out how
| precise the input approximations need to be in order to get a
| given precision for the final computation. The perspective
| presented here naturally leads to that as a matter of
| defining and establishing the arithmetic of oracles.
|
| As for playing around with physical representations of
| irrational numbers, keep in mind that there is no way to
| prove that, say, something that looks like a unit square to
| us is really a perfect square down to infinite precision. And
| without that, we can easily have that the unit square is only
| very close to such a figure, but is actually a rational
| rectangle with a rational diagonal that very closely
| approximates the square root of 2.
| ginnungagap wrote:
| This is irrelevant to your point, just change the numbers
| used as example, but we do not know whether pi+e is
| irrational! (Even though nobody believes it to be rational)
| auggierose wrote:
| > It is important to have a setup that respects that
| difference.
|
| No, it is not. Or rather, which differences matter depends on
| your application. A circle of radius 1 is a very natural thing
| to me. I don't think 3/4 is more natural.
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