13 ms·

God created the real numbers

136 points by Bogdanp 1y ago

bell-cot 1y ago
13 points and 18 comments here yesterday: https://news.ycombinator.com/item?id=45053007 https://news.ycombinator.com/item?id=45053007
andrewla 1y ago
I'm an enthusiastic Cantor skeptic, I lean very heavily constructivist to the point of almost being a finitist, but nonetheless I think the thesis of this article is basically correct. Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. At a micro level this is less true -- elementary particles specifically are a (mostly) discrete phenomenon, but representing the state even of a very simple system involves continuous quantities. But the Cantor vision of the real numbers is just wrong and completely unphysical. The idea of arbitrary precision is intrinsically broken in physical reality. Instead I am off the opinion that computation is the relevant process in the physical universe, so approximations to continuous quantities are where the "Eternal Nature" line lies, and the abstraction of the continuum is just that -- an abstraction of the idea of having perfect knowledge of the state of anything in the universe.
- empath75 1y ago
  > But the Cantor vision of the real numbers is just wrong and completely unphysical. They're unphysical, and yet the very physical human mind can work with them just fine. They're a perfectly logical construction from perfectly reasonable axioms. There are lots of objects in math which aren't physically realizable. Plato would have said that those sorts of objects are more real than anything which actually exists in "reality".
  - andrewla 1y ago
    There are two things being talked about here, and worth teasing them out. On the one hand, this article is talking about the hierarchy of "physicality" of various mathematical concepts, and they put Cantor's real numbers at the floor. I disagree with that specifically; two quantities are interestingly "unequal" only at the precision where an underlying process can distinguish them. Turing tells us that any underlying process must represent a computation, and that the power of computation is a law of the underlying reality of the universe (this is my view of the Universal Church-Turing Thesis, not necessarily the generally accepted variant). The other question is whether Cantor's conception of infinity is a useful one in mathematics. Here I think the answer is no. It leads to rabbit holes that are just uninteresting; trying to distinguish inifinities (continuum hypothesis) and leading us to counterintuitive and useless results. Fun to play with, like writing programs that can invoke a HaltingFunction oracle, but does not tell us anything that we can map back to reality. For example, the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful.
    - EthanHeilman 1y ago
      > On the one hand, this article is talking about the hierarchy of "physicality" of various mathematical concepts, and they put Cantor's real numbers at the floor. I disagree with that specifically I didn't mean to suggest that the reals are the floor of reality, rather that they are more floorlike than the integers. > The other question is whether Cantor's conception of infinity is a useful one in mathematics. Here I think the answer is no. Tools are created by transforming nature into something useful to humans. Is Cantor's conception of infinity more natural? I can't really say, but the uselessness and confusion seems more like nature than technology.
    - nh23423fefe 1y ago
      > the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful it leads to the idea that measuring 2 sets via a bijection is a better idea than measuring via containment
      - andrewla 1y ago
        That a bijection exists is incredibly useful. But the idea of "measuring" infinite sets in the cardinality sense is not very interesting or useful.
        oskaralund 1y ago
        Saying that two sets have the same cardinality is equivalent to saying there is a bijection between them. I don't understand how the latter can be useful but not the former?
        sfpotter 1y ago
        It isn't very interesting or useful... to you.
    - SabrinaJewson 1y ago
      > For example, the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful. I am not sure what you are arguing here. We’ve been teaching this to all undergraduate mathematicians for the last century; are you trying to make the point that this part of the curriculum is unnecessary, or that mathematics has not contributed to the wellbeing of society in the last hundred years? Both of these seem like rather difficult positions to defend.
  - shkkmo 1y ago
    > They're unphysical, and yet the very physical human mind can work with them just fine. Can it? We can only work with things we can name and the real numbers we can name are an infinitesimal fraction of the real numbers. (The nameable reals and sets of reals have the same cardinality as integers while the rest are a higher cardinality.)
    - SabrinaJewson 1y ago
      We can work with unnameable things very easily. Take, for instance, every known theorem that quantifies over all real numbers. If you try to argue that proving theorems about these real numbers does not constitute “working with” them, it seems you have chosen a rather deficient definition of “working with” that does not match with how that phrase is used in the real world.
      - shkkmo 1y ago
        I would argue that all of those theorems work with nameable sets of real numbers but not with any unnamable real numbers themselves.
  - Eddy_Viscosity2 1y ago
    The human mind can't work with a real number any more than it can infinity. We box them into concepts and then work with those. An actual raw real number is unfathomable.
    - SabrinaJewson 1y ago
      I don’t know about you, I can work with it just fine. I know its properties. I can manipulate it. I can prove theorems about it. What more is there? In fact, if you are to argue that we cannot know a “raw” real number, I would point out that we can’t know a natural number either! Take 2: you can picture two apples, you can imagine second place, you can visualize its decimal representation in Arabic numerals, you can tell me all its arithmetical properties, you can write down its construction as a set in ZFC set theory… but can you really know the number – not a representation of the number, not its properties, but the number itself? Of course not: mathematical objects are their properties and nothing more. It doesn’t even make sense to consider the idea of a “raw” object.
      - Eddy_Viscosity2 1y ago
        You can hold a two in your head, but you can't hold a number with infinitely many decimal places. Any manipulations you do with the real 2 are done conceptually whereas with the natural 2, its done concretely.
        drdeca 1y ago
        The decimal places are just a way of representing it.
        Eddy_Viscosity2 1y ago
        The infinite number of decimal places is the definitional feature of a real number. No matter how's it represented they are still there and cannot be contained in our brains. We can say pi and hold the concept of pi in our heads, but not the actual number.
  - tialaramex 1y ago
    > They're unphysical, and yet the very physical human mind can work with them just fine Nah, you're likely thinking of the rationals, which are basically just two integers in a halloween costume. Ooh a third, big deal. The overwhelming majority of the reals are completely batshit and you're not working with them "just fine" except in some very hand wavy sense.
    - ysofunny 1y ago
      the rationals are 3 naturals with in a "2,1" structure. the first 2 naturals form an integer. that integer and a 3rd natural constitute a real (but this 3rd natural best be bigger than zero, else we're in trouble) what I choose to focus after observing the "unphysical" nature of numbers. is the sense of natural opposition (bordering on alternation) between "mathematical true" and "physical true". both are claiming to be really real Reality. in the mathematical realm, finite things are "impossible", they become "zero", negible in the presence of infinities. it's impossible for the primes to be finite (by contradiction). it's impossible for things (numbers or functions of mathematical objects) to be finite. but in the physical reality, it's the "infinite things" which become impossible. the "decimal point" (i.e. scientific notation i.e. positional numeral systems) is truly THE wonder of the world. for some reason I want something better than such a system... so I'm still learning about categories
      - tialaramex 1y ago
        Huh?
- NooneAtAll3 1y ago
  if we are arguing that natural numbers are made from abstraction, then we must apply that to real numbers as well - quantum values are complex numbers, that only become real once we start asking "what is position of the thing" or "what's its velocity"
- blueplanet200 1y ago
  >I'm an enthusiastic Cantor skeptic A skeptic in what way? He said a lot.
  - andrewla 1y ago
    Here I'm referring to the cloud of things that Hilbert called "Cantor's Paradise". Basically everything around the notion of cardinality of infinities.
    - blueplanet200 1y ago
      Please say more, I don't see how you can be _skeptical_ of those ideas. Math is math, if you start with ZFC axioms you get uncountable infinites. Maybe you don't start with those axioms. But that has nothing to do with truth, it's just a different mathematical setting.
      - andrewla 1y ago
        I loosely identify with the schools of intuitinalism/construtivism/finitism. Primary idea is that the Law of the Excluded Middle is not meaningful. So yes, generally not starting with ZFC. I can't speak to "truth" in that sense. The skepticism here is skepticism of the utility of the ideas stemming from Cantor's Paradise. It ends up in a very naval-gazing place where you prove obviously false things (like Banach-Tarski) from the axioms but have no way to map these wildly non-constructive ideas back into the real world. Or where you construct a version of the reals where the reals that we can produce via any computation is a set of measure 0 in the reals.
        CyLith 1y ago
        I don't understand why you believe Banach-Tarski to be obviously false. All that BT tells me is that matter is not modeled by a continuum since matter is composed of discrete atoms. This says nothing of the falsity of BT or the continuum.
        1y ago
- chasd00 1y ago
  > The idea of arbitrary precision is intrinsically broken in physical reality. you said a lot and i probably don't understand but doesn't pi contradict this? pi definitely exists in physical reality, wherever there is a circle, and seems to be have a never ending supply of decimal points.
  - LegionMammal978 1y ago
    > wherever there is a circle, Is there a circle in physical reality? Or only approximate circles, or things we model as circles? In any case, a believer in computation as reality would say that any digit of π has the potential to exist, as the result of a definite computation, but that the entirety does not actually exist apart from the process used to compute it.
  - Dylan16807 1y ago
    Can you name a physical thing that is a circle even to the baseline precision level of a 64 bit float?
    - IAmBroom 1y ago
      A black hole.
      - andrewla 1y ago
        A non-rotating black hole. Or a rotating black hole with zero charge. Or a rotating black hole with non-zero charge no external magnetic fields. Or a rotating black hole with non-zero charge with non-time-varying external magnetic fields. Or a wart on a frog on a bump on the log on a hole on the bottom of the sea.
      - tromp 1y ago
        A black hole is no more a perfect sphere than a sun is. Would gravity from the nearest other black hole not have a deforming effect of at least 2^-64 ?
      - Kranar 1y ago
- alphazard 1y ago
  > Nature and the universe is all about continuous quantities One could argue that nature always deals in discrete quantities and we have models that accurately predict these quantities. Then we use math that humans clearly created (limits) to produce similar models, except they imagine continuous inputs.
  - adrian_b 1y ago
    The quantity of matter and the quantity of electricity are discrete, but work, time and space are continuous, like also any quantities derived from them. There have been attempts to create discrete models of time and space, but nothing useful has resulted from those attempts. Most quantities encountered in nature include some dependency on work/energy, time or space, so nature deals mostly in continuous quantities, or more precisely the models that we can use to predict what happens in nature are still based mostly on continuous quantities, despite the fact that about a century and a half have passed since the discreteness of matter and electricity has been confirmed.
    - gpm 1y ago
      > but work, time and space are continuous I'm under the impression that all our theories of time and space (and thus work) break down at the scale of 1 plank unit and smaller. Which isn't proof that they aren't continuous, but I don't see how you could assert that they are either.
      - dhosek 1y ago
        Matter and energy are discrete. The continuity or discreteness of time and space are unknown. There are arguments for both cases, but nobody really knows for sure. It’s fairly easy to go from integers to many subsets of the reals (rationals are straightforward, constructible numbers not too hard, algebraic numbers more of a challenge), but the idea that the reals are, well real, depends on a continuity of spacetime that we can’t prove exists.
        adrian_b 1y ago
- NoahZuniga 1y ago
  You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). The standard construction for the real numbers is to start with the rationals and "fill in all the holes". So why even bother with filling in the holes and instead just declare God created the rationals?
  - IAmBroom 1y ago
    > You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). Citation needed. Especially since there are well-established math proofs of irrational numbers.
    - NoahZuniga 1y ago
      The argument is essentially that you can only measure things to finite precision. And for any measurement you've made at this finite precision, there exist both infinitely rational and irrational numbers. So it's impossible to rule out that the actual value you measured is one of those infinitely many rational numbers.
      - oskaralund 1y ago
        This argument feels like it's assuming the conclusion. If in principle it is only possible to measure quantities to finite precision, then it follows logically that we couldn't tell the difference between a rational and real universe. The question is, is the premise true here?
        BalinKing 1y ago
        AFAIK it would take an infinite amount of time to measure something to infinite precision, at least by the usual ways we’d think to do so…. I suppose one could assume a universe where that somehow isn’t the case, but (to my knowledge) that’s firmly in science-fiction territory.
- kazinator 1y ago
  > representing the state even of a very simple system involves continuous quantities. But that's tatamount to the belief that the minutest particle of the universe requires the equivalent of an infinite number of bits of state.
- wizardforhire 1y ago
  1/137
- daxfohl 1y ago
  But what if the expansion of the universe is due to some banach-tarski process?
- tomasson 1y ago
  “ Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. ” Hard disagree. This is the problem with math disconnected from physics. The real world is composed of quanta and spectra, i.e. reality is NOT continuous!
  - griffzhowl 1y ago
    Only bound states, like electrons confined to atomic orbitals, have quantized energies. Free electrons (or any particles) can have a continuous range of energies. Quantum mechanics (and general relativity) is still based on contiuous space and time, hence a continuous range of possible velocities and (kinetic) energies
    - svnt 1y ago
      Energies, yes, but the concept of energy quanta is inverted to e.g. time and length in that we have a maximum, not a minimum, where our understanding/models are limited, right?
athrowaway3z 1y ago
Can't say that I'm completely in the headspace to follow the argument, but wanted to add my 2 cents from a few years ago. Integers come into existence long before god - as the only presumption required is a difference between one thing and another (or nothing). The integers also create infinite gaps. The primes. So no - I do not think reals are closer to the divine. They require we import infinity twice to be defined, and I'm undecided on whether our reality has unbounded 'precision' like that - or if 'just' an infinite number of discrete units.
- IAmBroom 1y ago
  Depends on your view of God. If God existed before creation, there were not two things to compare. I'm not even sure "nothing" existed - maybe God was smart enough to avoid creating "null" values. Caveat: former Catholic; 50+ years of fervent atheism.
  - bandie91 1y ago
    there is no before and after in God. God is above/around creation. creation includes time. and also God the Son is IN creation.
- foobarian 1y ago
  I find primes spooky. They seem to be a concept that exists regardless of reality or universe. How does such a incontrovertible structure arise? ps. Various numerology phenomena have a similar vibe, and no wonder so many people who go off the deep end tend to get trapped by them. Maybe I will be one of them as I become old and senile :-D
  - ysofunny 1y ago
    > How does such a incontrovertible structure arise? yes, I also enjoy trying to answer this question. what is such an structure even mean? how could it be that simply defining numbers, obersving addition, and generalizing it away into multiplication would yield this natural structure? It all begins with zero. the predecessor of One, the best known number. zero can be assumed by anyone. the surprise is how all zeros are the same zero. (by uniqueness of emptyset; but as I hope you can see, I'm a crank. a nutjob. I'll stop
LegionMammal978 1y ago
Personally, I like to split the difference: the physical continuum definitely exists, to whatever extent any physical thing exists, but the real number line (and indeed the completed inductive set of integers) may just be a human-constructed fiction. The physical continuum is not necessarily identical to the real continuum; the latter is just a very useful model that lets us do human things like calculus. (And the discrepancy might not be in the physical continuum being simpler than the mathematical reals, as some here postulate, but rather in the continuum being far stranger than the reals, in ways we may never observe nor comprehend.)
zarzavat 1y ago
God created the rational numbers. The universe requires infinite divisibility, i.e. a dense set. It doesn't require infinite precision, i.e. a complete set. Our equations for the universe require a complete set, but that would be confusing the map with the territory. There is no physical evidence for uncountable infinities, those are purely in the imagination of man.
- adrian_b 1y ago
  The physical evidence is quite irrelevant in this case, and there also is no evidence that uncountable infinities do not exist. This is a problem of modeling optimization. The models based on uncountable "real" numbers are logically consistent and simple to use, so they are adequate for predicting what happens in natural or artificial systems. All attempts to avoid the uncountable infinities produce models that are both more complicated and also incomplete, as they do not cover all the applications of traditional infinitesimal calculus, topology and geometry. Unless someone will succeed to present a theory that avoids uncountable infinities while being as simple as the classic theory and being applicable to all the former uses, I see such attempts as interesting, but totally impractical.
  - zarzavat 1y ago
    The real numbers require infinite storage and infinite computation. There are both distinctly unphysical concepts. The real numbers are a useful mathematical trick that make it possible to prove results in calculus. What you surrender in return for being able to prove statements is to give up the ability to compute expressions. This may be a worthwhile trade-off for physicists but for the universe (which does many computations and zero proofs) it's quite a burden.
- andrewla 1y ago
  Why are rationals special? They represent an exactness in a similarly unphysical way as the integers. The rationals are infinitely precise. 1/3 is not the same as 0.33333 or 0.33333333 or 0.3. The real numbers exist and are approximable, either by rationals or by decimal expansion. The idea of approximability and computability are the critical things, not the specific representation.
tboyd47 1y ago
> If the something under examination causes a sense of existential nausea, disorientation, and a deep feeling that is can't possibly work like that, it is divine. This is a Jewish and Christian conception of God. How can this be true when so many things that give us comfort in the natural world: fresh fruit, shade trees, sunshine and warm sand between our toes, etc., were not created by man? Even in mathematics itself: how improbable, how ludicrous, how miraculous is it that the 3rd, 4th, and 5th natural numbers -- numbers you could discover by looking at your own hands -- have the amazing property of demonstrating the Pythagorean theorem? The Islamic ideal of God (Allah) is so much more balanced. God created both the integers AND the reals. He created everything, some things for our comfort and rest, some things to drive us close to madness, and a lot of stuff in between. Peel back enough layers of causality and all of creation has the stamp of the divine.
- thechao 1y ago
  > This is a Muslim conception of God. How can this be true when so many things that give us comfort in the natural world: fresh fruit, shade trees, sunshine and warm sand between our toes, etc., were not created by man? ... > The Jewish [Christian] ideal of God (YHVH) is so much more balanced. There's enough bigotry out there. Let's not make assumptions about people's beliefs.
  - tboyd47 1y ago
    Is it bigoted to discuss nuanced differences between belief systems?
- BalinKing 1y ago
  I want to push back on this, because the Christian conception of God definitely includes the idea that God created all good and comforting things, and is indeed their ultimate source. Like, just because God is transcendent[0] does not mean He cannot create things that are perfectly approachable, understandable, and enjoyable. [0] Jesus being human changes the calculus quite a lot, of course, as elaborated in e.g. Hebrews 4:14–16. God, who was fully transcendent, became human, hence why Jesus is also called Immanuel/Emmanuel (lit. “God with us”) in the Bible.
getnormality 1y ago
All math is just a system of ideas, specifically rules that people made up and follow because it's useful. I'm so used to thinking this way that I don't understand what all the fuss is about, mathematical objects being "real". Ideas are real but they're not real in the way that rocks are. Whenever there's a mysterious pattern in nature, people have felt the need to assert that some immaterial "thing" makes it so. But this just creates another mystery: what is the relationship between the material and the immaterial realm? What governs that? (Calling one or more of the immaterial entities "God" doesn't really make it any less mysterious.) If we add entities to our model of reality to answer questions and all it does is create more and more esoteric questions, we should take some advice from Occam's Shovel: when you're in a hole, stop digging.
- cernocky 1y ago
  Ideas are real in the way rocks are if we are concerned with their informational being. They are real informationally - ideas and math participate in forming the world. Nowadays, LLMs, Search and other apps probably affect the world even more than any common rock. Which is more real?
  - getnormality 1y ago
    I don't know what is meant by the informational being of a rock.
    - cernocky 1y ago
      one way to think about rock is to acknowledge it as an informational entity. an entity which is likely more passive then lets say a human or an app, yet by simply being part of the environment, it changes what can be done in the environment. if it wasn't there and if it didn't had a certain shape, the opportunities of other actors in the environment would surely be different. after all rock can be used as a tool and even as a computer. if its still not intuitive, think about Aeolian Harp which is a passive statue, yet a musical instrument, or think how you could encode a perceptron or a simple neural net into a stone (through which a water or air would flow for example). now, even if any ordinary rock doesn't exactly encode neural net, it should be more clear that it still affects information flow. does it help?
zamalek 1y ago
When I truly grokked complex numbers, I felt as though real numbers were a lie - though I would now say that it was a convenient omission. There are many things that are more naturally described using complex numbers - waves (which much of reality boils down to) immediately come to mind. Even if something does align better with real numbers, it's still just x+0i. Maybe I'll change my mind ~when~ if I finally grok quaternions.
- ysofunny 1y ago
  I think one quaternion contains all complex numbers I thought this when watching the 3b1b + ben eater collaboration on quaternion visualizers. von Neumann would say you'll never grok quaternions. but merely get used to them.
- znkr 1y ago
  > Even if something does align better with real numbers, it's still just x+0i Beware, it’s not always useful to work in complex numbers, you sometimes want to do something different for reals and complex numbers. The prime example here is complex analysis. Defining differentiation is based on limits, on the complex plane there are a lot more directions to approach a limit vs just two on the real line. This has some interesting implications. For example, any function differentiable on the complex plane is infinitely differentiable.
ljsprague 1y ago
His footnotes are out of order.
- EthanHeilman 1y ago
  I'm glad someone noticed.
  - grg0 1y ago
    Determining the real order is left as an exercise for the reader.
    - BeetleB 1y ago
      Can they be well ordered?
  - ljsprague 1y ago
    Are they ordered by importance or laziness?
chuckadams 1y ago
Quoth the old Fortran chestnut: "GOD IS REAL (unless declared as an integer)"
ajuc 1y ago
> the principle that there is more reality in a cause than in its effect What a weird idea. Why would that be?
- grg0 1y ago
  Not that I am well-versed in this, but it seems this is the cause-effect equivalent of the whole containing the part and not the other way around (or something greater can contain something smaller, but not conversely.)
entaloneralie 1y ago
God is a FRACTRAN program.
ChrisArchitect 1y ago
Previously yesterday: https://news.ycombinator.com/item?id=45053007 https://news.ycombinator.com/item?id=45053007
state_less 1y ago
So we're celebrating the real numbers, but maybe we should hoist up the illusory numbers? Back in the day, they thought that some numbers were "imaginary" numbers (e.g. sqrt(-1)) and nowadays, engineers use those imaginary numbers all the time and they feel as real as the reals. So, here's to math keeping our imagination limber and extending our ideas of what's real.
stephc_int13 1y ago
The abstract nature of real numbers and infinity is easily forgotten, sometimes leading to famous paradox or intellectual fallacies like the simulation hypothesis. Just models, useful but flawed abstractions.
schuyler2d 1y ago
A friend of mine argued that "math is invented" rather than discovered. This seemed wrong to me and in arguing against it I found https://plato.stanford.edu/entries/nominalism-mathematics/ https://plato.stanford.edu/entries/nominalism-mathematics/ At least at this stage I think it relates to whether you believe "the universe"/reality is a sort of momentary collection of the currently-existing things. Vs seeing reality as the set of all things that might obstruct "me" or any entity from doing something. To me, even if the wall is invisible it's still a wall
- spauldo 1y ago
  I personally consider it invented. The universe is what it is. Math is a model we've collectively built that tries to work within the underlying rules of how logic works in the universe. Like all models, any disagreement with observation means our model is wrong. But honestly, the whole question is akin to asking how many angels can dance on the head of a pin. On a separate subject, that site you linked does something strange with scrolling on Firefox mobile. Hey web devs - stop screwing around with things like scrolling! Browser devs implement scrolling in a consistent manner. You aren't making the user experience better with your silly JavaScript tricks!
_2d30 1y ago
Since the Integers are contained in the Reals, the idea that the Reals are "Eternal Nature" and the Integers are "Created by Humans" is surely false.
- galangalalgol 1y ago
  Given that reality is quantized (isn't it?) maybe it is humans that made up the reals and integers were god.
martindale 1y ago
Poelstra probably has a lot to say on the subject — but I think the natural order is the simpler and more "divine" variant.
curtisszmania 1y ago
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egorelik 1y ago
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gsf_emergency_2 1y ago
For Erdos, it's the Supreme Fascist who created the real numbers to see if we would give up or fight amongst ourselves trying (or, in most cases, not trying) to make them less weird
moi2388 1y ago
I am a finitist and constructionist at heart. Sure, mathematical abstractions and infinite structures are fun to play around with.. But go ahead and actually provide me the list of all naturals. You can not. Ever.
- lo_zamoyski 1y ago
  This might be conflating two concerns, though, which is the reality of numbers and their actual existence. Compare Plato with Aristotle. Plato held that the all forms exist in some third realm, so numbers would be counted among them. Aristotle, however, held that forms exist in particular instantiations or in minds that abstracted them from reality. (Aquinas could be said to synthesize both views in the sense that forms exist in particulars and in minds, but also have their origin in God, thus making God a sort of third realm, in a way. Neo-Platonists would view the "mind of God" similarly.) Now, in the Aristotelian view, numbers are quantities abstracted from concrete reality (indeed, quantity is one of the categories), but they are not substantial forms, as you will not see instances of numbers as substances in the world. They're abstractions of accidental forms. Furthermore, a form needn't be instantiated actually, but can exist potentially. This is how he resolves Zeno's paradoxes. You can divide a length an infinite number of times - or in a CS context, you can apply the successor function indefinitely - but only potentially; as a matter of actuality, you have not divided a length an infinite number of times. So, for Aristotle, you have a finite plurality of things that are potentially infinitely divisible, or a finite series of actions that can be potentially infinitely repeated or whatever. For a contemporary realist, Aristotelian treatment of math, James Franklin is worth checking out [0]. [0] https://web.maths.unsw.edu.au/~jim/structmath.html https://web.maths.unsw.edu.au/~jim/structmath.html
  - moi2388 1y ago
    If something can exist theoretically but not practically, your theory is wrong. But go ahead and divide a length an infinite number of times then. And actually infinite, not as it tends to infinity.
    - lo_zamoyski 1y ago
tempodox 1y ago
You mean, she didn’t create the complex numbers?
- ndsipa_pomu 1y ago
  She must have as the universe seems to run on rules that need negative and complex numbers (also irrationals). Just because humans took a while to discover them doesn't mean that they weren't always existing.
  - tempodox 1y ago
    Agreed. I was just wondering about the reductive title, since the reals are a strict subset of the complex.
jojomodding 1y ago
So by constructing the reals out of sets of rationals, themselves constructed out of integers, have we considered heresy? Is that good? Bad? I don't quite get what the text is about.
metalman 1y ago
I cant agree less. Taking the basic concept of dividing the universe into things that are real, and constructs and blowing it up into a whole hiarchy of values , putting numbers at the top, is only a construct, built on constructs. The current state of science is one where there are a few islands where things work internaly, like how elements are created through certain processes in stars, it's mostly good, but there are exceptions, and then there is a gulf, a vast hand wavy conjectural gulf between that and exactly where the helium came from, or when, or if it did. It is the eternal always almost of bieng poised on the brink of a revelation thats great for suckering in a few more to toil and grind away at the secrets of reality, which they do, and it is from those endevours that some of the best patentable ideas come from.
Xcelerate 1y ago
Everyone likes to debate the philosophy of whether the reals are “real”, but for me there is a much more practical question at hand: does the existence of something within a mathematical theory (i.e., derivability of a “∃ [...]” sentence) reflect back on our ability to predict the result of symbolic manipulations of arbitrary finite strings according to an arbitrary finite rule set over an arbitrary finite period of time? For AC and CH, the answer is provably “no” as these axioms have been shown to say nothing about the behavior of halting problems, which any question about the manipulation of symbols can be phrased in terms of (well, any specific question—more general cases move up the arithmetical hierarchy). If it’s not reflective in this precise sense, then the derivation of, e.g., a set-theoretic ∃ in some instances has no effect on any prediction of known physics (i.e., we are aware of no method of falsification).
akssri 1y ago
Theory of Computation wasn't around when all this "exciting" stuff was developed in Mathematics. Given their non-constructive nature "real" numbers are unsurprisingly totally incompatible with computation. Chaitin has a great paper on this and shows how Cantor's constructions were reflected a half-century later by Turing. https://arxiv.org/abs/math/0411418 https://arxiv.org/abs/math/0411418 Except of-course, while "hyper-Turing" machines that can do magic "post-Turing" "post-Halting" computation are seen as absurd fictions, real-numbers are seen as "normal" and "obvious" and "common-sensical"! It was amusing sometime back to see people pooh-pooh the likes of Hava Siegelmann for being funded for their "super-Turing" machines with "real-number" computation, without realizing that the core issue is the "real"-number itself! I've always found this quite strange, but I've realized that this is almost blasphemy (people in STEM, and esp. their "allies", aren't as enlightened etc. as they pretend to be tbh). Some historicans of mathematics claim (C. K. Raju for eg.) that this comes from the insertion of Greek-Christian theological bent in the development of modern mathematics. Anyone who has taken measure theory etc. and then gone on to do "practical" numerical stuff, and then realizes the pointlessness of much of this hard/abstract construction dealing with "scary" monsters that can't even be computed, would perhaps wholeheartedly agree. edit: The post has a great link to a note on Cantor's theology, https://apeironcentre.org/theology-of-georg-cantor/ https://apeironcentre.org/theology-of-georg-cantor/
- i2go 1y ago
  > Given their non-constructive nature "real" numbers are unsurprisingly totally incompatible with computation. It is funny you say that when Turing defined Turing machines to compute real numbers (like π for example). In its original definition, a number was computable if its Turing machine did not stop. Which makes sense since π does not have a finite decimal expansion. Today, we usually define Turing machines to decide problems and a problem is decidable if for every input its Turing machine stops with a ``yes'' or ``no'' answer. I guess this is what makes people think what you said in the quote above. Maybe this definition is more intuitive but this conclusion from it could not be more wrong. Think about it for a second, if the computable numbers were countable there would be no uncomputable problem (Turing actually used the classic cantor diagonal argument to prove that there were uncomputable numbers)
tomasson 1y ago
Commutative? Did the real numbers create God? Certainly Turing and Godel showed that computation is not universal (complete). Looking at one element of number theory in isolation seems unproductive? Arithmetic space is staggeringly complex, but structured, layered. IMHO number theory is like a hall of mirrors without a defined interface with physics. See Yang Mills mass gap.
scoofy 1y ago
Don’t question the axioms… it’s a waste of time and the axioms enjoy it.
TomONeil 1y ago
Math is what underlies the foundation of the universe so yes numbers did arise from God's realm. Measurement came first then the Universe!
ngcc_hk 1y ago
Any elaboration to help us know the structure around us is useful and interesting. Might lead to useless results like even and odd numbers. Still say using bijection and diagonal rules illuminate something about our thinking of number system. And might be we have inconsistent maths but nevertheless still interesting.

136 points by Bogdanp 1y ago

13 points and 18 comments here yesterday: https://news.ycombinator.com/item?id=45053007 https://news.ycombinator.com/item?id=45053007

andrewla 1y ago

I'm an enthusiastic Cantor skeptic, I lean very heavily constructivist to the point of almost being a finitist, but nonetheless I think the thesis of this article is basically correct. Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. At a micro level this is less true -- elementary particles specifically are a (mostly) discrete phenomenon, but representing the state even of a very simple system involves continuous quantities. But the Cantor vision of the real numbers is just wrong and completely unphysical. The idea of arbitrary precision is intrinsically broken in physical reality. Instead I am off the opinion that computation is the relevant process in the physical universe, so approximations to continuous quantities are where the "Eternal Nature" line lies, and the abstraction of the continuum is just that -- an abstraction of the idea of having perfect knowledge of the state of anything in the universe.

empath75 1y ago
> But the Cantor vision of the real numbers is just wrong and completely unphysical. They're unphysical, and yet the very physical human mind can work with them just fine. They're a perfectly logical construction from perfectly reasonable axioms. There are lots of objects in math which aren't physically realizable. Plato would have said that those sorts of objects are more real than anything which actually exists in "reality".
- andrewla 1y ago
  There are two things being talked about here, and worth teasing them out. On the one hand, this article is talking about the hierarchy of "physicality" of various mathematical concepts, and they put Cantor's real numbers at the floor. I disagree with that specifically; two quantities are interestingly "unequal" only at the precision where an underlying process can distinguish them. Turing tells us that any underlying process must represent a computation, and that the power of computation is a law of the underlying reality of the universe (this is my view of the Universal Church-Turing Thesis, not necessarily the generally accepted variant). The other question is whether Cantor's conception of infinity is a useful one in mathematics. Here I think the answer is no. It leads to rabbit holes that are just uninteresting; trying to distinguish inifinities (continuum hypothesis) and leading us to counterintuitive and useless results. Fun to play with, like writing programs that can invoke a HaltingFunction oracle, but does not tell us anything that we can map back to reality. For example, the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful.
  - EthanHeilman 1y ago
    > On the one hand, this article is talking about the hierarchy of "physicality" of various mathematical concepts, and they put Cantor's real numbers at the floor. I disagree with that specifically I didn't mean to suggest that the reals are the floor of reality, rather that they are more floorlike than the integers. > The other question is whether Cantor's conception of infinity is a useful one in mathematics. Here I think the answer is no. Tools are created by transforming nature into something useful to humans. Is Cantor's conception of infinity more natural? I can't really say, but the uselessness and confusion seems more like nature than technology.
  - nh23423fefe 1y ago
    > the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful it leads to the idea that measuring 2 sets via a bijection is a better idea than measuring via containment
    - andrewla 1y ago
      That a bijection exists is incredibly useful. But the idea of "measuring" infinite sets in the cardinality sense is not very interesting or useful.
      - oskaralund 1y ago
        Saying that two sets have the same cardinality is equivalent to saying there is a bijection between them. I don't understand how the latter can be useful but not the former?
      - sfpotter 1y ago
        It isn't very interesting or useful... to you.
  - SabrinaJewson 1y ago
    > For example, the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful. I am not sure what you are arguing here. We’ve been teaching this to all undergraduate mathematicians for the last century; are you trying to make the point that this part of the curriculum is unnecessary, or that mathematics has not contributed to the wellbeing of society in the last hundred years? Both of these seem like rather difficult positions to defend.
- shkkmo 1y ago
  > They're unphysical, and yet the very physical human mind can work with them just fine. Can it? We can only work with things we can name and the real numbers we can name are an infinitesimal fraction of the real numbers. (The nameable reals and sets of reals have the same cardinality as integers while the rest are a higher cardinality.)
  - SabrinaJewson 1y ago
    We can work with unnameable things very easily. Take, for instance, every known theorem that quantifies over all real numbers. If you try to argue that proving theorems about these real numbers does not constitute “working with” them, it seems you have chosen a rather deficient definition of “working with” that does not match with how that phrase is used in the real world.
    - shkkmo 1y ago
      I would argue that all of those theorems work with nameable sets of real numbers but not with any unnamable real numbers themselves.
- Eddy_Viscosity2 1y ago
  The human mind can't work with a real number any more than it can infinity. We box them into concepts and then work with those. An actual raw real number is unfathomable.
  - SabrinaJewson 1y ago
    I don’t know about you, I can work with it just fine. I know its properties. I can manipulate it. I can prove theorems about it. What more is there? In fact, if you are to argue that we cannot know a “raw” real number, I would point out that we can’t know a natural number either! Take 2: you can picture two apples, you can imagine second place, you can visualize its decimal representation in Arabic numerals, you can tell me all its arithmetical properties, you can write down its construction as a set in ZFC set theory… but can you really know the number – not a representation of the number, not its properties, but the number itself? Of course not: mathematical objects are their properties and nothing more. It doesn’t even make sense to consider the idea of a “raw” object.
    - Eddy_Viscosity2 1y ago
      You can hold a two in your head, but you can't hold a number with infinitely many decimal places. Any manipulations you do with the real 2 are done conceptually whereas with the natural 2, its done concretely.
      - drdeca 1y ago
        The decimal places are just a way of representing it.
        Eddy_Viscosity2 1y ago
        The infinite number of decimal places is the definitional feature of a real number. No matter how's it represented they are still there and cannot be contained in our brains. We can say pi and hold the concept of pi in our heads, but not the actual number.
- tialaramex 1y ago
  > They're unphysical, and yet the very physical human mind can work with them just fine Nah, you're likely thinking of the rationals, which are basically just two integers in a halloween costume. Ooh a third, big deal. The overwhelming majority of the reals are completely batshit and you're not working with them "just fine" except in some very hand wavy sense.
  - ysofunny 1y ago
    the rationals are 3 naturals with in a "2,1" structure. the first 2 naturals form an integer. that integer and a 3rd natural constitute a real (but this 3rd natural best be bigger than zero, else we're in trouble) what I choose to focus after observing the "unphysical" nature of numbers. is the sense of natural opposition (bordering on alternation) between "mathematical true" and "physical true". both are claiming to be really real Reality. in the mathematical realm, finite things are "impossible", they become "zero", negible in the presence of infinities. it's impossible for the primes to be finite (by contradiction). it's impossible for things (numbers or functions of mathematical objects) to be finite. but in the physical reality, it's the "infinite things" which become impossible. the "decimal point" (i.e. scientific notation i.e. positional numeral systems) is truly THE wonder of the world. for some reason I want something better than such a system... so I'm still learning about categories
    - tialaramex 1y ago
      Huh?
NooneAtAll3 1y ago
if we are arguing that natural numbers are made from abstraction, then we must apply that to real numbers as well - quantum values are complex numbers, that only become real once we start asking "what is position of the thing" or "what's its velocity"
blueplanet200 1y ago
>I'm an enthusiastic Cantor skeptic A skeptic in what way? He said a lot.
- andrewla 1y ago
  Here I'm referring to the cloud of things that Hilbert called "Cantor's Paradise". Basically everything around the notion of cardinality of infinities.
  - blueplanet200 1y ago
    Please say more, I don't see how you can be _skeptical_ of those ideas. Math is math, if you start with ZFC axioms you get uncountable infinites. Maybe you don't start with those axioms. But that has nothing to do with truth, it's just a different mathematical setting.
    - andrewla 1y ago
      I loosely identify with the schools of intuitinalism/construtivism/finitism. Primary idea is that the Law of the Excluded Middle is not meaningful. So yes, generally not starting with ZFC. I can't speak to "truth" in that sense. The skepticism here is skepticism of the utility of the ideas stemming from Cantor's Paradise. It ends up in a very naval-gazing place where you prove obviously false things (like Banach-Tarski) from the axioms but have no way to map these wildly non-constructive ideas back into the real world. Or where you construct a version of the reals where the reals that we can produce via any computation is a set of measure 0 in the reals.
      - CyLith 1y ago
        I don't understand why you believe Banach-Tarski to be obviously false. All that BT tells me is that matter is not modeled by a continuum since matter is composed of discrete atoms. This says nothing of the falsity of BT or the continuum.
        1y ago
chasd00 1y ago
> The idea of arbitrary precision is intrinsically broken in physical reality. you said a lot and i probably don't understand but doesn't pi contradict this? pi definitely exists in physical reality, wherever there is a circle, and seems to be have a never ending supply of decimal points.
- LegionMammal978 1y ago
  > wherever there is a circle, Is there a circle in physical reality? Or only approximate circles, or things we model as circles? In any case, a believer in computation as reality would say that any digit of π has the potential to exist, as the result of a definite computation, but that the entirety does not actually exist apart from the process used to compute it.
- Dylan16807 1y ago
  Can you name a physical thing that is a circle even to the baseline precision level of a 64 bit float?
  - IAmBroom 1y ago
    A black hole.
    - andrewla 1y ago
      A non-rotating black hole. Or a rotating black hole with zero charge. Or a rotating black hole with non-zero charge no external magnetic fields. Or a rotating black hole with non-zero charge with non-time-varying external magnetic fields. Or a wart on a frog on a bump on the log on a hole on the bottom of the sea.
    - tromp 1y ago
      A black hole is no more a perfect sphere than a sun is. Would gravity from the nearest other black hole not have a deforming effect of at least 2^-64 ?
    - Kranar 1y ago
alphazard 1y ago
> Nature and the universe is all about continuous quantities One could argue that nature always deals in discrete quantities and we have models that accurately predict these quantities. Then we use math that humans clearly created (limits) to produce similar models, except they imagine continuous inputs.
- adrian_b 1y ago
  The quantity of matter and the quantity of electricity are discrete, but work, time and space are continuous, like also any quantities derived from them. There have been attempts to create discrete models of time and space, but nothing useful has resulted from those attempts. Most quantities encountered in nature include some dependency on work/energy, time or space, so nature deals mostly in continuous quantities, or more precisely the models that we can use to predict what happens in nature are still based mostly on continuous quantities, despite the fact that about a century and a half have passed since the discreteness of matter and electricity has been confirmed.
  - gpm 1y ago
    > but work, time and space are continuous I'm under the impression that all our theories of time and space (and thus work) break down at the scale of 1 plank unit and smaller. Which isn't proof that they aren't continuous, but I don't see how you could assert that they are either.
    - dhosek 1y ago
      Matter and energy are discrete. The continuity or discreteness of time and space are unknown. There are arguments for both cases, but nobody really knows for sure. It’s fairly easy to go from integers to many subsets of the reals (rationals are straightforward, constructible numbers not too hard, algebraic numbers more of a challenge), but the idea that the reals are, well real, depends on a continuity of spacetime that we can’t prove exists.
      - adrian_b 1y ago
NoahZuniga 1y ago
You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). The standard construction for the real numbers is to start with the rationals and "fill in all the holes". So why even bother with filling in the holes and instead just declare God created the rationals?
- IAmBroom 1y ago
  > You know it wouldn't be possible for us to tell the difference between a rational universe (one where all quantities are rational numbers) and a real universe (one where you can have irrational quantities). Citation needed. Especially since there are well-established math proofs of irrational numbers.
  - NoahZuniga 1y ago
    The argument is essentially that you can only measure things to finite precision. And for any measurement you've made at this finite precision, there exist both infinitely rational and irrational numbers. So it's impossible to rule out that the actual value you measured is one of those infinitely many rational numbers.
    - oskaralund 1y ago
      This argument feels like it's assuming the conclusion. If in principle it is only possible to measure quantities to finite precision, then it follows logically that we couldn't tell the difference between a rational and real universe. The question is, is the premise true here?
      - BalinKing 1y ago
        AFAIK it would take an infinite amount of time to measure something to infinite precision, at least by the usual ways we’d think to do so…. I suppose one could assume a universe where that somehow isn’t the case, but (to my knowledge) that’s firmly in science-fiction territory.
kazinator 1y ago
> representing the state even of a very simple system involves continuous quantities. But that's tatamount to the belief that the minutest particle of the universe requires the equivalent of an infinite number of bits of state.
wizardforhire 1y ago
1/137
daxfohl 1y ago
But what if the expansion of the universe is due to some banach-tarski process?
tomasson 1y ago
“ Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. ” Hard disagree. This is the problem with math disconnected from physics. The real world is composed of quanta and spectra, i.e. reality is NOT continuous!
- griffzhowl 1y ago
  Only bound states, like electrons confined to atomic orbitals, have quantized energies. Free electrons (or any particles) can have a continuous range of energies. Quantum mechanics (and general relativity) is still based on contiuous space and time, hence a continuous range of possible velocities and (kinetic) energies
  - svnt 1y ago
    Energies, yes, but the concept of energy quanta is inverted to e.g. time and length in that we have a maximum, not a minimum, where our understanding/models are limited, right?

athrowaway3z 1y ago

Can't say that I'm completely in the headspace to follow the argument, but wanted to add my 2 cents from a few years ago. Integers come into existence long before god - as the only presumption required is a difference between one thing and another (or nothing). The integers also create infinite gaps. The primes. So no - I do not think reals are closer to the divine. They require we import infinity twice to be defined, and I'm undecided on whether our reality has unbounded 'precision' like that - or if 'just' an infinite number of discrete units.

IAmBroom 1y ago
Depends on your view of God. If God existed before creation, there were not two things to compare. I'm not even sure "nothing" existed - maybe God was smart enough to avoid creating "null" values. Caveat: former Catholic; 50+ years of fervent atheism.
- bandie91 1y ago
  there is no before and after in God. God is above/around creation. creation includes time. and also God the Son is IN creation.
foobarian 1y ago
I find primes spooky. They seem to be a concept that exists regardless of reality or universe. How does such a incontrovertible structure arise? ps. Various numerology phenomena have a similar vibe, and no wonder so many people who go off the deep end tend to get trapped by them. Maybe I will be one of them as I become old and senile :-D
- ysofunny 1y ago
  > How does such a incontrovertible structure arise? yes, I also enjoy trying to answer this question. what is such an structure even mean? how could it be that simply defining numbers, obersving addition, and generalizing it away into multiplication would yield this natural structure? It all begins with zero. the predecessor of One, the best known number. zero can be assumed by anyone. the surprise is how all zeros are the same zero. (by uniqueness of emptyset; but as I hope you can see, I'm a crank. a nutjob. I'll stop

LegionMammal978 1y ago

Personally, I like to split the difference: the physical continuum definitely exists, to whatever extent any physical thing exists, but the real number line (and indeed the completed inductive set of integers) may just be a human-constructed fiction. The physical continuum is not necessarily identical to the real continuum; the latter is just a very useful model that lets us do human things like calculus. (And the discrepancy might not be in the physical continuum being simpler than the mathematical reals, as some here postulate, but rather in the continuum being far stranger than the reals, in ways we may never observe nor comprehend.)

zarzavat 1y ago

God created the rational numbers. The universe requires infinite divisibility, i.e. a dense set. It doesn't require infinite precision, i.e. a complete set. Our equations for the universe require a complete set, but that would be confusing the map with the territory. There is no physical evidence for uncountable infinities, those are purely in the imagination of man.

adrian_b 1y ago
The physical evidence is quite irrelevant in this case, and there also is no evidence that uncountable infinities do not exist. This is a problem of modeling optimization. The models based on uncountable "real" numbers are logically consistent and simple to use, so they are adequate for predicting what happens in natural or artificial systems. All attempts to avoid the uncountable infinities produce models that are both more complicated and also incomplete, as they do not cover all the applications of traditional infinitesimal calculus, topology and geometry. Unless someone will succeed to present a theory that avoids uncountable infinities while being as simple as the classic theory and being applicable to all the former uses, I see such attempts as interesting, but totally impractical.
- zarzavat 1y ago
  The real numbers require infinite storage and infinite computation. There are both distinctly unphysical concepts. The real numbers are a useful mathematical trick that make it possible to prove results in calculus. What you surrender in return for being able to prove statements is to give up the ability to compute expressions. This may be a worthwhile trade-off for physicists but for the universe (which does many computations and zero proofs) it's quite a burden.
andrewla 1y ago
Why are rationals special? They represent an exactness in a similarly unphysical way as the integers. The rationals are infinitely precise. 1/3 is not the same as 0.33333 or 0.33333333 or 0.3. The real numbers exist and are approximable, either by rationals or by decimal expansion. The idea of approximability and computability are the critical things, not the specific representation.

tboyd47 1y ago

> If the something under examination causes a sense of existential nausea, disorientation, and a deep feeling that is can't possibly work like that, it is divine. This is a Jewish and Christian conception of God. How can this be true when so many things that give us comfort in the natural world: fresh fruit, shade trees, sunshine and warm sand between our toes, etc., were not created by man? Even in mathematics itself: how improbable, how ludicrous, how miraculous is it that the 3rd, 4th, and 5th natural numbers -- numbers you could discover by looking at your own hands -- have the amazing property of demonstrating the Pythagorean theorem? The Islamic ideal of God (Allah) is so much more balanced. God created both the integers AND the reals. He created everything, some things for our comfort and rest, some things to drive us close to madness, and a lot of stuff in between. Peel back enough layers of causality and all of creation has the stamp of the divine.

thechao 1y ago
> This is a Muslim conception of God. How can this be true when so many things that give us comfort in the natural world: fresh fruit, shade trees, sunshine and warm sand between our toes, etc., were not created by man? ... > The Jewish [Christian] ideal of God (YHVH) is so much more balanced. There's enough bigotry out there. Let's not make assumptions about people's beliefs.
- tboyd47 1y ago
  Is it bigoted to discuss nuanced differences between belief systems?
BalinKing 1y ago
I want to push back on this, because the Christian conception of God definitely includes the idea that God created all good and comforting things, and is indeed their ultimate source. Like, just because God is transcendent[0] does not mean He cannot create things that are perfectly approachable, understandable, and enjoyable. [0] Jesus being human changes the calculus quite a lot, of course, as elaborated in e.g. Hebrews 4:14–16. God, who was fully transcendent, became human, hence why Jesus is also called Immanuel/Emmanuel (lit. “God with us”) in the Bible.

getnormality 1y ago

All math is just a system of ideas, specifically rules that people made up and follow because it's useful. I'm so used to thinking this way that I don't understand what all the fuss is about, mathematical objects being "real". Ideas are real but they're not real in the way that rocks are. Whenever there's a mysterious pattern in nature, people have felt the need to assert that some immaterial "thing" makes it so. But this just creates another mystery: what is the relationship between the material and the immaterial realm? What governs that? (Calling one or more of the immaterial entities "God" doesn't really make it any less mysterious.) If we add entities to our model of reality to answer questions and all it does is create more and more esoteric questions, we should take some advice from Occam's Shovel: when you're in a hole, stop digging.

cernocky 1y ago
Ideas are real in the way rocks are if we are concerned with their informational being. They are real informationally - ideas and math participate in forming the world. Nowadays, LLMs, Search and other apps probably affect the world even more than any common rock. Which is more real?
- getnormality 1y ago
  I don't know what is meant by the informational being of a rock.
  - cernocky 1y ago
    one way to think about rock is to acknowledge it as an informational entity. an entity which is likely more passive then lets say a human or an app, yet by simply being part of the environment, it changes what can be done in the environment. if it wasn't there and if it didn't had a certain shape, the opportunities of other actors in the environment would surely be different. after all rock can be used as a tool and even as a computer. if its still not intuitive, think about Aeolian Harp which is a passive statue, yet a musical instrument, or think how you could encode a perceptron or a simple neural net into a stone (through which a water or air would flow for example). now, even if any ordinary rock doesn't exactly encode neural net, it should be more clear that it still affects information flow. does it help?

zamalek 1y ago

When I truly grokked complex numbers, I felt as though real numbers were a lie - though I would now say that it was a convenient omission. There are many things that are more naturally described using complex numbers - waves (which much of reality boils down to) immediately come to mind. Even if something does align better with real numbers, it's still just x+0i. Maybe I'll change my mind ~when~ if I finally grok quaternions.

ysofunny 1y ago
I think one quaternion contains all complex numbers I thought this when watching the 3b1b + ben eater collaboration on quaternion visualizers. von Neumann would say you'll never grok quaternions. but merely get used to them.
znkr 1y ago
> Even if something does align better with real numbers, it's still just x+0i Beware, it’s not always useful to work in complex numbers, you sometimes want to do something different for reals and complex numbers. The prime example here is complex analysis. Defining differentiation is based on limits, on the complex plane there are a lot more directions to approach a limit vs just two on the real line. This has some interesting implications. For example, any function differentiable on the complex plane is infinitely differentiable.

ljsprague 1y ago

His footnotes are out of order.

EthanHeilman 1y ago
I'm glad someone noticed.
- grg0 1y ago
  Determining the real order is left as an exercise for the reader.
  - BeetleB 1y ago
    Can they be well ordered?
- ljsprague 1y ago
  Are they ordered by importance or laziness?

chuckadams 1y ago

Quoth the old Fortran chestnut: "GOD IS REAL (unless declared as an integer)"

ajuc 1y ago

> the principle that there is more reality in a cause than in its effect What a weird idea. Why would that be?

grg0 1y ago
Not that I am well-versed in this, but it seems this is the cause-effect equivalent of the whole containing the part and not the other way around (or something greater can contain something smaller, but not conversely.)

entaloneralie 1y ago

God is a FRACTRAN program.

ChrisArchitect 1y ago

Previously yesterday: https://news.ycombinator.com/item?id=45053007 https://news.ycombinator.com/item?id=45053007

state_less 1y ago

So we're celebrating the real numbers, but maybe we should hoist up the illusory numbers? Back in the day, they thought that some numbers were "imaginary" numbers (e.g. sqrt(-1)) and nowadays, engineers use those imaginary numbers all the time and they feel as real as the reals. So, here's to math keeping our imagination limber and extending our ideas of what's real.

stephc_int13 1y ago

The abstract nature of real numbers and infinity is easily forgotten, sometimes leading to famous paradox or intellectual fallacies like the simulation hypothesis. Just models, useful but flawed abstractions.

schuyler2d 1y ago

A friend of mine argued that "math is invented" rather than discovered. This seemed wrong to me and in arguing against it I found https://plato.stanford.edu/entries/nominalism-mathematics/ https://plato.stanford.edu/entries/nominalism-mathematics/ At least at this stage I think it relates to whether you believe "the universe"/reality is a sort of momentary collection of the currently-existing things. Vs seeing reality as the set of all things that might obstruct "me" or any entity from doing something. To me, even if the wall is invisible it's still a wall

spauldo 1y ago
I personally consider it invented. The universe is what it is. Math is a model we've collectively built that tries to work within the underlying rules of how logic works in the universe. Like all models, any disagreement with observation means our model is wrong. But honestly, the whole question is akin to asking how many angels can dance on the head of a pin. On a separate subject, that site you linked does something strange with scrolling on Firefox mobile. Hey web devs - stop screwing around with things like scrolling! Browser devs implement scrolling in a consistent manner. You aren't making the user experience better with your silly JavaScript tricks!

_2d30 1y ago

Since the Integers are contained in the Reals, the idea that the Reals are "Eternal Nature" and the Integers are "Created by Humans" is surely false.

galangalalgol 1y ago
Given that reality is quantized (isn't it?) maybe it is humans that made up the reals and integers were god.

martindale 1y ago

Poelstra probably has a lot to say on the subject — but I think the natural order is the simpler and more "divine" variant.

curtisszmania 1y ago

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egorelik 1y ago

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gsf_emergency_2 1y ago

For Erdos, it's the Supreme Fascist who created the real numbers to see if we would give up or fight amongst ourselves trying (or, in most cases, not trying) to make them less weird

moi2388 1y ago

I am a finitist and constructionist at heart. Sure, mathematical abstractions and infinite structures are fun to play around with.. But go ahead and actually provide me the list of all naturals. You can not. Ever.

lo_zamoyski 1y ago
This might be conflating two concerns, though, which is the reality of numbers and their actual existence. Compare Plato with Aristotle. Plato held that the all forms exist in some third realm, so numbers would be counted among them. Aristotle, however, held that forms exist in particular instantiations or in minds that abstracted them from reality. (Aquinas could be said to synthesize both views in the sense that forms exist in particulars and in minds, but also have their origin in God, thus making God a sort of third realm, in a way. Neo-Platonists would view the "mind of God" similarly.) Now, in the Aristotelian view, numbers are quantities abstracted from concrete reality (indeed, quantity is one of the categories), but they are not substantial forms, as you will not see instances of numbers as substances in the world. They're abstractions of accidental forms. Furthermore, a form needn't be instantiated actually, but can exist potentially. This is how he resolves Zeno's paradoxes. You can divide a length an infinite number of times - or in a CS context, you can apply the successor function indefinitely - but only potentially; as a matter of actuality, you have not divided a length an infinite number of times. So, for Aristotle, you have a finite plurality of things that are potentially infinitely divisible, or a finite series of actions that can be potentially infinitely repeated or whatever. For a contemporary realist, Aristotelian treatment of math, James Franklin is worth checking out [0]. [0] https://web.maths.unsw.edu.au/~jim/structmath.html https://web.maths.unsw.edu.au/~jim/structmath.html
- moi2388 1y ago
  If something can exist theoretically but not practically, your theory is wrong. But go ahead and divide a length an infinite number of times then. And actually infinite, not as it tends to infinity.
  - lo_zamoyski 1y ago

tempodox 1y ago

You mean, she didn’t create the complex numbers?

ndsipa_pomu 1y ago
She must have as the universe seems to run on rules that need negative and complex numbers (also irrationals). Just because humans took a while to discover them doesn't mean that they weren't always existing.
- tempodox 1y ago
  Agreed. I was just wondering about the reductive title, since the reals are a strict subset of the complex.

jojomodding 1y ago

So by constructing the reals out of sets of rationals, themselves constructed out of integers, have we considered heresy? Is that good? Bad? I don't quite get what the text is about.

metalman 1y ago

I cant agree less. Taking the basic concept of dividing the universe into things that are real, and constructs and blowing it up into a whole hiarchy of values , putting numbers at the top, is only a construct, built on constructs. The current state of science is one where there are a few islands where things work internaly, like how elements are created through certain processes in stars, it's mostly good, but there are exceptions, and then there is a gulf, a vast hand wavy conjectural gulf between that and exactly where the helium came from, or when, or if it did. It is the eternal always almost of bieng poised on the brink of a revelation thats great for suckering in a few more to toil and grind away at the secrets of reality, which they do, and it is from those endevours that some of the best patentable ideas come from.

Xcelerate 1y ago

Everyone likes to debate the philosophy of whether the reals are “real”, but for me there is a much more practical question at hand: does the existence of something within a mathematical theory (i.e., derivability of a “∃ [...]” sentence) reflect back on our ability to predict the result of symbolic manipulations of arbitrary finite strings according to an arbitrary finite rule set over an arbitrary finite period of time? For AC and CH, the answer is provably “no” as these axioms have been shown to say nothing about the behavior of halting problems, which any question about the manipulation of symbols can be phrased in terms of (well, any specific question—more general cases move up the arithmetical hierarchy). If it’s not reflective in this precise sense, then the derivation of, e.g., a set-theoretic ∃ in some instances has no effect on any prediction of known physics (i.e., we are aware of no method of falsification).

akssri 1y ago

Theory of Computation wasn't around when all this "exciting" stuff was developed in Mathematics. Given their non-constructive nature "real" numbers are unsurprisingly totally incompatible with computation. Chaitin has a great paper on this and shows how Cantor's constructions were reflected a half-century later by Turing. https://arxiv.org/abs/math/0411418 https://arxiv.org/abs/math/0411418 Except of-course, while "hyper-Turing" machines that can do magic "post-Turing" "post-Halting" computation are seen as absurd fictions, real-numbers are seen as "normal" and "obvious" and "common-sensical"! It was amusing sometime back to see people pooh-pooh the likes of Hava Siegelmann for being funded for their "super-Turing" machines with "real-number" computation, without realizing that the core issue is the "real"-number itself! I've always found this quite strange, but I've realized that this is almost blasphemy (people in STEM, and esp. their "allies", aren't as enlightened etc. as they pretend to be tbh). Some historicans of mathematics claim (C. K. Raju for eg.) that this comes from the insertion of Greek-Christian theological bent in the development of modern mathematics. Anyone who has taken measure theory etc. and then gone on to do "practical" numerical stuff, and then realizes the pointlessness of much of this hard/abstract construction dealing with "scary" monsters that can't even be computed, would perhaps wholeheartedly agree. edit: The post has a great link to a note on Cantor's theology, https://apeironcentre.org/theology-of-georg-cantor/ https://apeironcentre.org/theology-of-georg-cantor/

i2go 1y ago
> Given their non-constructive nature "real" numbers are unsurprisingly totally incompatible with computation. It is funny you say that when Turing defined Turing machines to compute real numbers (like π for example). In its original definition, a number was computable if its Turing machine did not stop. Which makes sense since π does not have a finite decimal expansion. Today, we usually define Turing machines to decide problems and a problem is decidable if for every input its Turing machine stops with a ``yes'' or ``no'' answer. I guess this is what makes people think what you said in the quote above. Maybe this definition is more intuitive but this conclusion from it could not be more wrong. Think about it for a second, if the computable numbers were countable there would be no uncomputable problem (Turing actually used the classic cantor diagonal argument to prove that there were uncomputable numbers)

tomasson 1y ago

Commutative? Did the real numbers create God? Certainly Turing and Godel showed that computation is not universal (complete). Looking at one element of number theory in isolation seems unproductive? Arithmetic space is staggeringly complex, but structured, layered. IMHO number theory is like a hall of mirrors without a defined interface with physics. See Yang Mills mass gap.

scoofy 1y ago

Don’t question the axioms… it’s a waste of time and the axioms enjoy it.

TomONeil 1y ago

Math is what underlies the foundation of the universe so yes numbers did arise from God's realm. Measurement came first then the Universe!

ngcc_hk 1y ago

Any elaboration to help us know the structure around us is useful and interesting. Might lead to useless results like even and odd numbers. Still say using bijection and diagonal rules illuminate something about our thinking of number system. And might be we have inconsistent maths but nevertheless still interesting.