17 ms·

How inevitable is the concept of numbers?

182 points by perardi 5y ago

paganel 5y ago
I can understand how natural numbers can be “constructed” (for lack of a better word) as a byproduct of counting, what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero. Maybe related, while I can also partially understand multiplication (syntactic sugar for adding) I could never understand multiplication by zero, meaning how come when you multiply a number (no matter how big) by zero you get zero as a result. Maybe there’s some Wittgenstein-like material somewhere that will better explain this, in which case I’ll very happy for some references.
- inglor_cz 5y ago
  No need for Wittgenstein. Fish have 0 legs. So how many legs do N fish have? N*0 = 0. Now division by zero, that is Devil's idea.
- ddlutz 5y ago
  Isn't multiplication by 0 natural? If i'm at a party and everybody wants 2 beers, then if there is 1 person I need 2 beers (1 * 2), 2 people need 4 beers (2 * 2), but 0 people need 0 beers (0 * 2).
  - paganel 5y ago
    “Zero people” and “needing” is an oxymoron, nothing (zero) can’t be associated to any natural thing (like “needing”), that’s what I was trying to say in my not so clear comment above. Writing this down I realized we’re still trying to write in fancier words what the pre-Socratics had a clear understanding of 2500+ years ago, and speaking of the Greeks is too bad that Wolfram didn’t mention Plato by name in the first few paragraphs, the chair example is practically taken from him (expecting a Parmenides quote would have probably been too much).
    - ezconnect 5y ago
      When you combine understanding language and numbers you will have a hard time.
    - babygoat 5y ago
      Say you have 5 bags with 5 apples in each. How many apples do you have? 5 x 5 = 25. If you have 5 empty bags, 5 x 0 = 0.
    - greydius 5y ago
      Noone needs a fancy car. Noone = zero people.
    - jcranmer 5y ago
      I think I understand what you're trying to say. Let me try to motivate algebra in less explicitly algebraic terms for you: Zero is an algebraic concept of nothing. While it refers to no physical thing, its existence in algebra is necessary to describe several mathematical laws, and several properties it has in algebra inherently derives from its algebraic concept of nothing. Let's define both addition and multiplication [1]. Addition is an abstract representation of combination: you combine a pile of 2 things and a pile of 3 things to get a pile 5 things. Now zero is the model for what you can combine with anything else that doesn't do anything: combining a pile of 0 things (that is, nothing) with a pile of 3 things leaves you a pile of 3 things. Negative numbers represent undoing a combination: combine a pile of 5 things with a pile of -3 things (i.e., "take 3 things from the pile") leaves with a pile of 2 things. Negative numbers and zero numbers may not necessarily have a direct physical analogue, but by introducing them for algebraic purposes, things actually become simpler: you use the same terminology and logic to deal with both adding and removing things, or perhaps coming to the conclusion that in the end there's no net effect. Now, multiplication is scaling. A half a pile of 2 things is 1 thing. Scaling and combining interact with each other, too. Taking half of a pile of 3 things and half of a pile of 5 things is the same taking half of a pile of 8 things. Or I can say that doubling a pile and then adding another of the original is the same as tripling a pile (i.e., 2x + x = 3x). This is where things get interesting. We can do nothing by adding a pile of something and immediately taking it away, leaving us with what we started (i.e., 0 = a·x - a·x). From above, we can also see that that is the same as adding a pile whose scale is 0 (i.e., a·x - a·x = (a - a)·x). Simplifying the equation a bit, we end up with 0 = 0·x: multiplying by 0 must yield 0 to make both addition and multiplication make sense. So the concept of nothing times anything yielding nothing isn't a requirement of nothing itself, but it's a requirement of how addition and multiplication works, and how nothing itself interacts with those operations. Incidentally, the deeper you dive into mathematics, the more important you realize the concept of 0--of nothing--actually is. The most powerful ways to describe operations are based on how they arrive at doing nothing in interestingly nontrivial ways. And things that don't have ways to do nothing tend not to be very interesting structures to look at. [1] I'm alluding to a vector spaces here, although by glossing over the difference between scalars and vectors, it could also be viewed as rings instead.
  - hanche 5y ago
    Nobody needs no beers, therefore everybody needs at least one beer? Language is funny.
  - Someone 5y ago
    Given how long it took to figure out that zero exists I don’t think it is natural, or at least a lot less natural than 1, 2, etc. https://en.wikipedia.org/wiki/History_of_ancient_numeral_systems https://en.wikipedia.org/wiki/History_of_ancient_numeral_sys...: “Abstract numerals, dissociated from the thing being counted, were invented about 3100 BC” https://en.wikipedia.org/wiki/0#History https://en.wikipedia.org/wiki/0#History: “By 1770 BC, the Egyptians had a symbol for zero in accounting texts” That’s over a thousand years, and that’s only the use of zeroes as a placeholder inside numbers. The use of a lone ‘zero’ symbol for the number zero seems to have taken over 2,000 more years (https://en.wikipedia.org/wiki/Brahmagupta#Zero https://en.wikipedia.org/wiki/Brahmagupta#Zero)
- edgyquant 5y ago
  Because multiplication is just the number of times you add something to 0. If you add 2 to 0 3 times you get 6, if you add 2 to 0 0 times you get zero.
  - paganel 5y ago
    I’ve mentioned in a comment above, I find nothing natural in doing operations related to zero i.e. related to nothingness, meaning that I see “adding” and “zero times” as an oxymoron (you cannot associate an action, “adding”, to nothingness, i.e. to “zero”).
    - orobinson 5y ago
      Well by that logic if you have nothing, there's nothing you can do to it, so you will always have nothing, which reconciles quite nicely with multiplying numbers by 0 resulting in 0.
      - mrkstu 5y ago
        Yep. Q: "How many times has it rained this week?" A: "It has rained zero times" Zero is a perfectly natural state.
        Someone 5y ago
        I think humans could predict solar eclipses and solve quite a few differential equations before they managed to give that answer. The normal answer to that question was “It hasn’t rained this week” or even “What do you mean? It hasn’t rained this week”. https://en.wikipedia.org/wiki/Brahmagupta#Zero https://en.wikipedia.org/wiki/Brahmagupta#Zero: ”The Brāhmasphuṭasiddhānta is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number” That book is from around AD 628 (https://en.wikipedia.org/wiki/Brāhmasphuṭasiddhānta https://en.wikipedia.org/wiki/Brāhmasphuṭasiddhānta)
        5y ago
- bena 5y ago
  It might be better to conceptualize negative numbers as numbers in a different direction or along a different axis. If I cover lunch for you, you would owe me $5. In a way, you now have -$5. You won't have $0 until you pay me back the $5 you owe. We can also say you have $5 of debt. That makes the number positive while still representing the amount. But when reconciling your books, it'll be subtracted from your total.
- agumonkey 5y ago
  I can envision relativity if our brain has relative anatomic constructs. That said the symbolic idea is a source of struggle even for teens.
- dogecoinbase 5y ago
  I would highly recommend Paul Benacerraf's article "What Numbers Could Not Be" for a philosophical if somewhat technical take on the subject (PDF link): http://michaeljohnsonphilosophy.com/wp-content/uploads/2015/08/Benacerraf.pdf http://michaeljohnsonphilosophy.com/wp-content/uploads/2015/... The thesis, broadly, revolves around contemplation about whether an number is an object, but the larger question is whether it's possible to have any small part of mathematics, even a single integer, which doesn't imply the whole (or at least, a significant structure).
  - paganel 5y ago
    Thanks a lot for the link/reference, that's the sort of stuff I was looking for.
- pharrington 5y ago
  How much do you add to five to get two? edit: another intuition is that ordering is a property of numbers. Positive numbers are ordered in one direction. Can numbers be ordered in other directions?
- gorgoiler 5y ago
  It’s fun to posit these ideas by going in the intuitive direction, then going in the reverse direction and seeing what happens. I recently found this to be helpful when investigation fixed and floating point numbers. If ”1011” means 2^3 + 2^1 + 2^0, which is 11 the “1011.101” means the same thing but with an additional 2^-1 + 2^-3 aka 5/8. Negative powers seem weird to begin with but it kind of just is there to just discover, due to the symmetry. This kind of arithmetic is far less fundamental than what you and the article are talking about — it is just representation really, in computers — but I think it is a good example of how if you can tread a path in one direction then turning around and coming back to where you started then carrying on in the other direction is a useful tool for teaching and learning.
- hcarvalhoalves 5y ago
  I believe the concept of negative number doesn't arise naturally from counting, you need equations – even though we teach it all at once to kids when introducing the number line.
  - feanaro 5y ago
    You don't really need equations, you just need missing elements, or debt. Do I have extra bricks, just enough (0 extra) of them or am I missing some? The case of missing bricks is naturally modelled with negative numbers.
- reggieband 5y ago
  I'm not sure if it is what you want, but Intuitionism [1] is one area that challenges modern fashions in Mathematical thinking. It suffered greatly under the formalist approach lead by David Hilbert and still has little main-stream support despite Gödel and his incompleteness proofs. Veritasium's latest video on Gödel's incompleteness [2] gives a pretty fair account of how we settled on the current fashionable foundations of Math (including nods to Cantor and Hilbert). For a more formal history there is a book "The Philosophy of Set Theory" [3] that sketches out how we got to where we are. I have always been unsatisfied with the current foundations of math and their obtuse basis in Set Theory. Although I must admit, Category Theory has alleviated that quite a lot, especially with the relaxing of equivalence compared to equality. 1. https://en.wikipedia.org/wiki/Intuitionism https://en.wikipedia.org/wiki/Intuitionism 2. https://www.youtube.com/watch?v=HeQX2HjkcNo https://www.youtube.com/watch?v=HeQX2HjkcNo 3. https://www.maa.org/press/maa-reviews/the-philosophy-of-set-theory-an-historical-introduction-to-cantors-paradise https://www.maa.org/press/maa-reviews/the-philosophy-of-set-...
  - eigenket 5y ago
    I am somewhat familiar with the standard foundation of maths in set theory, and slightly less familiar with the program to ground maths in Category theory (although I do know a fair bit of category theory). Maybe I am biased but I do not find the category theory foundations any less obtuse than the standard formulations. Like its pretty cool that you can do this stuff with category theory, but I think there is a reason the set theory was done first.
    - reggieband 5y ago
      I didn't meant to imply that Category Theory addressed the obtuseness of Set Theory. Rather, I was alluding to work in Category Theory that helps to redefine our ideas of equality and equivalence. A discussion of that is available in Quanta Magazine [1]. 1. https://www.quantamagazine.org/with-category-theory-mathematics-escapes-from-equality-20191010/ https://www.quantamagazine.org/with-category-theory-mathemat...
- saeranv 5y ago
  I prefer to think of natural numbers as a 1D vector. Negative numbers are therefore just a function of the direction of the vector. I don't have a good explanation of what zero is in this context... Would someone else have a way to explain it (i.e maybe related to the nullspace?).
- Ekaros 5y ago
  Start with then fingers. Then lose two. You have now 8 fingers. What is the number of lost fingers if not negative number.
  - falcor84 5y ago
    What do you mean? The number of "lost" fingers is 2.
- cabalamat 5y ago
  > I can understand how natural numbers can be “constructed” (for lack of a better word) as a byproduct of counting So you start off with zero, and the successor to zero, and the successor to that number, etc, and they're the counting numbers... > what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero. ...and there are all sorts of numbers that aren't counting numbers, but can be manipulated by the same rules of arithmetic. E.g.: h*2=1 -- fractions n+1=0 -- negative numbers r*r=2 -- irrational numbers i*i+1=0 -- imaginary numbers
- falcor84 5y ago
  As a kid, I recall that the thing that really made negative numbers click for me was underground floors, particularly when pressing on elevator buttons.
- joshuaissac 5y ago
  We can informally derive other types of number by extending operations on our existing set of natural numbers, seeking a "closure" for that operation, and seeing if we get consistent results. So extend the concept of differences between natural numbers, by subtracting a large number from a smaller number and defining the result as belonging to a new class of number. Similarly, get fractions by defining non-integer ratios between integers, real numbers by defining non-fractional limits of infinite sums of fractions, imaginary numbers of defining non-real roots of polynomials with real co-efficients, and so on. Each time, we have an operation that works for some subset of our numbers, so we look at what does not work, and see if we can make it work anyway by defining the result of such an operation as a new type of number. And then we repeat with a different operation. But there are things we cannot consistently "extend", like division by zero, or 0^0, so we leave just those out.
  - lupire 5y ago
    0^0 is near universally regarded as 1, except in specific domains. If you define your terms carefully, you usually get a reasonable answer. 0^-1 = 0^0/0 = 1/0 is indeterminate, as are all the negative powers of 0. https://www.maa.org/book/export/html/116806 https://www.maa.org/book/export/html/116806
- jasonwatkinspdx 5y ago
  So I'm no philosopher of math, but the intuitive way I think of it is that negative numbers are like borrowing a place holder. Think about how electrical charge and currents work at the physical level. There's electrons, which have negative charge, or holes, which have positive charge. Holes are just an empty place an electron can go, rather than an extant particle. Similarly, when we think of negative numbers in relation to counting numbers, we're just using a notational trick to keep track of a place holder or hole, a spot where a unary count can potentially go later to cancel it out. There's a pretty fascinating book named Quantum Computing Since Democritus by Scott Aaronson, one of the leaders in that field. The idea of the book is "Could the ancient greeks have discovered quantum mechanics?" Much of the higher level math in the book is past my familiarity, but the central theme is clear and intuitive: if you take ordinary probabilities, generalize them to allow negative probabilities, and then generalize those to allow complex numbers, out pops quantum mechanics quite naturally. Why do these two generalization steps make sense? What the heck is a negative probability of an event? It's exactly just notational borrowing in the same sense as above. Why generalize to complex numbers? That's more tricky, but I think of it from two directions: 1. It allows you to model partial constructive and destructive interference of probabilities, rather than just simple union or intersection. 2. Complex numbers are algebraically closed, while more simple numbers are not. So it feels natural that our ultimate number system to model nature would need to extend all this way. I realize some math heavy folks would find my way of thinking of this a bit hand wavy, but it really has helped me cut through the confusion and mystery. It also fits in very well with a bayesian perspective on probabilities. This, the "fields are what's real" perspective on physics, and bayesian epistemology in general have greatly simplified the way I think of these big idea question topics.
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- diegoperini 5y ago
  At t0, you say zero. At t1, you say one. ... At t7, you say seven. Positive seven is what you say while your clock says t7. Negative seven is what you plan and "will" say at t7 while your clock still says t0.
- TheOtherHobbes 5y ago
  The spatial metaphor is the number line. Positive operations and numbers move you to the right. Negative operations and numbers move you to the left. So a negative number is simply a count of a change in direction. This works very literally. Ten miles west followed by five miles east is five miles west. There's nothing mysterious or weird about the "absence of westness" when you've turned around and started off in the opposite direction. It generalises neatly to complex numbers where i is a rotation in the complex plane - instead of being pointlessly-weird-for-the-sake-of-it: "the square root of -1 which we've spent years telling you can't exist, and now we're telling you it can." Technically this is a form of basis vector. Beyond that things get complicated. But the idea is still valid - you define your direction markers (even if they're functions instead of constants) and then you can work out where you are in the space you're exploring, and what "movement", "counting" and "position" mean in that space. Multiplying by zero is always a null movement of no distance.
- dhosek 5y ago
  On multiplication, it's better to not think of multiplication as being repeated additions, but instead to think of it in terms of areas. Taking that approach, it becomes easier to consider, e.g., (x² + 2x - 1)×(3x² -4x + c) since we can write the terms along the sides of a rectangle and divide the rectangle into the products of the terms. One of the things I did in my math for liberal arts majors class I used to teach was also to do a fictionalized version of the expansion of the concept of numbers from *N* to the whole numbers to integers to rationals to algebraic numbers to reals to complex numbers. I say fictionalized because whole numbers and negative integers historically come after (positive) rationals, algebraic numbers and transcendental reals, but for pedagogical purposes, its easier to go in order of increasing supersets.
- mejutoco 5y ago
  From what I read, negative numbers took a while to be accepted for the same reason, so you are not alone ;) IMO negative numbers, complex numbers for circles, etc can be thought of as practical overloads. If we agree that the symbol minus means a debt, we can use it like that. We could agree, on *5 as meaning something, and if enough find it useful in 200 years we will teach it in school as obvious. I think the negative number is obviously useful (you can add up positive and negative numbers and get a balance, it has an easy way to be plotted, etc) P.S. another example is the Orbifold notation, from Conway. https://en.wikipedia.org/wiki/Orbifold_notation https://en.wikipedia.org/wiki/Orbifold_notation
SquibblesRedux 5y ago
While the post covers quite a bit of ground, it feels (to me) like it conflates knowledge representation, language, biological systems (i.e., the messiness of implementation), computability, and realism. Regarding numbers in particular, there are a practically uncountably infinite number of mathematical truths that apply equally to numbers or to other abstract (non-numerical) mathematical ideas. I would rather see depth in one area or another, rather than a conflation of ideas providing food for thought. Otherwise there are far too many variables to consider for a worthwhile analysis.
- contravariant 5y ago
  Because of the limitation of language there's only a countable number of mathematical truths that can be proven or written down. So for all practical purposes there's countably many.
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  - Aardwolf 5y ago
    Could you prove for every real value between 0 and 1 that it's greater than or equal to 0? That's an uncountable amount of proofs
    - kevinventullo 5y ago
      No, because proofs have to consist of a finite number of words. Thus there are only countably many proofs of anything. In particular, there are only countably many reals between 0 and 1 which can be expressed in a finite number of words.
      - c22 5y ago
        Are there a finite number of words? Language seems to grow and adapt to new concepts as needed, perhaps there is an infinity of linguistic descriptions available to us.
ezconnect 5y ago
When my two year old can't count the items in front of him he just says it's "many".
- a9h74j 5y ago
  IIRC (edit: and as alluded in the article) there are languages which count as "one, two, three, many". But one two and three are also innate (immediately recognizable) quantities are they not? So does a person using such a number system actually count, or recognize and categorize only?
  - perardi 5y ago
    This Lexicon Valley podcast goes into the “having a vocabulary for large numbers is actually weird” thing. https://slate.com/podcasts/lexicon-valley/2021/03/english-language-not-normal https://slate.com/podcasts/lexicon-valley/2021/03/english-la...
    - a9h74j 5y ago
      Great addition (no pun intended), thanks.
  - ezconnect 5y ago
    Depends on context maybe, Koreans have different number names depending on context.
  - desas 5y ago
    The piraha language is probably the most famous non-counting language. They appear to recognise - https://slate.com/human-interest/2013/10/piraha-cognitive-anumeracy-in-a-language-without-numbers.html https://slate.com/human-interest/2013/10/piraha-cognitive-an...
- agumonkey 5y ago
  One two infinity
ManBlanket 5y ago
‘If a lion could speak, we could not understand him’. - Wittgenstein So even things we understand as fundamental truths of the universe are so deeply rooted in human experience that it's possible another living entity could be so different than ourselves that the capacity to sympathize with one another about certain things may simply not exist. Would a lion understand a corsage? Would an alien know what 12 x 12 means? The universe is just one hella big ¯\_( ͡° ͜ʖ ͡°)_/¯
- slver 5y ago
  When we train neural networks, we don't tell them what to think and how to think about it. We mostly focus on overall structure, layers and we want useful outcomes. And yet we see lots of patterns that repeat ways in which we think. I think if a lion could speak, we'd understand him, because we're much closer to "a speaking lion" ourselves than we realize. Our intelligence didn't just happen at the last mile between ape and humans. Our intelligence has been emerging from the simplest animal there is throughout our entire evolutionary history. And even then we see birds, octopuses and so on use tools and solve problems very similarly to us, which have branched much earlier from us. There are many different types of minds that can evolve. But I believe the fundamentals will always be the same. They're fundamentals of information processing systems. And BTW, we know dolphins, parrots and so on can do basic math, like counting, and translating that count to another set of objects. So they understand numbers, that's what numbers are at their most basic. We marvel at how enormous the universe is, and how varied life in it can be, how different the minds of aliens may be. But in that seemingly humble pondering, there underlies something very arrogant. We think our mind is special. We think it's unique. And other minds will think extremely differently. I think we'll figure out sooner or later, that our mind is basically inevitable, the broad strokes will be replicated for every species throughout the universe. This doesn't mean it'll be immediately easy to figure out aliens and their culture. But those are the details, not the fundamentals.
  - coder-3 5y ago
    The way I see the universe is that all this complexity is just the interplay of a finite few axioms (the fundamental laws of physics). Given enough time and scale, the seemingly unique complexity converges back into a few (relatively speaking) patterns. If that wasn't the case the universe would be pure chaos.
slver 5y ago
I like that he keeps it open-minded, but I've thought a lot about this and to me the case for numbers being inevitable is decisively yes. Every adaptable system evolves to adapt to a changing environment by detecting "modes" (categories) and adapting to each mode. Then we start noticing categories come in instances. There's a tree, there are more trees. So now it's useful to count them... Then it's useful to have fractions. And measurements (like weight and length). Basic operations like + - emerge, and with that multiplication. And with that division... and so on. I think the fundamental properties of math are one of those things that will keep emerging for every thinking system. Not because numbers are fundamental to the universe. Rather they're fundamental aspect of intelligently adapting to (i.e. thinking about and making predictions about) the universe.
- mrfox321 5y ago
  Your argument is tautological. Do we really know if adaptation is responding to discrete classifications of environmental pressure? There could be other ways to describe how systems adapt. You are ascribing a lot to how you think it emerged. I think your opinions are great, but I just have a hard time having beliefs about things that are as fundamental as numbers.
  - swagasaurus-rex 5y ago
    Another perspective could be that we are creatures emergent out of the physical and chemical processes of the universe. There is some similarity or equivalence to some of these processes with mathematics, which allows us to model a subset of physics/chemistry with mathematics. Because physics/chemistry can create us using rules that follow math (or even some undiscovered rules), this same physics/chemistry gives us the faculties of reason that lets us think about and apply math in some universal way The physical, chemical, and mathematical substrate to which we are born allow us to even discover that the rules of math can be applied to some subset of the laws (or chaos?) of the universe.
    - mensetmanusman 5y ago
      Physics creates us, universe creates physics, … creates universe
agumonkey 5y ago
And memory
pjettter 5y ago
The word "finger" did not occur so I skipped it.
- alcover 5y ago
  It's certainly a big reason why we're so adept at counting but don't you think speech plays a big role too (and could be sufficient) ? I mean numbers are fingers but they're also a kind of song "one two three.." you map on things.
  - dane-pgp 5y ago
    > also a kind of song "one two three.." you map on things. Apparently it's as easy as "A, B, C...".
- dang 5y ago
  "Please don't post shallow dismissals, especially of other people's work. A good critical comment teaches us something." https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html
aaroninsf 5y ago
"The brain does much more than just recollect; it inter-compares, it synthesizes, it analyzes. It generates abstractions. The simplest thought like the concept of the number one has an elaborate logical underpinning; the brain has its own language for testing the structure and consistency of the world." Thank you, Carl. Sit down Stephen.
bluenose69 5y ago
My guess is that the aliens of which Wolfram speaks would know about numbers. After all, animals do (see e.g. https://www.bbc.com/future/article/20121128-animals-that-can-count https://www.bbc.com/future/article/20121128-animals-that-can...).
- typon 5y ago
  What if that's only a feature of animals that went through biological evolution on earth?
- djhonovak 5y ago
  Apes together count.
domrally 5y ago
We use the natural numbers as an abstraction to understand computation. Arithmetic and number theory have isomorphisms to other areas of mathematics. The proofs-as-programs equivalence shows that math and computation are the same. So i like to think numbers are how we perceive the computational aspect of reality. https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...
Bancakes 5y ago
NASA found it simpler to explain numbers with hydrogen atoms than peano arithmetic for their voyager.
- babelfish 5y ago
  Interesting. Any source or context on this?
  - Frost1x 5y ago
    Not OP but I'm guessing OP is referring to the Voyager Golden Record attached to the first Voyager probe, specifically the playback instructions using hydrogen atom to derive time units for playing the record: https://en.m.wikipedia.org/wiki/Voyager_Golden_Record#Playback https://en.m.wikipedia.org/wiki/Voyager_Golden_Record#Playba... Many of the ideas for the record came from Carl Sagan and a committee he lead working with NASA.
- mrfox321 5y ago
  Does "easier" equate to "universal". I think it does, just thinking about your statement out loud. Since to humans, explaining numbers with hydrogen would be harder, I think. The common ground between humans is larger, so we can rely on something less fundamental and abstract.
  - TeMPOraL 5y ago
    This was about finding some kind of common understanding to anchor the rest of the message to. Hydrogen, being the most abundant element in the universe, is likely to be known and studied by any civilization advanced enough to detect Voyager or its signals.
gmuslera 5y ago
If Mathematics is a language to describe reality, maybe numbers are not its whole alphabet. Inherent complexity, things that are not discrete, and emergent properties may not be described adequately with numbers, and maybe a different alphabet or even language is needed. Numbers may be (or not, it may depend on our biology) a good initial concept, but maybe something else may be developed, something more "correct" to deal with the tasks of describing reality, something like metric vs imperial units. The idea of integrating time to your vision of reality in the way it is used in the Ted Chiang's The story of your life (the movie Arrival was based on it) could be a good approach.
- eigenket 5y ago
  Almost all of modern mathematics is not about numbers, although a lot of the objects studied can eventually be related to numbers in some way.
  - TheOtherHobbes 5y ago
    It's not about numbers, but it's built on concepts which wouldn't exist without numbers as a foundational abstraction. There may be entire categories of representational concepts which we abstract poorly, or don't abstract at all, or possibly don't perceive at all - because from our point of view the relationships are so complex and remote they're effectively invisible.
    - darkscape 5y ago
      I've been thinking about this since I read about the Pirahã people. The concept of number enables engineering and business (accounting) (and perhaps timekeeping). Someone unfamiliar with numbers would find it very difficult to function in modern society and a group of such people wouldn't build modern civilization (unless they discovered numbers, which someone did at some point, of course). Now, here's an example of my crazy dream: imagine the possibility of currently inaccessible ways of perceiving or thinking, that, if they were to become available, would enable a four year old child to gain an understanding of, for example, elementary particle physics from zero to phd level in mere minutes. Of course, that way of thinking might be so different that said understanding might make no use of (our current) mathematics or the concept of elementary particle at all.
ffhhj 5y ago
Every number is a random number.
mosseater 5y ago
To me the concept of numbers is inherent in the dualistic minds we all have. If there is "me" and "other", there is 1 and 1, together making 2... a grouping of similar "others" is 3, and so on. It's simply just our nature.
breck 5y ago
All structures in our universe are trees^. Numbers are a metalanguage for describing those trees. It is possible that there are other independent universes that we can't perceive, but I'd expect any aliens that we eventually interact with to be operating only in the treeverse, and so also be fluent in a language like our numbers. ^ Perhaps there are other independent universes out there that we don't perceive, but human brains are trees and all structures we can perceive and communicate about also are trees. We live in a treeverse.
- eigenket 5y ago
  I genuinely don't understand what this means. How a beach ball a tree? What about the symmetry group of the beach ball?
  - breck 5y ago
    > How a beach ball a tree? _ / \ | | \ _ /
    - eigenket 5y ago
      That is neither a beach ball, nor a tree.
      - swagasaurus-rex 5y ago
        This is some serious science happening.
        breck 5y ago
        The search for nature's joints.
geijoenr 5y ago
"We take in some visual scene. But when we describe it in human language we’re always in effect coming up with a symbolic description of the scene." Human mind works in a qualitative manner, we need symbols to translate qualitative perception into a quantitative abstraction. This started as an economic and social organization need (geometry), but later evolved into Mathematics and got to overcome the shortcomings of natural language to describe reality (philosophy became science). Numbers are just symbols that map human perception to a reality that is inherently quantitative. I fail to grasp what Mr. Wolfram tries to explain here, but it looks to me as if he is regressing into philosophy.
amai 5y ago
"God made the integers, all else is the work of man. " Leopold Kronecker
- eigenket 5y ago
  A more modern mathematician might say something like "God made the empty set, all else is the work of man".
hyperpallium2 5y ago
Arguably, number came from money. Not just the counting, but the generalization/fungibility - money can be exchanged for anything (unlike barter); number can represent a quantity of anything. Trade and money have more direct survival advantages than number, an evolutionary gradient for improving the cognitive capacity supporting this generalization
- mrfox321 5y ago
  Shouldn't language predate money? Language or cave paintings or whatever, use discrete symbols to describe physical reality
  - hyperpallium2 5y ago
    Interesting! You can have 1:1 between symbols and objects, i.e. without generalization. Sidenote: the oldest writing found (so far) is of accounting...
novaRom 5y ago
Numbers is the consequence of conservation law. If you relax the definition of the natural number as something that can be stable at time and space, you will see another system is possible. Gosh, even periodic table is a good example of what is possible if no strict proton-oriented association is in mind.
marcodiego 5y ago
There is a brazilian tribe (piranhã) that knows no concept of quantity besides one, two an d many. Also, in their language, verbs are not flexed related to time. This is probably du e to their lifestyle that needs no long planning, discussions about the past or managing m ultiple instances of the same resources.
- Tainnor 5y ago
  Basically every single claim about Pirahã (not Piranha) needs to be taken with a huge grain of salt. There's just not enough people who have studied the language and the claims are so strong and unparalleled that we really ought to have more evidence between making any conclusive statements.
  - marcodiego 5y ago
    Hmmm... I stand corrected. Although I lived in the Amazon region for a few years (1993 to 2002 in Rondônia), never formally studied any native language and mostly blindly believed the wikipedia article on Pirahã people[0] and probably mixed things from the Hopi language time controversy. Thanks for such info. Ironically, my English was mostly learnt from people from Europe who came there to conduct researches. This is amazing and sad at the same time. But the fact that some tribes can whistle names, words and entire phrases and use such skill to communicate while hunting in the jungle is something I was told from an native speaker, can't remember what tribe or language it was; probably Caripuna or Suruí. [0] https://en.wikipedia.org/wiki/Pirah%C3%A3_language#Verbs https://en.wikipedia.org/wiki/Pirah%C3%A3_language#Verbs [1] https://en.wikipedia.org/wiki/Hopi_time_controversy https://en.wikipedia.org/wiki/Hopi_time_controversy
    - Tainnor 5y ago
      I'm not claiming that what Everett et al. are saying is wrong, just that the claims are so spectacular that I'd like to see more evidence before I believe them.
- osrec 5y ago
  Interesting. I would have thought that just by looking at their fingers and toes they'd arrive at a set of 1 to 10 at least...
kazinator 5y ago
The concept of symbols is inevitable.
deleted 5y ago
[deleted]
codeismath 5y ago
I think numbers are wonderful and cool abstract thingies. We grow up being taught them (in my case Arabic numerals in base 10) and they become such part of our being that we start to think that we know what “5” is. Then I was introduced to Roman Numerals. Didn’t like those much. That one taught me that 4 is more related to 5 than it is 3. Then some of us learn a base 2 number system or base 16 number system and for a brief moment we question whether we actually knew what “5” was. And then we may learn about Peano and the successor(s…) of zero. Did we really know what “5” was then? What is five-ness really? Perhaps fiveness is a set within a set within a set… Every interaction our human minds have with fiveness is with a representation or encoding of 5. With each new representation, I’ve experienced fiveness slightly differently. But I like computers and math. For me, the best to experience oneness with the fiveness is through computation with Lambda Calculus and Church Numerals. Very fond of that encoding.
- routerl 5y ago
  In fact, you're talking about digits and numerals, not numbers. But this actually drives your point home, I think.
- lisper 5y ago
  > What is five-ness really? It is everything that has some relation to something that has that same relation to something that has that same relation to something that has that same relation to something that has that same relation to some unique thing that does NOT have that same relation to anything.
  - function_seven 5y ago
    Is this the spoken form of a Church Numeral or something?
    - lisper 5y ago
      Yes. Exactly. (You could also see it as a rendering of some of the Peano axioms.)
  - 5y ago
hypertele-Xii 5y ago
Saying that the universe is irreducible with pockets of reducibility is a total oxymoron. It's not "irreducible" if parts of it are reducible. "This wall is impenetrable, except for the holes there. Don't mind them I'm making a point here!"
- DudeInBasement 5y ago
  It is if I'm an idiot
- tlb 5y ago
  He's using a technical definition of irreducibility [https://en.wikipedia.org/wiki/Irreducibility https://en.wikipedia.org/wiki/Irreducibility] for which the claim is true. It doesn't mean "cannot be expressed more concisely at all".
taotau 5y ago
I've always thought of numbers, and the broader system of mathematics as equivalent to the Newtonian understanding of physics - an adequate tool to explain observed reality - but the edge cases that break the system are starting to pile up, and we are just waiting for someone to discover the Quantum/General Relativity theory of mathematics.
- rrao 5y ago
  As someone who has recently started diving deeper into math recently, that is very interesting and something I was totally unaware of. Could you point me to some of these edge cases?
alephnan 5y ago
As late as the development of any resource gathering society
benlivengood 5y ago
Repetition or iteration is fundamental to mathematics. Symbolic mathematics is iteratively applying axioms and schemas to elements of a language. Computation is iteratively applying rules or functions to sets. The correlation between numbers and iterations is strong; zero iterations is identity, one iteration is the unit of computing/application/doing-something. More iterations yield ordinal numbers, and there we are at numbers being inevitable. In theory a universe could be entirely continuous with no need to "compute" in any discrete steps but quantization in our universe suggests that numbers also have a place in correspondence to quantized aspects of reality.
dhosek 5y ago
When I used to teach a class called “Mathematics for Liberal Arts Majors,” the first class began with having students “draw six.¹” I refused to give more explicit instructions. It was fascinating to see the results that came out of this, along with the ideas about numbers that it sparked. 1. I have to confess that the choice of six was not arbitrary.
- andai 5y ago
  Why six?
  - havernator 5y ago
    If any students hear the word wrong, you'll get some really funny drawings. (my guess)
    - wyqydsyq 5y ago
      Or if any of the students are from New Zealand
      - Mandatum 5y ago
        NZ = Sux Aus = Sex
        wyqydsyq 5y ago
        Most Kiwi accents I've heard noticeably pronounced other vowels as "i". e.g. "dick" instead of "deck", "tinnis" instead of "tennis"
  - dhosek 5y ago
    Mathematically, 6 is the most interesting small number. For example, one might represent it as XXXXXX XXX XXX XX XX XX X XX XXX or the sides of a cube, hexagons or six-spoked wheels. Occasionally someone would draw a rectangular prism with sides of length 1, 2 and 3. I often had students who might also draw 六 or 陸 or ٦ or ৬ (I had a diverse student body).
galaxyLogic 5y ago
Numbers are not fundamental. What is fundamental is laws of arithmetic. 1 + 2 == 2 + 1. Laws like that can only be expressed in some language, which we call "numbers". But only by knowing and using such laws "numbers" become useful. So in some weird sense numbers allow us to define the laws and the laws define numbers. No?
d--b 5y ago
I guess one of the most fundamental difference between Wolfram’s model for fundamental physics and traditional physics is that Wolfram’s doesn’t have the concept of measure or of continuum at the fundamental level. Space and time, according to Wolfram’s model of the universe, are emerging properties of ‘the network’. Without such things as space and measures, there is no numbers in the fundamental “equations” that drive the system. While I am not a big fan of Wolfram’s physics (I think it’s a bit backwards that he took something that he knows very well and somehow finds that it’s how the universe works), I kind of like the idea that space and time emerge from something more computational. I think that’s what the article should have been focused on...
- Barrin92 5y ago
  >I think it’s a bit backwards that he took something that he knows very well and somehow finds that it’s how the universe works That's in a sense how every model works. All theoretical models of the world are human inventions that aid in making sense of the world in terms familiar to us and nothing more. If you get good results from thinking the world is made of atoms, then the world is made out of atoms. When someone comes along and explains the same thing with strings, then the world is made out of strings. Models are just 'manners of speaking'. It might very well be that you have a dozen entirely different, but equally accurate fundamental ways to talk about a thing.
  - d--b 5y ago
    Well yes in a way. But it somehow makes the claim more dubious. It’s like when you go see an osteopath and they tell you that your stomach problem is due to your bad posture...
- ganafagol 5y ago
  It's so sad to see that he is taking an interesting question and turns it into a sales pitch of his computational universe theory. It's as if Stephen Wolframs mental horizon starts getting more and more restricted over the years since he tries to frame everything he sees in terms of his physical theory.
ultrablack 5y ago
Numbers are inevitable. How many people are we? How many berries do we need to collect for each. Later: How many seeds do we need to sow. How big a foundation is needed for this building… How many suns until the weather gets warmer again? I suppose you could do without if you lived in a cave with endless food readily available outside.
- chmod775 5y ago
  > How many people are we? We are Steven, Simon, and Sarah. > How many berries do we need to collect for each. A handful for each. > How big a foundation is needed for this building. What building? You haven't even built the foundation yet. And so on. You can do without numbers. And in fact people did all of the above without measuring quantities in any meaningful way in the history of humanity.
daxfohl 5y ago
Odd, I was thinking about this earlier today. I came to the conclusion that an axiomatization of math sans numbers doesn't make sense because you have to have a certain number of axioms. I'm certain a counterargument could be made though.
- LightMachine 5y ago
  Well [Kind](https://github.com/uwu-tech/kind https://github.com/uwu-tech/kind) has only one "axiom" (the lambda), so, I don't know? This is a really inspiring thought and I'm glad people are debating it for the content and not attacking Wolfram just because.
avereveard 5y ago
Eh idk something that moves uses energy and something that uses energy needs to replenish said energy. A natural question becomes how many recharge traveling a path needs, of course you can think about it as positions, but the stops themselves are intrinsically countable
dsubburam 5y ago
There's the related concept of 'sortals', the preconditions for countability. From the Stanford encyclopedia of philosophy[1]: The three main ideas are that a sortal tells us what the essence of a thing is tells us how to count things of that kind, which requires knowing which things are different and which are the same tells us when something continues to exist, and when it goes out of existence [1] https://plato.stanford.edu/entries/sortals/ https://plato.stanford.edu/entries/sortals/
ganafagol 5y ago
Starts his article in the very first sentence with referencing another article written by himself where he praises his achievement of having invented interstellar travel during a hollywood movie shoot for which he was the science advisor and which showcased the language invented by him and called by his name. Stephen Wolfram is so self-centred, I sometimes wonder if it's actually a stage character invented by a really talented comedian.
thecounter 5y ago
In the beginning was the primordial distinction between darkness and light, nothing and something, before and after. Its name was the Unit, its faces named zero and one. And with the Unit came the Successor, the primordial operation, for what is cleaved may ever be joined. From one comes two, from two comes three, and ever on until forever and always, with each successor given a name as a number. And with these numbers came a set comprising them, and the set was Natural and good, and from it came many wondrous things. For from repetition of the Successor came Addition, and the set was closed under Addition, and the Counter saw that this was good. And from repetition of Addition came Multiplication, and the set was closed under Multiplication, and the Counter saw that this was good. And from repetition of Multiplication came Exponentiation, and the set was closed under Exponentiation, and the Counter saw that this was good. But if a thing can be done it can be undone. What is given can be taken away. If there is Addition there must be Subtraction. A shadow fell over the face of the Counter for under Subtraction the set was not closed. Yet the set of Natural numbers had its closure under Subtraction, and this closure was another set named Integers, and the Counter saw that the Integers were good. But if Addition of a Natural number has an inverse, so too must Multiplication by a Natural number, and this inverse was Division. A shadow passed again across the face of the Counter for under Division by a Natural number the set of Integers was not closed. Yet the set of Integers had its closure under Division by a Natural number, and the closure was another set named Rational numbers, and the Counter saw that the Rational numbers were good and rejoiced at their scope, for between any two Rational numbers was an infinity of other Rational numbers, each with its own name. But if Addition and Multiplication by Natural numbers have inverses, so too must Exponentiation, and indeed, so must the combination of Addition, Multiplication, and Exponentiation in a polynomial with Integer coefficients, and this inverse was the finding of Roots. A shadow passed again across the face of the Counter for almost never were the Roots of polynomials Rational. Yet the Roots of polynomials with Integer coefficients gave rise to a new set, the set of Algebraic numbers, and the Counter saw that the Algebraic numbers were good and rejoiced at their scope, for the Algebraic numbers have complexities that delight and amaze, and each has its own name. And yet. Almost no number is Algebraic. Almost every number belongs instead to a Transcendental realm where there are many terrors and almost nothing can be named.
- raphyjake 5y ago
deleted 5y ago
[deleted]
Razengan 5y ago
One of my favorite thought games is trying to imagine an alien species who started off with a different way of counting/measuring things. Say, maybe going with complex numbers right away.
sunstone 5y ago
Goedel showed that 'less than', 'equal to', and 'more than' (ie pairing) are essential. Individual natural numbers are cute but not compelling. Even if God created them.
doonesbury 5y ago
It's inevitable.

182 points by perardi 5y ago

I can understand how natural numbers can be “constructed” (for lack of a better word) as a byproduct of counting, what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero. Maybe related, while I can also partially understand multiplication (syntactic sugar for adding) I could never understand multiplication by zero, meaning how come when you multiply a number (no matter how big) by zero you get zero as a result. Maybe there’s some Wittgenstein-like material somewhere that will better explain this, in which case I’ll very happy for some references.

inglor_cz 5y ago
No need for Wittgenstein. Fish have 0 legs. So how many legs do N fish have? N*0 = 0. Now division by zero, that is Devil's idea.
ddlutz 5y ago
Isn't multiplication by 0 natural? If i'm at a party and everybody wants 2 beers, then if there is 1 person I need 2 beers (1 * 2), 2 people need 4 beers (2 * 2), but 0 people need 0 beers (0 * 2).
- paganel 5y ago
  “Zero people” and “needing” is an oxymoron, nothing (zero) can’t be associated to any natural thing (like “needing”), that’s what I was trying to say in my not so clear comment above. Writing this down I realized we’re still trying to write in fancier words what the pre-Socratics had a clear understanding of 2500+ years ago, and speaking of the Greeks is too bad that Wolfram didn’t mention Plato by name in the first few paragraphs, the chair example is practically taken from him (expecting a Parmenides quote would have probably been too much).
  - ezconnect 5y ago
    When you combine understanding language and numbers you will have a hard time.
  - babygoat 5y ago
    Say you have 5 bags with 5 apples in each. How many apples do you have? 5 x 5 = 25. If you have 5 empty bags, 5 x 0 = 0.
  - greydius 5y ago
    Noone needs a fancy car. Noone = zero people.
  - jcranmer 5y ago
    I think I understand what you're trying to say. Let me try to motivate algebra in less explicitly algebraic terms for you: Zero is an algebraic concept of nothing. While it refers to no physical thing, its existence in algebra is necessary to describe several mathematical laws, and several properties it has in algebra inherently derives from its algebraic concept of nothing. Let's define both addition and multiplication [1]. Addition is an abstract representation of combination: you combine a pile of 2 things and a pile of 3 things to get a pile 5 things. Now zero is the model for what you can combine with anything else that doesn't do anything: combining a pile of 0 things (that is, nothing) with a pile of 3 things leaves you a pile of 3 things. Negative numbers represent undoing a combination: combine a pile of 5 things with a pile of -3 things (i.e., "take 3 things from the pile") leaves with a pile of 2 things. Negative numbers and zero numbers may not necessarily have a direct physical analogue, but by introducing them for algebraic purposes, things actually become simpler: you use the same terminology and logic to deal with both adding and removing things, or perhaps coming to the conclusion that in the end there's no net effect. Now, multiplication is scaling. A half a pile of 2 things is 1 thing. Scaling and combining interact with each other, too. Taking half of a pile of 3 things and half of a pile of 5 things is the same taking half of a pile of 8 things. Or I can say that doubling a pile and then adding another of the original is the same as tripling a pile (i.e., 2x + x = 3x). This is where things get interesting. We can do nothing by adding a pile of something and immediately taking it away, leaving us with what we started (i.e., 0 = a·x - a·x). From above, we can also see that that is the same as adding a pile whose scale is 0 (i.e., a·x - a·x = (a - a)·x). Simplifying the equation a bit, we end up with 0 = 0·x: multiplying by 0 must yield 0 to make both addition and multiplication make sense. So the concept of nothing times anything yielding nothing isn't a requirement of nothing itself, but it's a requirement of how addition and multiplication works, and how nothing itself interacts with those operations. Incidentally, the deeper you dive into mathematics, the more important you realize the concept of 0--of nothing--actually is. The most powerful ways to describe operations are based on how they arrive at doing nothing in interestingly nontrivial ways. And things that don't have ways to do nothing tend not to be very interesting structures to look at. [1] I'm alluding to a vector spaces here, although by glossing over the difference between scalars and vectors, it could also be viewed as rings instead.
- hanche 5y ago
  Nobody needs no beers, therefore everybody needs at least one beer? Language is funny.
- Someone 5y ago
  Given how long it took to figure out that zero exists I don’t think it is natural, or at least a lot less natural than 1, 2, etc. https://en.wikipedia.org/wiki/History_of_ancient_numeral_systems https://en.wikipedia.org/wiki/History_of_ancient_numeral_sys...: “Abstract numerals, dissociated from the thing being counted, were invented about 3100 BC” https://en.wikipedia.org/wiki/0#History https://en.wikipedia.org/wiki/0#History: “By 1770 BC, the Egyptians had a symbol for zero in accounting texts” That’s over a thousand years, and that’s only the use of zeroes as a placeholder inside numbers. The use of a lone ‘zero’ symbol for the number zero seems to have taken over 2,000 more years (https://en.wikipedia.org/wiki/Brahmagupta#Zero https://en.wikipedia.org/wiki/Brahmagupta#Zero)
edgyquant 5y ago
Because multiplication is just the number of times you add something to 0. If you add 2 to 0 3 times you get 6, if you add 2 to 0 0 times you get zero.
- paganel 5y ago
  I’ve mentioned in a comment above, I find nothing natural in doing operations related to zero i.e. related to nothingness, meaning that I see “adding” and “zero times” as an oxymoron (you cannot associate an action, “adding”, to nothingness, i.e. to “zero”).
  - orobinson 5y ago
    Well by that logic if you have nothing, there's nothing you can do to it, so you will always have nothing, which reconciles quite nicely with multiplying numbers by 0 resulting in 0.
    - mrkstu 5y ago
      Yep. Q: "How many times has it rained this week?" A: "It has rained zero times" Zero is a perfectly natural state.
      - Someone 5y ago
        I think humans could predict solar eclipses and solve quite a few differential equations before they managed to give that answer. The normal answer to that question was “It hasn’t rained this week” or even “What do you mean? It hasn’t rained this week”. https://en.wikipedia.org/wiki/Brahmagupta#Zero https://en.wikipedia.org/wiki/Brahmagupta#Zero: ”The Brāhmasphuṭasiddhānta is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number” That book is from around AD 628 (https://en.wikipedia.org/wiki/Brāhmasphuṭasiddhānta https://en.wikipedia.org/wiki/Brāhmasphuṭasiddhānta)
        5y ago
bena 5y ago
It might be better to conceptualize negative numbers as numbers in a different direction or along a different axis. If I cover lunch for you, you would owe me $5. In a way, you now have -$5. You won't have $0 until you pay me back the $5 you owe. We can also say you have $5 of debt. That makes the number positive while still representing the amount. But when reconciling your books, it'll be subtracted from your total.
agumonkey 5y ago
I can envision relativity if our brain has relative anatomic constructs. That said the symbolic idea is a source of struggle even for teens.
dogecoinbase 5y ago
I would highly recommend Paul Benacerraf's article "What Numbers Could Not Be" for a philosophical if somewhat technical take on the subject (PDF link): http://michaeljohnsonphilosophy.com/wp-content/uploads/2015/08/Benacerraf.pdf http://michaeljohnsonphilosophy.com/wp-content/uploads/2015/... The thesis, broadly, revolves around contemplation about whether an number is an object, but the larger question is whether it's possible to have any small part of mathematics, even a single integer, which doesn't imply the whole (or at least, a significant structure).
- paganel 5y ago
  Thanks a lot for the link/reference, that's the sort of stuff I was looking for.
pharrington 5y ago
How much do you add to five to get two? edit: another intuition is that ordering is a property of numbers. Positive numbers are ordered in one direction. Can numbers be ordered in other directions?
gorgoiler 5y ago
It’s fun to posit these ideas by going in the intuitive direction, then going in the reverse direction and seeing what happens. I recently found this to be helpful when investigation fixed and floating point numbers. If ”1011” means 2^3 + 2^1 + 2^0, which is 11 the “1011.101” means the same thing but with an additional 2^-1 + 2^-3 aka 5/8. Negative powers seem weird to begin with but it kind of just is there to just discover, due to the symmetry. This kind of arithmetic is far less fundamental than what you and the article are talking about — it is just representation really, in computers — but I think it is a good example of how if you can tread a path in one direction then turning around and coming back to where you started then carrying on in the other direction is a useful tool for teaching and learning.
hcarvalhoalves 5y ago
I believe the concept of negative number doesn't arise naturally from counting, you need equations – even though we teach it all at once to kids when introducing the number line.
- feanaro 5y ago
  You don't really need equations, you just need missing elements, or debt. Do I have extra bricks, just enough (0 extra) of them or am I missing some? The case of missing bricks is naturally modelled with negative numbers.
reggieband 5y ago
I'm not sure if it is what you want, but Intuitionism [1] is one area that challenges modern fashions in Mathematical thinking. It suffered greatly under the formalist approach lead by David Hilbert and still has little main-stream support despite Gödel and his incompleteness proofs. Veritasium's latest video on Gödel's incompleteness [2] gives a pretty fair account of how we settled on the current fashionable foundations of Math (including nods to Cantor and Hilbert). For a more formal history there is a book "The Philosophy of Set Theory" [3] that sketches out how we got to where we are. I have always been unsatisfied with the current foundations of math and their obtuse basis in Set Theory. Although I must admit, Category Theory has alleviated that quite a lot, especially with the relaxing of equivalence compared to equality. 1. https://en.wikipedia.org/wiki/Intuitionism https://en.wikipedia.org/wiki/Intuitionism 2. https://www.youtube.com/watch?v=HeQX2HjkcNo https://www.youtube.com/watch?v=HeQX2HjkcNo 3. https://www.maa.org/press/maa-reviews/the-philosophy-of-set-theory-an-historical-introduction-to-cantors-paradise https://www.maa.org/press/maa-reviews/the-philosophy-of-set-...
- eigenket 5y ago
  I am somewhat familiar with the standard foundation of maths in set theory, and slightly less familiar with the program to ground maths in Category theory (although I do know a fair bit of category theory). Maybe I am biased but I do not find the category theory foundations any less obtuse than the standard formulations. Like its pretty cool that you can do this stuff with category theory, but I think there is a reason the set theory was done first.
  - reggieband 5y ago
    I didn't meant to imply that Category Theory addressed the obtuseness of Set Theory. Rather, I was alluding to work in Category Theory that helps to redefine our ideas of equality and equivalence. A discussion of that is available in Quanta Magazine [1]. 1. https://www.quantamagazine.org/with-category-theory-mathematics-escapes-from-equality-20191010/ https://www.quantamagazine.org/with-category-theory-mathemat...
saeranv 5y ago
I prefer to think of natural numbers as a 1D vector. Negative numbers are therefore just a function of the direction of the vector. I don't have a good explanation of what zero is in this context... Would someone else have a way to explain it (i.e maybe related to the nullspace?).
Ekaros 5y ago
Start with then fingers. Then lose two. You have now 8 fingers. What is the number of lost fingers if not negative number.
- falcor84 5y ago
  What do you mean? The number of "lost" fingers is 2.
cabalamat 5y ago
> I can understand how natural numbers can be “constructed” (for lack of a better word) as a byproduct of counting So you start off with zero, and the successor to zero, and the successor to that number, etc, and they're the counting numbers... > what I could never understood on a deeper level are negative numbers, I can’t see how a number i.e. a count can be lower than zero. ...and there are all sorts of numbers that aren't counting numbers, but can be manipulated by the same rules of arithmetic. E.g.: h*2=1 -- fractions n+1=0 -- negative numbers r*r=2 -- irrational numbers i*i+1=0 -- imaginary numbers
falcor84 5y ago
As a kid, I recall that the thing that really made negative numbers click for me was underground floors, particularly when pressing on elevator buttons.
joshuaissac 5y ago
We can informally derive other types of number by extending operations on our existing set of natural numbers, seeking a "closure" for that operation, and seeing if we get consistent results. So extend the concept of differences between natural numbers, by subtracting a large number from a smaller number and defining the result as belonging to a new class of number. Similarly, get fractions by defining non-integer ratios between integers, real numbers by defining non-fractional limits of infinite sums of fractions, imaginary numbers of defining non-real roots of polynomials with real co-efficients, and so on. Each time, we have an operation that works for some subset of our numbers, so we look at what does not work, and see if we can make it work anyway by defining the result of such an operation as a new type of number. And then we repeat with a different operation. But there are things we cannot consistently "extend", like division by zero, or 0^0, so we leave just those out.
- lupire 5y ago
  0^0 is near universally regarded as 1, except in specific domains. If you define your terms carefully, you usually get a reasonable answer. 0^-1 = 0^0/0 = 1/0 is indeterminate, as are all the negative powers of 0. https://www.maa.org/book/export/html/116806 https://www.maa.org/book/export/html/116806
jasonwatkinspdx 5y ago
So I'm no philosopher of math, but the intuitive way I think of it is that negative numbers are like borrowing a place holder. Think about how electrical charge and currents work at the physical level. There's electrons, which have negative charge, or holes, which have positive charge. Holes are just an empty place an electron can go, rather than an extant particle. Similarly, when we think of negative numbers in relation to counting numbers, we're just using a notational trick to keep track of a place holder or hole, a spot where a unary count can potentially go later to cancel it out. There's a pretty fascinating book named Quantum Computing Since Democritus by Scott Aaronson, one of the leaders in that field. The idea of the book is "Could the ancient greeks have discovered quantum mechanics?" Much of the higher level math in the book is past my familiarity, but the central theme is clear and intuitive: if you take ordinary probabilities, generalize them to allow negative probabilities, and then generalize those to allow complex numbers, out pops quantum mechanics quite naturally. Why do these two generalization steps make sense? What the heck is a negative probability of an event? It's exactly just notational borrowing in the same sense as above. Why generalize to complex numbers? That's more tricky, but I think of it from two directions: 1. It allows you to model partial constructive and destructive interference of probabilities, rather than just simple union or intersection. 2. Complex numbers are algebraically closed, while more simple numbers are not. So it feels natural that our ultimate number system to model nature would need to extend all this way. I realize some math heavy folks would find my way of thinking of this a bit hand wavy, but it really has helped me cut through the confusion and mystery. It also fits in very well with a bayesian perspective on probabilities. This, the "fields are what's real" perspective on physics, and bayesian epistemology in general have greatly simplified the way I think of these big idea question topics.
- deleted 5y ago
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diegoperini 5y ago
At t0, you say zero. At t1, you say one. ... At t7, you say seven. Positive seven is what you say while your clock says t7. Negative seven is what you plan and "will" say at t7 while your clock still says t0.
TheOtherHobbes 5y ago
The spatial metaphor is the number line. Positive operations and numbers move you to the right. Negative operations and numbers move you to the left. So a negative number is simply a count of a change in direction. This works very literally. Ten miles west followed by five miles east is five miles west. There's nothing mysterious or weird about the "absence of westness" when you've turned around and started off in the opposite direction. It generalises neatly to complex numbers where i is a rotation in the complex plane - instead of being pointlessly-weird-for-the-sake-of-it: "the square root of -1 which we've spent years telling you can't exist, and now we're telling you it can." Technically this is a form of basis vector. Beyond that things get complicated. But the idea is still valid - you define your direction markers (even if they're functions instead of constants) and then you can work out where you are in the space you're exploring, and what "movement", "counting" and "position" mean in that space. Multiplying by zero is always a null movement of no distance.
dhosek 5y ago
On multiplication, it's better to not think of multiplication as being repeated additions, but instead to think of it in terms of areas. Taking that approach, it becomes easier to consider, e.g., (x² + 2x - 1)×(3x² -4x + c) since we can write the terms along the sides of a rectangle and divide the rectangle into the products of the terms. One of the things I did in my math for liberal arts majors class I used to teach was also to do a fictionalized version of the expansion of the concept of numbers from *N* to the whole numbers to integers to rationals to algebraic numbers to reals to complex numbers. I say fictionalized because whole numbers and negative integers historically come after (positive) rationals, algebraic numbers and transcendental reals, but for pedagogical purposes, its easier to go in order of increasing supersets.
mejutoco 5y ago
From what I read, negative numbers took a while to be accepted for the same reason, so you are not alone ;) IMO negative numbers, complex numbers for circles, etc can be thought of as practical overloads. If we agree that the symbol minus means a debt, we can use it like that. We could agree, on *5 as meaning something, and if enough find it useful in 200 years we will teach it in school as obvious. I think the negative number is obviously useful (you can add up positive and negative numbers and get a balance, it has an easy way to be plotted, etc) P.S. another example is the Orbifold notation, from Conway. https://en.wikipedia.org/wiki/Orbifold_notation https://en.wikipedia.org/wiki/Orbifold_notation

SquibblesRedux 5y ago

While the post covers quite a bit of ground, it feels (to me) like it conflates knowledge representation, language, biological systems (i.e., the messiness of implementation), computability, and realism. Regarding numbers in particular, there are a practically uncountably infinite number of mathematical truths that apply equally to numbers or to other abstract (non-numerical) mathematical ideas. I would rather see depth in one area or another, rather than a conflation of ideas providing food for thought. Otherwise there are far too many variables to consider for a worthwhile analysis.

contravariant 5y ago
Because of the limitation of language there's only a countable number of mathematical truths that can be proven or written down. So for all practical purposes there's countably many.
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- Aardwolf 5y ago
  Could you prove for every real value between 0 and 1 that it's greater than or equal to 0? That's an uncountable amount of proofs
  - kevinventullo 5y ago
    No, because proofs have to consist of a finite number of words. Thus there are only countably many proofs of anything. In particular, there are only countably many reals between 0 and 1 which can be expressed in a finite number of words.
    - c22 5y ago
      Are there a finite number of words? Language seems to grow and adapt to new concepts as needed, perhaps there is an infinity of linguistic descriptions available to us.

ezconnect 5y ago

When my two year old can't count the items in front of him he just says it's "many".

a9h74j 5y ago
IIRC (edit: and as alluded in the article) there are languages which count as "one, two, three, many". But one two and three are also innate (immediately recognizable) quantities are they not? So does a person using such a number system actually count, or recognize and categorize only?
- perardi 5y ago
  This Lexicon Valley podcast goes into the “having a vocabulary for large numbers is actually weird” thing. https://slate.com/podcasts/lexicon-valley/2021/03/english-language-not-normal https://slate.com/podcasts/lexicon-valley/2021/03/english-la...
  - a9h74j 5y ago
    Great addition (no pun intended), thanks.
- ezconnect 5y ago
  Depends on context maybe, Koreans have different number names depending on context.
- desas 5y ago
  The piraha language is probably the most famous non-counting language. They appear to recognise - https://slate.com/human-interest/2013/10/piraha-cognitive-anumeracy-in-a-language-without-numbers.html https://slate.com/human-interest/2013/10/piraha-cognitive-an...
agumonkey 5y ago
One two infinity

ManBlanket 5y ago

‘If a lion could speak, we could not understand him’. - Wittgenstein So even things we understand as fundamental truths of the universe are so deeply rooted in human experience that it's possible another living entity could be so different than ourselves that the capacity to sympathize with one another about certain things may simply not exist. Would a lion understand a corsage? Would an alien know what 12 x 12 means? The universe is just one hella big ¯\_( ͡° ͜ʖ ͡°)_/¯

slver 5y ago
When we train neural networks, we don't tell them what to think and how to think about it. We mostly focus on overall structure, layers and we want useful outcomes. And yet we see lots of patterns that repeat ways in which we think. I think if a lion could speak, we'd understand him, because we're much closer to "a speaking lion" ourselves than we realize. Our intelligence didn't just happen at the last mile between ape and humans. Our intelligence has been emerging from the simplest animal there is throughout our entire evolutionary history. And even then we see birds, octopuses and so on use tools and solve problems very similarly to us, which have branched much earlier from us. There are many different types of minds that can evolve. But I believe the fundamentals will always be the same. They're fundamentals of information processing systems. And BTW, we know dolphins, parrots and so on can do basic math, like counting, and translating that count to another set of objects. So they understand numbers, that's what numbers are at their most basic. We marvel at how enormous the universe is, and how varied life in it can be, how different the minds of aliens may be. But in that seemingly humble pondering, there underlies something very arrogant. We think our mind is special. We think it's unique. And other minds will think extremely differently. I think we'll figure out sooner or later, that our mind is basically inevitable, the broad strokes will be replicated for every species throughout the universe. This doesn't mean it'll be immediately easy to figure out aliens and their culture. But those are the details, not the fundamentals.
- coder-3 5y ago
  The way I see the universe is that all this complexity is just the interplay of a finite few axioms (the fundamental laws of physics). Given enough time and scale, the seemingly unique complexity converges back into a few (relatively speaking) patterns. If that wasn't the case the universe would be pure chaos.

slver 5y ago

I like that he keeps it open-minded, but I've thought a lot about this and to me the case for numbers being inevitable is decisively yes. Every adaptable system evolves to adapt to a changing environment by detecting "modes" (categories) and adapting to each mode. Then we start noticing categories come in instances. There's a tree, there are more trees. So now it's useful to count them... Then it's useful to have fractions. And measurements (like weight and length). Basic operations like + - emerge, and with that multiplication. And with that division... and so on. I think the fundamental properties of math are one of those things that will keep emerging for every thinking system. Not because numbers are fundamental to the universe. Rather they're fundamental aspect of intelligently adapting to (i.e. thinking about and making predictions about) the universe.

mrfox321 5y ago
Your argument is tautological. Do we really know if adaptation is responding to discrete classifications of environmental pressure? There could be other ways to describe how systems adapt. You are ascribing a lot to how you think it emerged. I think your opinions are great, but I just have a hard time having beliefs about things that are as fundamental as numbers.
- swagasaurus-rex 5y ago
  Another perspective could be that we are creatures emergent out of the physical and chemical processes of the universe. There is some similarity or equivalence to some of these processes with mathematics, which allows us to model a subset of physics/chemistry with mathematics. Because physics/chemistry can create us using rules that follow math (or even some undiscovered rules), this same physics/chemistry gives us the faculties of reason that lets us think about and apply math in some universal way The physical, chemical, and mathematical substrate to which we are born allow us to even discover that the rules of math can be applied to some subset of the laws (or chaos?) of the universe.
  - mensetmanusman 5y ago
    Physics creates us, universe creates physics, … creates universe

agumonkey 5y ago

And memory

pjettter 5y ago

The word "finger" did not occur so I skipped it.

alcover 5y ago
It's certainly a big reason why we're so adept at counting but don't you think speech plays a big role too (and could be sufficient) ? I mean numbers are fingers but they're also a kind of song "one two three.." you map on things.
- dane-pgp 5y ago
  > also a kind of song "one two three.." you map on things. Apparently it's as easy as "A, B, C...".
dang 5y ago
"Please don't post shallow dismissals, especially of other people's work. A good critical comment teaches us something." https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html

aaroninsf 5y ago

"The brain does much more than just recollect; it inter-compares, it synthesizes, it analyzes. It generates abstractions. The simplest thought like the concept of the number one has an elaborate logical underpinning; the brain has its own language for testing the structure and consistency of the world." Thank you, Carl. Sit down Stephen.

bluenose69 5y ago

My guess is that the aliens of which Wolfram speaks would know about numbers. After all, animals do (see e.g. https://www.bbc.com/future/article/20121128-animals-that-can-count https://www.bbc.com/future/article/20121128-animals-that-can...).

typon 5y ago
What if that's only a feature of animals that went through biological evolution on earth?
djhonovak 5y ago
Apes together count.

domrally 5y ago

We use the natural numbers as an abstraction to understand computation. Arithmetic and number theory have isomorphisms to other areas of mathematics. The proofs-as-programs equivalence shows that math and computation are the same. So i like to think numbers are how we perceive the computational aspect of reality. https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...

Bancakes 5y ago

NASA found it simpler to explain numbers with hydrogen atoms than peano arithmetic for their voyager.

babelfish 5y ago
Interesting. Any source or context on this?
- Frost1x 5y ago
  Not OP but I'm guessing OP is referring to the Voyager Golden Record attached to the first Voyager probe, specifically the playback instructions using hydrogen atom to derive time units for playing the record: https://en.m.wikipedia.org/wiki/Voyager_Golden_Record#Playback https://en.m.wikipedia.org/wiki/Voyager_Golden_Record#Playba... Many of the ideas for the record came from Carl Sagan and a committee he lead working with NASA.
mrfox321 5y ago
Does "easier" equate to "universal". I think it does, just thinking about your statement out loud. Since to humans, explaining numbers with hydrogen would be harder, I think. The common ground between humans is larger, so we can rely on something less fundamental and abstract.
- TeMPOraL 5y ago
  This was about finding some kind of common understanding to anchor the rest of the message to. Hydrogen, being the most abundant element in the universe, is likely to be known and studied by any civilization advanced enough to detect Voyager or its signals.

gmuslera 5y ago

If Mathematics is a language to describe reality, maybe numbers are not its whole alphabet. Inherent complexity, things that are not discrete, and emergent properties may not be described adequately with numbers, and maybe a different alphabet or even language is needed. Numbers may be (or not, it may depend on our biology) a good initial concept, but maybe something else may be developed, something more "correct" to deal with the tasks of describing reality, something like metric vs imperial units. The idea of integrating time to your vision of reality in the way it is used in the Ted Chiang's The story of your life (the movie Arrival was based on it) could be a good approach.

eigenket 5y ago
Almost all of modern mathematics is not about numbers, although a lot of the objects studied can eventually be related to numbers in some way.
- TheOtherHobbes 5y ago
  It's not about numbers, but it's built on concepts which wouldn't exist without numbers as a foundational abstraction. There may be entire categories of representational concepts which we abstract poorly, or don't abstract at all, or possibly don't perceive at all - because from our point of view the relationships are so complex and remote they're effectively invisible.
  - darkscape 5y ago
    I've been thinking about this since I read about the Pirahã people. The concept of number enables engineering and business (accounting) (and perhaps timekeeping). Someone unfamiliar with numbers would find it very difficult to function in modern society and a group of such people wouldn't build modern civilization (unless they discovered numbers, which someone did at some point, of course). Now, here's an example of my crazy dream: imagine the possibility of currently inaccessible ways of perceiving or thinking, that, if they were to become available, would enable a four year old child to gain an understanding of, for example, elementary particle physics from zero to phd level in mere minutes. Of course, that way of thinking might be so different that said understanding might make no use of (our current) mathematics or the concept of elementary particle at all.

ffhhj 5y ago

Every number is a random number.

mosseater 5y ago

To me the concept of numbers is inherent in the dualistic minds we all have. If there is "me" and "other", there is 1 and 1, together making 2... a grouping of similar "others" is 3, and so on. It's simply just our nature.

breck 5y ago

All structures in our universe are trees^. Numbers are a metalanguage for describing those trees. It is possible that there are other independent universes that we can't perceive, but I'd expect any aliens that we eventually interact with to be operating only in the treeverse, and so also be fluent in a language like our numbers. ^ Perhaps there are other independent universes out there that we don't perceive, but human brains are trees and all structures we can perceive and communicate about also are trees. We live in a treeverse.

eigenket 5y ago
I genuinely don't understand what this means. How a beach ball a tree? What about the symmetry group of the beach ball?
- breck 5y ago
  > How a beach ball a tree? _ / \ | | \ _ /
  - eigenket 5y ago
    That is neither a beach ball, nor a tree.
    - swagasaurus-rex 5y ago
      This is some serious science happening.
      - breck 5y ago
        The search for nature's joints.

geijoenr 5y ago

"We take in some visual scene. But when we describe it in human language we’re always in effect coming up with a symbolic description of the scene." Human mind works in a qualitative manner, we need symbols to translate qualitative perception into a quantitative abstraction. This started as an economic and social organization need (geometry), but later evolved into Mathematics and got to overcome the shortcomings of natural language to describe reality (philosophy became science). Numbers are just symbols that map human perception to a reality that is inherently quantitative. I fail to grasp what Mr. Wolfram tries to explain here, but it looks to me as if he is regressing into philosophy.

amai 5y ago

"God made the integers, all else is the work of man. " Leopold Kronecker

eigenket 5y ago
A more modern mathematician might say something like "God made the empty set, all else is the work of man".

hyperpallium2 5y ago

Arguably, number came from money. Not just the counting, but the generalization/fungibility - money can be exchanged for anything (unlike barter); number can represent a quantity of anything. Trade and money have more direct survival advantages than number, an evolutionary gradient for improving the cognitive capacity supporting this generalization

mrfox321 5y ago
Shouldn't language predate money? Language or cave paintings or whatever, use discrete symbols to describe physical reality
- hyperpallium2 5y ago
  Interesting! You can have 1:1 between symbols and objects, i.e. without generalization. Sidenote: the oldest writing found (so far) is of accounting...

novaRom 5y ago

Numbers is the consequence of conservation law. If you relax the definition of the natural number as something that can be stable at time and space, you will see another system is possible. Gosh, even periodic table is a good example of what is possible if no strict proton-oriented association is in mind.

marcodiego 5y ago

There is a brazilian tribe (piranhã) that knows no concept of quantity besides one, two an d many. Also, in their language, verbs are not flexed related to time. This is probably du e to their lifestyle that needs no long planning, discussions about the past or managing m ultiple instances of the same resources.

Tainnor 5y ago
Basically every single claim about Pirahã (not Piranha) needs to be taken with a huge grain of salt. There's just not enough people who have studied the language and the claims are so strong and unparalleled that we really ought to have more evidence between making any conclusive statements.
- marcodiego 5y ago
  Hmmm... I stand corrected. Although I lived in the Amazon region for a few years (1993 to 2002 in Rondônia), never formally studied any native language and mostly blindly believed the wikipedia article on Pirahã people[0] and probably mixed things from the Hopi language time controversy. Thanks for such info. Ironically, my English was mostly learnt from people from Europe who came there to conduct researches. This is amazing and sad at the same time. But the fact that some tribes can whistle names, words and entire phrases and use such skill to communicate while hunting in the jungle is something I was told from an native speaker, can't remember what tribe or language it was; probably Caripuna or Suruí. [0] https://en.wikipedia.org/wiki/Pirah%C3%A3_language#Verbs https://en.wikipedia.org/wiki/Pirah%C3%A3_language#Verbs [1] https://en.wikipedia.org/wiki/Hopi_time_controversy https://en.wikipedia.org/wiki/Hopi_time_controversy
  - Tainnor 5y ago
    I'm not claiming that what Everett et al. are saying is wrong, just that the claims are so spectacular that I'd like to see more evidence before I believe them.
osrec 5y ago
Interesting. I would have thought that just by looking at their fingers and toes they'd arrive at a set of 1 to 10 at least...

kazinator 5y ago

The concept of symbols is inevitable.

deleted 5y ago

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codeismath 5y ago

I think numbers are wonderful and cool abstract thingies. We grow up being taught them (in my case Arabic numerals in base 10) and they become such part of our being that we start to think that we know what “5” is. Then I was introduced to Roman Numerals. Didn’t like those much. That one taught me that 4 is more related to 5 than it is 3. Then some of us learn a base 2 number system or base 16 number system and for a brief moment we question whether we actually knew what “5” was. And then we may learn about Peano and the successor(s…) of zero. Did we really know what “5” was then? What is five-ness really? Perhaps fiveness is a set within a set within a set… Every interaction our human minds have with fiveness is with a representation or encoding of 5. With each new representation, I’ve experienced fiveness slightly differently. But I like computers and math. For me, the best to experience oneness with the fiveness is through computation with Lambda Calculus and Church Numerals. Very fond of that encoding.

routerl 5y ago
In fact, you're talking about digits and numerals, not numbers. But this actually drives your point home, I think.
lisper 5y ago
> What is five-ness really? It is everything that has some relation to something that has that same relation to something that has that same relation to something that has that same relation to something that has that same relation to some unique thing that does NOT have that same relation to anything.
- function_seven 5y ago
  Is this the spoken form of a Church Numeral or something?
  - lisper 5y ago
    Yes. Exactly. (You could also see it as a rendering of some of the Peano axioms.)
- 5y ago

hypertele-Xii 5y ago

Saying that the universe is irreducible with pockets of reducibility is a total oxymoron. It's not "irreducible" if parts of it are reducible. "This wall is impenetrable, except for the holes there. Don't mind them I'm making a point here!"

DudeInBasement 5y ago
It is if I'm an idiot
tlb 5y ago
He's using a technical definition of irreducibility [https://en.wikipedia.org/wiki/Irreducibility https://en.wikipedia.org/wiki/Irreducibility] for which the claim is true. It doesn't mean "cannot be expressed more concisely at all".

taotau 5y ago

I've always thought of numbers, and the broader system of mathematics as equivalent to the Newtonian understanding of physics - an adequate tool to explain observed reality - but the edge cases that break the system are starting to pile up, and we are just waiting for someone to discover the Quantum/General Relativity theory of mathematics.

rrao 5y ago
As someone who has recently started diving deeper into math recently, that is very interesting and something I was totally unaware of. Could you point me to some of these edge cases?

alephnan 5y ago

As late as the development of any resource gathering society

benlivengood 5y ago

Repetition or iteration is fundamental to mathematics. Symbolic mathematics is iteratively applying axioms and schemas to elements of a language. Computation is iteratively applying rules or functions to sets. The correlation between numbers and iterations is strong; zero iterations is identity, one iteration is the unit of computing/application/doing-something. More iterations yield ordinal numbers, and there we are at numbers being inevitable. In theory a universe could be entirely continuous with no need to "compute" in any discrete steps but quantization in our universe suggests that numbers also have a place in correspondence to quantized aspects of reality.

dhosek 5y ago

When I used to teach a class called “Mathematics for Liberal Arts Majors,” the first class began with having students “draw six.¹” I refused to give more explicit instructions. It was fascinating to see the results that came out of this, along with the ideas about numbers that it sparked. 1. I have to confess that the choice of six was not arbitrary.

andai 5y ago
Why six?
- havernator 5y ago
  If any students hear the word wrong, you'll get some really funny drawings. (my guess)
  - wyqydsyq 5y ago
    Or if any of the students are from New Zealand
    - Mandatum 5y ago
      NZ = Sux Aus = Sex
      - wyqydsyq 5y ago
        Most Kiwi accents I've heard noticeably pronounced other vowels as "i". e.g. "dick" instead of "deck", "tinnis" instead of "tennis"
- dhosek 5y ago
  Mathematically, 6 is the most interesting small number. For example, one might represent it as XXXXXX XXX XXX XX XX XX X XX XXX or the sides of a cube, hexagons or six-spoked wheels. Occasionally someone would draw a rectangular prism with sides of length 1, 2 and 3. I often had students who might also draw 六 or 陸 or ٦ or ৬ (I had a diverse student body).

galaxyLogic 5y ago

Numbers are not fundamental. What is fundamental is laws of arithmetic. 1 + 2 == 2 + 1. Laws like that can only be expressed in some language, which we call "numbers". But only by knowing and using such laws "numbers" become useful. So in some weird sense numbers allow us to define the laws and the laws define numbers. No?

d--b 5y ago

I guess one of the most fundamental difference between Wolfram’s model for fundamental physics and traditional physics is that Wolfram’s doesn’t have the concept of measure or of continuum at the fundamental level. Space and time, according to Wolfram’s model of the universe, are emerging properties of ‘the network’. Without such things as space and measures, there is no numbers in the fundamental “equations” that drive the system. While I am not a big fan of Wolfram’s physics (I think it’s a bit backwards that he took something that he knows very well and somehow finds that it’s how the universe works), I kind of like the idea that space and time emerge from something more computational. I think that’s what the article should have been focused on...

Barrin92 5y ago
>I think it’s a bit backwards that he took something that he knows very well and somehow finds that it’s how the universe works That's in a sense how every model works. All theoretical models of the world are human inventions that aid in making sense of the world in terms familiar to us and nothing more. If you get good results from thinking the world is made of atoms, then the world is made out of atoms. When someone comes along and explains the same thing with strings, then the world is made out of strings. Models are just 'manners of speaking'. It might very well be that you have a dozen entirely different, but equally accurate fundamental ways to talk about a thing.
- d--b 5y ago
  Well yes in a way. But it somehow makes the claim more dubious. It’s like when you go see an osteopath and they tell you that your stomach problem is due to your bad posture...
ganafagol 5y ago
It's so sad to see that he is taking an interesting question and turns it into a sales pitch of his computational universe theory. It's as if Stephen Wolframs mental horizon starts getting more and more restricted over the years since he tries to frame everything he sees in terms of his physical theory.

ultrablack 5y ago

Numbers are inevitable. How many people are we? How many berries do we need to collect for each. Later: How many seeds do we need to sow. How big a foundation is needed for this building… How many suns until the weather gets warmer again? I suppose you could do without if you lived in a cave with endless food readily available outside.

chmod775 5y ago
> How many people are we? We are Steven, Simon, and Sarah. > How many berries do we need to collect for each. A handful for each. > How big a foundation is needed for this building. What building? You haven't even built the foundation yet. And so on. You can do without numbers. And in fact people did all of the above without measuring quantities in any meaningful way in the history of humanity.

daxfohl 5y ago

Odd, I was thinking about this earlier today. I came to the conclusion that an axiomatization of math sans numbers doesn't make sense because you have to have a certain number of axioms. I'm certain a counterargument could be made though.

LightMachine 5y ago
Well [Kind](https://github.com/uwu-tech/kind https://github.com/uwu-tech/kind) has only one "axiom" (the lambda), so, I don't know? This is a really inspiring thought and I'm glad people are debating it for the content and not attacking Wolfram just because.

avereveard 5y ago

Eh idk something that moves uses energy and something that uses energy needs to replenish said energy. A natural question becomes how many recharge traveling a path needs, of course you can think about it as positions, but the stops themselves are intrinsically countable

dsubburam 5y ago

There's the related concept of 'sortals', the preconditions for countability. From the Stanford encyclopedia of philosophy[1]: The three main ideas are that a sortal tells us what the essence of a thing is tells us how to count things of that kind, which requires knowing which things are different and which are the same tells us when something continues to exist, and when it goes out of existence [1] https://plato.stanford.edu/entries/sortals/ https://plato.stanford.edu/entries/sortals/

ganafagol 5y ago

Starts his article in the very first sentence with referencing another article written by himself where he praises his achievement of having invented interstellar travel during a hollywood movie shoot for which he was the science advisor and which showcased the language invented by him and called by his name. Stephen Wolfram is so self-centred, I sometimes wonder if it's actually a stage character invented by a really talented comedian.

thecounter 5y ago

In the beginning was the primordial distinction between darkness and light, nothing and something, before and after. Its name was the Unit, its faces named zero and one. And with the Unit came the Successor, the primordial operation, for what is cleaved may ever be joined. From one comes two, from two comes three, and ever on until forever and always, with each successor given a name as a number. And with these numbers came a set comprising them, and the set was Natural and good, and from it came many wondrous things. For from repetition of the Successor came Addition, and the set was closed under Addition, and the Counter saw that this was good. And from repetition of Addition came Multiplication, and the set was closed under Multiplication, and the Counter saw that this was good. And from repetition of Multiplication came Exponentiation, and the set was closed under Exponentiation, and the Counter saw that this was good. But if a thing can be done it can be undone. What is given can be taken away. If there is Addition there must be Subtraction. A shadow fell over the face of the Counter for under Subtraction the set was not closed. Yet the set of Natural numbers had its closure under Subtraction, and this closure was another set named Integers, and the Counter saw that the Integers were good. But if Addition of a Natural number has an inverse, so too must Multiplication by a Natural number, and this inverse was Division. A shadow passed again across the face of the Counter for under Division by a Natural number the set of Integers was not closed. Yet the set of Integers had its closure under Division by a Natural number, and the closure was another set named Rational numbers, and the Counter saw that the Rational numbers were good and rejoiced at their scope, for between any two Rational numbers was an infinity of other Rational numbers, each with its own name. But if Addition and Multiplication by Natural numbers have inverses, so too must Exponentiation, and indeed, so must the combination of Addition, Multiplication, and Exponentiation in a polynomial with Integer coefficients, and this inverse was the finding of Roots. A shadow passed again across the face of the Counter for almost never were the Roots of polynomials Rational. Yet the Roots of polynomials with Integer coefficients gave rise to a new set, the set of Algebraic numbers, and the Counter saw that the Algebraic numbers were good and rejoiced at their scope, for the Algebraic numbers have complexities that delight and amaze, and each has its own name. And yet. Almost no number is Algebraic. Almost every number belongs instead to a Transcendental realm where there are many terrors and almost nothing can be named.

raphyjake 5y ago

deleted 5y ago

[deleted]

Razengan 5y ago

One of my favorite thought games is trying to imagine an alien species who started off with a different way of counting/measuring things. Say, maybe going with complex numbers right away.

sunstone 5y ago

Goedel showed that 'less than', 'equal to', and 'more than' (ie pairing) are essential. Individual natural numbers are cute but not compelling. Even if God created them.

doonesbury 5y ago

It's inevitable.