[HN Gopher] How inevitable is the concept of numbers?
___________________________________________________________________
How inevitable is the concept of numbers?
Author : perardi
Score : 77 points
Date : 2021-05-25 17:46 UTC (5 hours ago)
(HTM) web link (writings.stephenwolfram.com)
(TXT) w3m dump (writings.stephenwolfram.com)
| mosseater wrote:
| 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.
| bluenose69 wrote:
| 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...).
| typon wrote:
| What if that's only a feature of animals that went through
| biological evolution on earth?
| agumonkey wrote:
| And memory
| domrally wrote:
| 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_correspon...
| geijoenr wrote:
| "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.
| Bancakes wrote:
| NASA found it simpler to explain numbers with hydrogen atoms than
| peano arithmetic for their voyager.
| mrfox321 wrote:
| 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.
| babelfish wrote:
| Interesting. Any source or context on this?
| Frost1x wrote:
| 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#Playba...
|
| Many of the ideas for the record came from Carl Sagan and a
| committee he lead working with NASA.
| paganel wrote:
| 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 wrote:
| 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.
| cabalamat wrote:
| > 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
| diegoperini wrote:
| 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.
| falcor84 wrote:
| 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.
| reggieband wrote:
| 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 Godel and his incompleteness proofs. Veritasium's
| latest video on Godel'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
|
| 2. https://www.youtube.com/watch?v=HeQX2HjkcNo
|
| 3. https://www.maa.org/press/maa-reviews/the-philosophy-of-
| set-...
| eigenket wrote:
| 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 wrote:
| 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-
| mathemat...
| saeranv wrote:
| 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 wrote:
| Start with then fingers. Then lose two. You have now 8 fingers.
| What is the number of lost fingers if not negative number.
| falcor84 wrote:
| What do you mean? The number of "lost" fingers is 2.
| dogecoinbase wrote:
| 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/...
|
| 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 wrote:
| Thanks a lot for the link/reference, that's the sort of stuff
| I was looking for.
| agumonkey wrote:
| I can envision relativity if our brain has relative anatomic
| constructs. That said the symbolic idea is a source of struggle
| even for teens.
| pharrington wrote:
| 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?
| ddlutz wrote:
| 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 wrote:
| "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).
| babygoat wrote:
| 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.
| ezconnect wrote:
| When you combine understanding language and numbers you
| will have a hard time.
| greydius wrote:
| Noone needs a fancy car. Noone = zero people.
| jcranmer wrote:
| 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.
| Someone wrote:
| 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_sys.
| ..: _"Abstract numerals, dissociated from the thing being
| counted, were invented about 3100 BC"_
|
| 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)
| hanche wrote:
| Nobody needs no beers, therefore everybody needs at least one
| beer?
|
| Language is funny.
| joshuaissac wrote:
| 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.
| edgyquant wrote:
| 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 wrote:
| 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").
| InfiniteRand wrote:
| I think historically there were a number of cultures who
| had trouble with the mathematical concept of zero, so it's
| not universal
| _Nat_ wrote:
| > 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").
|
| Seems related to [this
| problem](https://stackoverflow.com/questions/47783926/why-
| are-loops-a...).
|
| This is, a proper loop is like do while
| (condition) { // ... body ....
| }
|
| , where each iteration checks for the condition. But some
| folks might feel like a loop should always just do
| something first and then consider if it should repeat, i.e.
| do { // ... body .... }
| while (condition)
|
| From that sort of perspective, zero might seem kinda
| unnatural and contrived, sorta like if
| (!zero) { do {
| // ... body .... } while
| (condition) }
|
| And maybe the fallacy is the idea that 1-times-anything is
| equal to anything.
|
| But 1-times- _X_ isn 't actually just _X_ -- it 's just
| that, in basic math, folks tend to ignore " _pure_ "
| functions under the fiction of an abstract universe. But
| this fiction isn't respected when we actually do
| computation; for example, if you write a program
| Print(1*x);
|
| without optimization, then the computer'll actually have to
| make a call to calculate 1-times-x; this is, it's not fully
| equivalent to Print(x);
|
| .
|
| Point being that, when you consider a multiplication-
| product, you have to acknowledge that, computationally,
| 1-times-anything isn't _just_ the " _anything_ "; there's
| also a structural difference.
|
| And so if you feel that the concept of zero is unnatural,
| it might be because you're thinking of multiplication as
| being a check-after-first-iteration thing, whereas it's a
| computationally distinct procedure (even if it produces
| superficially similar results).
|
| ---
|
| PS: This might be a hard discussion to have without TeX and
| proper formatting. Plain-text is too low-bandwidth for
| general discussions involving stuff like math.
| orobinson wrote:
| 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 wrote:
| Yep.
|
| Q: "How many _times_ has it rained this week? "
|
| A: "It has rained zero _times_ "
|
| Zero is a perfectly natural state.
| jasonwatkinspdx wrote:
| 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.
| gorgoiler wrote:
| 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 wrote:
| 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 wrote:
| 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.
| bena wrote:
| 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.
| novaRom wrote:
| 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.
| gmuslera wrote:
| 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 wrote:
| 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.
| ska wrote:
| > If Mathematics is a language to describe reality, maybe
| numbers are not its whole alphabet.
|
| Who would claim they are? Mathematics initially co-evolved with
| our ideas of numbers, but many other important objects have
| been thought about for a very long time now.
| breck wrote:
| 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 wrote:
| I genuinely don't understand what this means. How a beach ball
| a tree? What about the symmetry group of the beach ball?
| breck wrote:
| > How a beach ball a tree? _ /
| \ | | \ _ /
| eigenket wrote:
| That is neither a beach ball, nor a tree.
| pjettter wrote:
| The word "finger" did not occur so I skipped it.
| dang wrote:
| " _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
| alcover wrote:
| 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.
| amai wrote:
| "God made the integers, all else is the work of man. " Leopold
| Kronecker
| eigenket wrote:
| A more modern mathematician might say something like "God made
| the empty set, all else is the work of man".
| aaroninsf wrote:
| "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.
| kazinator wrote:
| The concept of _symbols_ is inevitable.
| SquibblesRedux wrote:
| 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 wrote:
| 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.
| [deleted]
| Aardwolf wrote:
| 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
| joe_the_user wrote:
| It's not. It's a single proof about a set, a set that's
| assumed to be uncountable in standard ZF set theory.
|
| The "axiom system" that (supposedly) contain a countable
| number of axioms. But these too are constructs of set
| theory. We still create proofs one by one of theories about
| axiom systems with infinite axiom - so we have a
| countable/enumerable set of such theories.
|
| The proof systems to we can see or touch involve this
| enumerable properties. Perhaps you could change that with
| an analogue computer that a person could input "any"
| "quantity" into. But that's outside math as things stand.
| Aardwolf wrote:
| Do you mean the proof that 0.25 >= 0 and the proof that
| 1/e >= 0 count as the same one, because there's a more
| general proof that a set of values including those is >=
| 0? But then where do you draw the line? When do you
| consider 2 proofs different enough to count as different
| ones?
| eigenket wrote:
| I think you have a slightly stricter definition of "a
| proof" than me. I would consider a proof that all the
| numbers in (0,1) are positive to also be a proof that the
| number 0.5 is positive, as well as the number 1/e, and
| Champernowne's constant.
|
| Since the original question was about uncountably many
| mathematical truths I would say we have one proof that
| proves uncountably many mathematical truths.
| kevinventullo wrote:
| 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.
| joe_the_user wrote:
| _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._
|
| I understand that many of those well-developed fields which
| exist on their own terms, have standard methods, standard
| questions and standard approaches to moving towards answers.
|
| The article jump between these fields to ask and grope for an
| answer to a simple question that in many ways can't be asked or
| answered in these fields.
|
| One thing to consider is that present day computers can follow
| the mechanical production of mathematical propositions close to
| completely. But computers have a lot of trouble producing or
| following arguments like this, in "natural language", which
| have a definite logic to them but whose operation is not based
| on only explicit, codified rules.
|
| Edit: To me, this sort of speculation is what philosophy
| actually should be doing. The questions that are "ill-defined
| but compelling" are the questions that have lead significant
| intellectual progress. How Zeno's paradox lead (or at least
| related) to the invention of calculus, how Einstein's thought
| experiments lead to relativity, etc.
| ska wrote:
| > conflates knowledge representation, language, biological
| systems [...], computability, and realism.
|
| You have pretty concisely described Wolfram's wheelhouse.
| ezconnect wrote:
| When my two year old can't count the items in front of him he
| just says it's "many".
| a9h74j wrote:
| 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 wrote:
| 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-
| la...
| desas wrote:
| The piraha language is probably the most famous non-counting
| language. They appear to recognise - https://slate.com/human-
| interest/2013/10/piraha-cognitive-an...
| ezconnect wrote:
| Depends on context maybe, Koreans have different number names
| depending on context.
| agumonkey wrote:
| One two infinity
| marcodiego wrote:
| There is a brazilian tribe (piranha) 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 wrote:
| Basically every single claim about Piraha (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.
| osrec wrote:
| 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...
| hyperpallium2 wrote:
| 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 wrote:
| Shouldn't language predate money?
|
| Language or cave paintings or whatever, use discrete symbols to
| describe physical reality
| ffhhj wrote:
| Every number is a random number.
| ManBlanket wrote:
| '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 -\\_( deg [?]? deg)_/-
| slver wrote:
| 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 wrote:
| 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.
| xyzzy123 wrote:
| Related to this I have often wondered if childhood amnesia is
| (by a crude analogy) a "format problem".
|
| In this analogy, the lion is our younger selves. Perhaps our
| brain changes so much that we cannot understand (or access)
| what we were thinking or experiencing anymore.
|
| If we imagine that long term memories are encoded in a way that
| they can later be interpreted actively by the structures of the
| brain, we could expect that major developmental changes in
| brain function might impact the ability to interpret long term
| memory formed before the change.
|
| Physical trauma, the major reorganisations of infancy and
| language acquisition all prompt changes in organisation and
| processing that could be what renders earlier memories
| inaccessible.
|
| This (admittedly hand-wavey) idea seems to mesh well well with
| the observed details of the phenomenon. Frequently accessed
| memories may be re-encoded, leading to some continuity.
| Children themselves experience fairly continuous long-term
| memory. Types of memories where relevant processing might not
| have changed as much (smell, visual memory) might be more
| resilient. We would expect age of language acquisition to play
| a part. Etc.
| slver wrote:
| 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 wrote:
| 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.
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