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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 leve

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
  [deleted]
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

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