6 ms·

> This is why, Prof. Wildberger says he "doesn't believe in irrational numbers." Oh boy, I hope that they missed a joke or misquoted.

by knappa 1y ago

GTP 1y ago
Never show him a square whose side is 1, or he wouldn't believe in its diagonal :D
- greesil 1y ago
  If you watch it closely, you can see the hypotenuse converging.
rssoconnor 1y ago
I presume he is the same one who authored "Divine Proportions: Rational Trigonometry to Universal Geometry" (https://www.goodreads.com/book/show/791022.Divine_Proportions https://www.goodreads.com/book/show/791022.Divine_Proportion...) so I suspect he is not misquoted. I actually quite liked "Divine Proportions". As far as I know Wildberger is eccentric, but not exactly a crackpot.
- deleted 1y ago
  [deleted]
superidiot1932 1y ago
No, he really doesn't believe in them.
abetusk 1y ago
Here's a possible good faith interpretation: Irrational numbers exist as a limiting process, not as a static atomic unit. That is, irrational numbers are a verb, not a noun. We use irrational numbers as nouns, when convenient, but this is an abuse, in some sense. When we want some digit of sqrt(2), say, we need to interrogate an algorithm to get it. We talk about how much time it takes to extract the amount of precision we want. At best sqrt(2) can be thought of as an abstract symbol that, when we multiply it by itself, is 2. That is, an algebraic manipulation that we can reduce to an integer under certain circumstances, but it doesn't "exist" the same way that an integer or a rational exists.
- DietaryNonsense 1y ago
  Put another way: irrational numbers are functions, not values. You can run the function as long as you want and get more digits but the function won't halt.
  - tromp 1y ago
    That would make more sense if you use the word computation instead of function.
- zmgsabst 1y ago
  > That is, an algebraic manipulation that we can reduce to an integer under certain circumstances, but it doesn't "exist" the same way that an integer or a rational exists. This depends on your interpretation: some view the reals as completions of that process, in which those “verbs” are “nouns”. But you can construct a coherent theory in which this is not the case — and nobody is much fussed, because mathematics is full of weird theories and interpretations. And both integers and rationals are defined by their relations, eg, integers are equivalence classes of pairs of naturals and rationals as equivalence classes of pairs of integers — where the class obeys some algebraic manipulation properties. If you feel there’s some great difference in sequences (and where you find that difference, eg, allowing only constructibles) is a matter of perspective.
wannabebarista 1y ago
He is a sort of strong finitist [0]. Roughly, this means that many usual mathematical objects that rely on notions of infinity, e.g., irrational numbers, do not exist. I came across his YouTube channel [1] years ago as a undergrad and became really confused about some ideas in logic as a result. [0] https://en.wikipedia.org/wiki/Ultrafinitism https://en.wikipedia.org/wiki/Ultrafinitism [1] https://www.youtube.com/@njwildberger/playlists https://www.youtube.com/@njwildberger/playlists
zitterbewegung 1y ago
In mathematics there are fringe philosophies but you still just need proofs that are kosher.
BeetleB 1y ago
I suspect that in reality it's transcendental numbers be doesn't believe in. Usually such folks believe in irrational but algebraic numbers.
jrdres 1y ago
Wildberger has always been this way. Way back in 2007, Marc Chu-Carroll's "Good Math Bad Math" highlighted Wildberger: "This isn’t the typical wankish crackpottery, but rather a deep and interesting bit of crackpottery." In brief, Wildberger is clearly educated, but also clearly rejects axioms that mathematicians accepted a long time ago (infinite sets in this case): "Dirty Rotten Infinite Sets and the Foundations of Math" http://www.goodmath.org/blog/2007/10/15/dirty-rotten-infinite-sets-and-the-foundations-of-math/ http://www.goodmath.org/blog/2007/10/15/dirty-rotten-infinit... Wildberger also wrote a book on geometry with nothing allowed but rationals. (Or something like that.)
feoren 1y ago
All models are wrong; some models are useful. I'm not sure I'd say I "believe in" the number 7, either. But the number 7 is very useful. It's reasonable to debate how useful irrational numbers are. That is: there are models of math where irrational numbers are "outlawed" in some sense, and it sounds like Prof. Wildberger studies those models. It's a "fringe" model though; most of math is better off simply accepting irrational numbers as they are. Clearly some of the most widely useful math involves exp(x), sin(x), and pi, all of which are either completely outlawed or require laborious workarounds if you're not allowing true irrational numbers. Still, it's a good idea to have some people studying these fringe models to get a better understanding of math as a whole. In this case, I find the argument "but you can't calculate it!" unconvincing, since every computer will have rational numbers they can't exactly calculate as well. Our computers can't calculate the exact value of 1/3 either; so what? If we're worried about computing things, we should consider whether we can calculate things to arbitrary precision or not within reasonable time. In that sense, pi behaves no worse than 1/3.
- jostylr 1y ago
  > since every computer will have rational numbers they can't exactly calculate as well It might be better worded as "can't calculate a decimal version of every rational number". One can work quite easily nowadays with exact representations of rational numbers on computer. With Bigint stuff, it is easy to have very large (for human purposes) numerators and denominators. To what extent practical calculations could be done with exact rational arithmetic, I am not sure of though I suspect it is largely not an issue as precision of inputs is presumably a limiting factor. Wildberger has specific objections to the usual definitions of real numbers and they vary based on the definition. For decimals, it is the idea that doing arithmetic with an infinite decimal is difficult even with a simple example such as 1/9*1/9 which is multiplying .111... times itself, leading to sums of 1s that carryover and create a repeating pattern that is not self-evident from the decimal itself. For Cauchy sequences, he objects to the absurd lack of uniqueness, particularly that given any finite sequence, one can prepend that sequence to the start of any Cauchy sequence. So a Cauchy sequence for pi could start with a trillion elements of a sequence converging to square root 2. This can be fixed up with tighter notions of a Cauchy sequence though that makes the arithmetic much more cumbersome. For Dedekind cuts, his issue seems mostly with a lack of explicit examples beyond roots. I think that is the weakest critique. Inspired by his objections, I came up with a version of real numbers using intervals. Usually such approaches use a family of overlapping, notionally shrinking intervals. I maximized it to include all intervals that include the real number and came up with axioms for it that allow one to skirt around the issue that this is defining the real number. My work on this is hosted on GitHub: https://github.com/jostylr/Reals-as-Oracles https://github.com/jostylr/Reals-as-Oracles

> This is why, Prof. Wildberger says he "doesn't believe in irrational numbers." Oh boy, I hope that they missed a joke or misquoted.

by knappa 1y ago

> This is why, Prof. Wildberger says he "doesn't believe in irrational numbers." Oh boy, I hope that they missed a joke or misquoted.

GTP 1y ago

Never show him a square whose side is 1, or he wouldn't believe in its diagonal :D

greesil 1y ago
If you watch it closely, you can see the hypotenuse converging.

rssoconnor 1y ago

I presume he is the same one who authored "Divine Proportions: Rational Trigonometry to Universal Geometry" (https://www.goodreads.com/book/show/791022.Divine_Proportions https://www.goodreads.com/book/show/791022.Divine_Proportion...) so I suspect he is not misquoted. I actually quite liked "Divine Proportions". As far as I know Wildberger is eccentric, but not exactly a crackpot.

deleted 1y ago
[deleted]

superidiot1932 1y ago

No, he really doesn't believe in them.

abetusk 1y ago

Here's a possible good faith interpretation: Irrational numbers exist as a limiting process, not as a static atomic unit. That is, irrational numbers are a verb, not a noun. We use irrational numbers as nouns, when convenient, but this is an abuse, in some sense. When we want some digit of sqrt(2), say, we need to interrogate an algorithm to get it. We talk about how much time it takes to extract the amount of precision we want. At best sqrt(2) can be thought of as an abstract symbol that, when we multiply it by itself, is 2. That is, an algebraic manipulation that we can reduce to an integer under certain circumstances, but it doesn't "exist" the same way that an integer or a rational exists.

DietaryNonsense 1y ago
Put another way: irrational numbers are functions, not values. You can run the function as long as you want and get more digits but the function won't halt.
- tromp 1y ago
  That would make more sense if you use the word computation instead of function.
zmgsabst 1y ago
> That is, an algebraic manipulation that we can reduce to an integer under certain circumstances, but it doesn't "exist" the same way that an integer or a rational exists. This depends on your interpretation: some view the reals as completions of that process, in which those “verbs” are “nouns”. But you can construct a coherent theory in which this is not the case — and nobody is much fussed, because mathematics is full of weird theories and interpretations. And both integers and rationals are defined by their relations, eg, integers are equivalence classes of pairs of naturals and rationals as equivalence classes of pairs of integers — where the class obeys some algebraic manipulation properties. If you feel there’s some great difference in sequences (and where you find that difference, eg, allowing only constructibles) is a matter of perspective.

wannabebarista 1y ago

He is a sort of strong finitist [0]. Roughly, this means that many usual mathematical objects that rely on notions of infinity, e.g., irrational numbers, do not exist. I came across his YouTube channel [1] years ago as a undergrad and became really confused about some ideas in logic as a result. [0] https://en.wikipedia.org/wiki/Ultrafinitism https://en.wikipedia.org/wiki/Ultrafinitism [1] https://www.youtube.com/@njwildberger/playlists https://www.youtube.com/@njwildberger/playlists

zitterbewegung 1y ago

In mathematics there are fringe philosophies but you still just need proofs that are kosher.

BeetleB 1y ago

I suspect that in reality it's transcendental numbers be doesn't believe in. Usually such folks believe in irrational but algebraic numbers.

jrdres 1y ago

Wildberger has always been this way. Way back in 2007, Marc Chu-Carroll's "Good Math Bad Math" highlighted Wildberger: "This isn’t the typical wankish crackpottery, but rather a deep and interesting bit of crackpottery." In brief, Wildberger is clearly educated, but also clearly rejects axioms that mathematicians accepted a long time ago (infinite sets in this case): "Dirty Rotten Infinite Sets and the Foundations of Math" http://www.goodmath.org/blog/2007/10/15/dirty-rotten-infinite-sets-and-the-foundations-of-math/ http://www.goodmath.org/blog/2007/10/15/dirty-rotten-infinit... Wildberger also wrote a book on geometry with nothing allowed but rationals. (Or something like that.)

feoren 1y ago

All models are wrong; some models are useful. I'm not sure I'd say I "believe in" the number 7, either. But the number 7 is very useful. It's reasonable to debate how useful irrational numbers are. That is: there are models of math where irrational numbers are "outlawed" in some sense, and it sounds like Prof. Wildberger studies those models. It's a "fringe" model though; most of math is better off simply accepting irrational numbers as they are. Clearly some of the most widely useful math involves exp(x), sin(x), and pi, all of which are either completely outlawed or require laborious workarounds if you're not allowing true irrational numbers. Still, it's a good idea to have some people studying these fringe models to get a better understanding of math as a whole. In this case, I find the argument "but you can't calculate it!" unconvincing, since every computer will have rational numbers they can't exactly calculate as well. Our computers can't calculate the exact value of 1/3 either; so what? If we're worried about computing things, we should consider whether we can calculate things to arbitrary precision or not within reasonable time. In that sense, pi behaves no worse than 1/3.

jostylr 1y ago
> since every computer will have rational numbers they can't exactly calculate as well It might be better worded as "can't calculate a decimal version of every rational number". One can work quite easily nowadays with exact representations of rational numbers on computer. With Bigint stuff, it is easy to have very large (for human purposes) numerators and denominators. To what extent practical calculations could be done with exact rational arithmetic, I am not sure of though I suspect it is largely not an issue as precision of inputs is presumably a limiting factor. Wildberger has specific objections to the usual definitions of real numbers and they vary based on the definition. For decimals, it is the idea that doing arithmetic with an infinite decimal is difficult even with a simple example such as 1/9*1/9 which is multiplying .111... times itself, leading to sums of 1s that carryover and create a repeating pattern that is not self-evident from the decimal itself. For Cauchy sequences, he objects to the absurd lack of uniqueness, particularly that given any finite sequence, one can prepend that sequence to the start of any Cauchy sequence. So a Cauchy sequence for pi could start with a trillion elements of a sequence converging to square root 2. This can be fixed up with tighter notions of a Cauchy sequence though that makes the arithmetic much more cumbersome. For Dedekind cuts, his issue seems mostly with a lack of explicit examples beyond roots. I think that is the weakest critique. Inspired by his objections, I came up with a version of real numbers using intervals. Usually such approaches use a family of overlapping, notionally shrinking intervals. I maximized it to include all intervals that include the real number and came up with axioms for it that allow one to skirt around the issue that this is defining the real number. My work on this is hosted on GitHub: https://github.com/jostylr/Reals-as-Oracles https://github.com/jostylr/Reals-as-Oracles