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Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

theguardian.com

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Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#31

Post by 7thaccount »

michaelsbradley wrote: post #5
7thaccount wrote: post #4

Earlier quoted context omitted.

How do we do things like electrical engineering without imaginary numbers? Is this method an actual improvement?

imaginary numbers are not the same thing as irrational numbers

I never said they were, but could've sworn that the Wikipedia page or parent comment did (I can't find it now and am questioning my sanity). I couldn't understand how he could try to get rid of them, although this isn't surprising as mathematics is basically magic to me once you get past calculus. I guess this is only about removing irrationals though.

Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#32

Post by vlovich123 »

kragen wrote: post #30
LPisGood wrote: post #25

Earlier quoted context omitted.

To an ultrafinitist, there is no such thing as a number that is inexpressible.

Right, but to be clear, it's not that ultrafinitists like Wildberger believe that they can express all the real numbers; rather, they believe that those inexpressible real numbers don't actually exist.

How does that work for calculus which regularly looks at the limits of functions as x approaches infinity and has very real real world applications that stem from such algorithms?

Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#33

Post by numpy-thagoras »

dr_dshiv wrote: post #29
numpy-thagoras wrote: post #21

Alright, I'll bite: To defend Wildberger a bit (because I am an ultrafinitist) I'd like to state first that Wildberger has poor personal PR ability. Now, as programmers here, you are all natural ultrafinitists as you work with finite quantities (computer systems) and use numerical methods to accurately approximate real numbers. An ultrafinitist says that that's really all there is to it. The extra axiomatic fluff abo…

So what is the length of the diagonal of a unit square, if not square root of 2? It can’t be rational—how is that rationalized by Wildberger?

Read Wildberger if you want to know what he thinks.

I can tell you that it is the output of a function, not a distinct entity that exists on its own independently of the computation.

The whole point is that as a theory for the foundations of mathematics, you do not need to assume numbers with infinitely long decimal expansions in order to do math.

Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#34

Post by ants_everywhere »

numpy-thagoras wrote: post #21

Alright, I'll bite: To defend Wildberger a bit (because I am an ultrafinitist) I'd like to state first that Wildberger has poor personal PR ability. Now, as programmers here, you are all natural ultrafinitists as you work with finite quantities (computer systems) and use numerical methods to accurately approximate real numbers. An ultrafinitist says that that's really all there is to it. The extra axiomatic fluff abo…

I like to imagine at the end of the human race when the sun explodes or whatever, some angelic being will tally up all the numbers ever used by humans and confirm that there are only finitely many of them. Then they'll chalk a tally on the scoreboard in favor of the ultrafinitists.

As long as someone isn't a crank (e.g. they aren't creating false proofs) I enjoy the occasional outsider.

Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#35

Post by numpy-thagoras »

ants_everywhere wrote: post #34
numpy-thagoras wrote: post #21

Alright, I'll bite: To defend Wildberger a bit (because I am an ultrafinitist) I'd like to state first that Wildberger has poor personal PR ability. Now, as programmers here, you are all natural ultrafinitists as you work with finite quantities (computer systems) and use numerical methods to accurately approximate real numbers. An ultrafinitist says that that's really all there is to it. The extra axiomatic fluff abo…

I like to imagine at the end of the human race when the sun explodes or whatever, some angelic being will tally up all the numbers ever used by humans and confirm that there are only finitely many of them. Then they'll chalk a tally on the scoreboard in favor of the ultrafinitists. As long as someone isn't a crank (e.g. they aren't creating false proofs) I enjoy the occasional outsider.

Heh that's about the only place and time when we'll know for sure, and until then, it's just high-grade banter :)

Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#37

Post by empath75 »

MarkusQ wrote: post #14
empath75 wrote: post #12

Earlier quoted context omitted.

You don't. He basically defines numbers like pi and e not as numbers, but as iterative functions, which you can run to whatever level of accuracy that you want. It's sort of a silly argument, because _all_ numbers can be treated like the output of a function, including the real numbers, so he has basically smuggled in all reals through the back door, because any real number can just be thought of as a function with i…

You can't get all the reals that way. The reals that can be produced by an algorithm make up a vanishingly small (e.g. countable) subset. Almost all of the reals are inexpressible.

What I described isn't really an algorithm, it's just taking the digits of a number, let's say:

foo=3.14159265...

Where after 5 is some continuing sequence of decimals.

The series of functions is literally just:

foo(0) = 3 foo(1) = 3.1 foo(2) = 3.14...

And to be clear, it's not just like, an algorithm that estimates pi, it's literally just a list of return values that is infinitely long that return more and more digits of whatever the number is. That is actually how he defines pi.

https://youtu.be/lcIbCZR0HbU?si=3YxcHfPlCFrlr5h3&t=2080

pi _happens_ to be computable, and there are more efficient functions that will produce those numbers, but you could do the same thing with an incomputable number, you just need a definition for the number which is infinitely long.

To be clear, I don't think any of this is a good idea, just pointing out that if he's going to allow that kind of definition of pi (ie, admit a definition that is just an infinite list of decimal representations), you can just do the same thing with any real number you like. He of course will say that he's _not_ allowing any _infinite list_, only an arbitrary long one.

Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#38

Post by griffzhowl »

7thaccount wrote: post #31
michaelsbradley wrote: post #5

Earlier quoted context omitted.

imaginary numbers are not the same thing as irrational numbers

I never said they were, but could've sworn that the Wikipedia page or parent comment did (I can't find it now and am questioning my sanity). I couldn't understand how he could try to get rid of them, although this isn't surprising as mathematics is basically magic to me once you get past calculus. I guess this is only about removing irrationals though.

It depends on the particular construction. You could construct the "complex rational field" by adding i to the rationals with the rule i^2 = -1. That seems to be what Wildberger is ok with. The standard complex numbers involve adding i to the real numbers, which Wildberger doesn't like.

I don't know how you'd do electrical engineering with the rational complex field, because electrical engineering and physics in general involves a lot of irrational quantities and calculus, and the standard foundations of these concepts use real numbers.

It's really up to finitists to show that there are problems with these methods and that they have a better way of doing things, because so far the standard way seems to work very well.

Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#39

Post by dhosek »

kragen wrote: post #17
LeifCarrotson wrote: post #6

Earlier quoted context omitted.

Thanks for the context - I was baffled at first how the Guardian would run with the tagline "a trignometric table more accurate than any". But it's because the sine of 60 degrees is said by modern tables to be equal to sqrt(3) / 2, which Wildberger doesn't "believe in", he prefers to state that the square of the sine is actually 3 / 4 and that this is "more accurate". The actual paper is at [1]: [1] https://doi.org/1…

Well, no, if you look at a trigonometric table, it doesn't say sin 60° = √3/2, because that isn't a useful value for calculation. It'll say something like 0.866025. But that has an error of a little more than 0.0000004. Instead Wildberger prefers saying that the spread (sin²) is ¾, which has no error. It is more accurate. There's no debate about this, except from margalabargala. The news from this paper (thanks for t…

> √3/2 … that isn't a useful value for calculation

Some tables do indeed have that value and it is a very useful value for calculation, one that can be symbolically manipulated to get you an exact number (albeit one likely expressed in radicals) for your work. When I used to teach algebra, it was a struggle to get students to let go of the decimal approximations that came out of their calculators and embrace expressions that weren’t simple decimals but were exact representations of the numbers at hand. (Then there’s really fun things like the fact that, e.g., √2 + √3 can also be written as √(5+2√6) (assuming I didn’t make an arithmetic error there)).

Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#40

Post by birn559 »

numpy-thagoras wrote: post #33
dr_dshiv wrote: post #29

Earlier quoted context omitted.

So what is the length of the diagonal of a unit square, if not square root of 2? It can’t be rational—how is that rationalized by Wildberger?

Read Wildberger if you want to know what he thinks. I can tell you that it is the output of a function, not a distinct entity that exists on its own independently of the computation. The whole point is that as a theory for the foundations of mathematics, you do not need to assume numbers with infinitely long decimal expansions in order to do math.

> I can tell you that it is the output of a function, not a distinct entity that exists on its own independently of the computation.

Could you elaborate? What is the output of that function if not an entity in it's own? Having studied math with philosophiy minor long time ago I am curious.

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