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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)

#61

Post by MarkusQ »

empath75 wrote: post #37
MarkusQ wrote: post #14

Earlier quoted context omitted.

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 digi…

That's the key point though, this list isn't infinitely long, and all the numbers in it are rational. And it is an algorithm (specifically, a lookup table).

All the numbers you get this way are going to be rational, and if you require them to be finite, you can't even identify them with any irrational numbers. At least with the computable numbers you get an infinite set of irrational numbers along with the rationals, while still never touching the vast majority of all numbers (the remaining, incomputable irrationals).

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

#63

Post by drdec »

adrian_b wrote: post #56
drdec wrote: post #45

Earlier quoted context omitted.

If you do not accept that space is infinitely divisible, then the diagonal of a unit square does not actually exist in the space.

A long time has passed since the paradoxes of Zeno of Elea, so now there really is no reason for not accepting that space is infinitely divisible. The error of Zeno of Elea was that he did not understand the symmetry between zero and infinity (or he pretended to not understand it). Because of this error, Zeno considered that infinity is stronger than zero, so he believed or pretended to believe that zero times infini…

> For now, there exists no evidence whatsoever that the physical space and time are not infinitely divisible.

What is your evidence for it? You want to make a claim about something being infinite, it is up to you to provide evidence.

> recognizing that zero times infinity can be any number and also zero or infinity.

This statement makes no sense in formal mathematics. Multiplication is a function, which means for each set of inputs there is one output. I imagine you are trying to say something about limits here, but the language you are using is very imprecise.

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

#64

Post by alexey-salmin »

griffzhowl wrote: post #57
alexey-salmin wrote: post #50

Earlier quoted context omitted.

> Even if you did that, you should show somehwere this finitist foundation disagrees with the results obtained by the standard foundation, otherwise there's no reason to think the standard foundation is in error. Well these are probably easy to find even now? E.g the Banach-Tarsky paradox is unlikely to be provable in finitist math which is somewhat of an improvement.

I was thinking more about applications in physics where calculus and irrational quantities are used all the time. At more advanced levels the theories are based on differential geometry and operators on Hilbert space. I'm not sure if fully worked out finitist versions of these even exist. Where finitist versions do exist, they're often technically more difficult to use than the standard versions, which is the opposit…

I'm not a finitist myself but my understanding is that it has to do as much with physics as does ZFC, which is very little. The math used in physics works on practice and did work long before the question of foundations even came up.

The problem that bothers some mathematicians is that despite working well math still lacks a solid foundation. Furthermore it's basically proven that these foundations can't even exist, or at least for the mainstream version of math. This is where non-mainstream versions pop up. The denial of uncountable sets does help you resolve some of the paradoxes. Not all unfortunately, even the countable sets already lead to things like incompleteness theorems. Well, one can dream.

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

#65

Post by griffzhowl »

alexey-salmin wrote: post #64
griffzhowl wrote: post #57

Earlier quoted context omitted.

I was thinking more about applications in physics where calculus and irrational quantities are used all the time. At more advanced levels the theories are based on differential geometry and operators on Hilbert space. I'm not sure if fully worked out finitist versions of these even exist. Where finitist versions do exist, they're often technically more difficult to use than the standard versions, which is the opposit…

I'm not a finitist myself but my understanding is that it has to do as much with physics as does ZFC, which is very little. The math used in physics works on practice and did work long before the question of foundations even came up. The problem that bothers some mathematicians is that despite working well math still lacks a solid foundation. Furthermore it's basically proven that these foundations can't even exist,…

> Furthermore it's basically proven that these foundations can't even exist,

What are you referring to? The current working foundation is ZFC but there are equivalent type theoretical foundations like what Lean and other proof-checking software uses. I guess you know that, but that's why I don't know what you mean by saying this

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

#66

Post by alexey-salmin »

griffzhowl wrote: post #65
alexey-salmin wrote: post #64

Earlier quoted context omitted.

I'm not a finitist myself but my understanding is that it has to do as much with physics as does ZFC, which is very little. The math used in physics works on practice and did work long before the question of foundations even came up. The problem that bothers some mathematicians is that despite working well math still lacks a solid foundation. Furthermore it's basically proven that these foundations can't even exist,…

> Furthermore it's basically proven that these foundations can't even exist, What are you referring to? The current working foundation is ZFC but there are equivalent type theoretical foundations like what Lean and other proof-checking software uses. I guess you know that, but that's why I don't know what you mean by saying this

I'm referring to the failure of Hilbert's program. All the incompleteness, undefinability and undecidability results arise when and only when some sort of infinite objects are present so I can definitely see the allure of finitism.

ZFC is a working foundation of math but it's unknown whether it's consistent or arithmetically sound and important statements like CH are independent from it. It's a "working foundation" but not a "true foundation" which alas cannot exist.

As mentioned above I'm personally not a finitist and think that math without infinite and uncountable sets is intellectually poorer. I don't mind however developing further a finitist subset of math and see what's provable (and describable) in it, much like there's value in proving theorems in ZF instead of ZFC whenever possible.

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