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New evidence that Cantor plagiarized Dedekind?

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Re: New evidence that Cantor plagiarized Dedekind?

#91

Post by stbede »

singlow wrote: post #75
ChocMontePy wrote: post #72

Earlier quoted context omitted.

I'm at the library so I checked your book. You said in there: > However, by October 1933, the issue was straightened out and she was aboard the Bremen, sailing for the United States. Since she died on 14 April 1935, it was 18 months rather than 2 years. That sounds like a rather pedantic correction on your part. That pedanticism is a bad sign and puts your "correction" about the cancer in doubt.

Complications after surgery to remove a cancerous tumor?

So she died of cancer

Re: New evidence that Cantor plagiarized Dedekind?

#92

Post by vessenes »

bhk wrote: post #73
vessenes wrote: post #62

Earlier quoted context omitted.

You’re thinking of this with the benefit of dedekind in your schooling - whether or not your calculus class told you about him. Density - a gapless number line - was neither obvious nor easy to prove; the construction is usually elided even in most undergraduate calculus unless you take actual calculus “real analysis” courses. The issue is this: for any given number you choose, I claim: you cannot tell me a number “t…

I think you are getting away from my point, which pertains to what the article said, which is that mathematicians thought there were "gaps". What mathematician? Can I see the original quote? The linguistic sleight-of-hand is what I challenge. What is this "gap" in which there are no numbers? - A reader would naturally assume the word refers to a range. But if that is the meaning, then mathematicians never believed th…

I don’t know the answers to all of your questions - but I believe you’d benefit from some mathematical history books around the formalization of the real analysis; I’m not the best person to give you that history.

A couple comments, though - first, all mathematics is linguistics and arguably it is all sleight of hand - that said the word “gaps” that you’ve rightly pointed out is vague is a journalists word standing in for a variety of concepts at different times.

The existence of the irrationals themselves were a secret in ancient greece - and hence known for thousands of years, but the structure of the irrationals has not been well understood until quite recently.

To talk precisely about these gaps, if you’re not a mathematical historian, you have to borrow terminology from the tools that were used to describe and formalize the irrationals -> if former concepts about the lines sound hand-wavy to you, it is because they WERE handwavy. And this handwaviness is about infinity as well, the two are intimately connected. In modern terms, the measure of the rationals across any subset of the (real) number line is zero - that is the meaning of the “gaps”. There is, between any two rationals, a great unending sea where if you were to choose a point completely at random, the odds of that point being another rational is zero.

EDIT: for a light but engaging read about topics like this, David Foster Wallace’s Everything and More is excellent.

Re: New evidence that Cantor plagiarized Dedekind?

#93

Post by thaumasiotes »

jerf wrote: post #79
thaumasiotes wrote: post #77

Earlier quoted context omitted.

> he’d worked out a proof that the algebraic numbers (the numbers you get as solutions to algebra problems) could be counted I can't say I'm fully comfortable with that characterization of the algebraic numbers. The definition itself does suggest a proof that they are countable: 1. The number of symbols that can appear in a well-defined algebra problem is finite. (For example, if we define algebra problems as being p…

You can also get to computable numbers through a similar argument, substituting something Turing-complete for algebra. You definitely do get to learn some interesting things about numbers from computable numbers. The differences between the computables and the full reals are much more subtle than the differences between the rationals and the reals.

> The differences between the computables and the full reals are much more subtle than the differences between the rationals and the reals.

How so?

Using the definition of computable numbers where you provide input of n and the output is every digit of the number up to n places past the decimal point, we can rephrase that definition like so:

A computable number c is one with the following property:

A Turing machine exists which, provided with a tolerance δ, will exhibit a rational number q

Clearly, a suitable rational will always exist, since rationals can be found within any distance of any real.

But for some particular real, we might not be able to find that rational through the use of a fixed Turing machine, in which case the real would be noncomputable. This suggests to me that there is a wider gap between the computables and the reals, where the approximation of a real number is limited by the need to describe it with a Turing machine, than there is between the rationals and the reals, where we can use the same approximation, but without that limitation.

(Obviously the rationals are a subset of the computables, but if we're considering a relationship to the real numbers, the rationals seem to have one that is closer and more direct...? The relationship of a computable number to a real number is defined through intermediary rational numbers.)

Re: New evidence that Cantor plagiarized Dedekind?

#94

Post by jerf »

thaumasiotes wrote: post #93
jerf wrote: post #79

Earlier quoted context omitted.

You can also get to computable numbers through a similar argument, substituting something Turing-complete for algebra. You definitely do get to learn some interesting things about numbers from computable numbers. The differences between the computables and the full reals are much more subtle than the differences between the rationals and the reals.

> The differences between the computables and the full reals are much more subtle than the differences between the rationals and the reals. How so? Using the definition of computable numbers where you provide input of n and the output is every digit of the number up to n places past the decimal point, we can rephrase that definition like so: A computable number c is one with the following property: A Turing machine e…

"This suggests to me that there is a wider gap between the computables and the reals"

I didn't say "wider". I said more subtle. It doesn't take much mathematical intuition and training to understand the rationals versus the reals. Understanding the computables versus the reals is a lot more tricky and takes a lot more thought. The simple arguments that show the difference between the rationals and the reals require a lot of very careful adjustment if you want to translate them to the computables versus the reals.

I agree the gap up to the reals is still bigger than rational -> computable. The reals are weird. This is a meme but there's a lot of truth in it: https://www.reddit.com/media?url=https%3A%2F%2Fpreview.redd....

Re: New evidence that Cantor plagiarized Dedekind?

#95

Post by bhk »

vessenes wrote: post #92
bhk wrote: post #73

Earlier quoted context omitted.

I think you are getting away from my point, which pertains to what the article said, which is that mathematicians thought there were "gaps". What mathematician? Can I see the original quote? The linguistic sleight-of-hand is what I challenge. What is this "gap" in which there are no numbers? - A reader would naturally assume the word refers to a range. But if that is the meaning, then mathematicians never believed th…

I don’t know the answers to all of your questions - but I believe you’d benefit from some mathematical history books around the formalization of the real analysis; I’m not the best person to give you that history. A couple comments, though - first, all mathematics is linguistics and arguably it is all sleight of hand - that said the word “gaps” that you’ve rightly pointed out is vague is a journalists word standing i…

Thanks for the references.

I think you will agree that the bulk of your comment employs a post-set-theory nomenclature.

Regarding "if you were to choose a point completely at random, the odds of that point being another rational is zero", I ponder the question of how one might casually "choose" a value with infinite entropy.

Re: New evidence that Cantor plagiarized Dedekind?

#96

Post by dhosek »

thaumasiotes wrote: post #77
AdAbsurdum wrote: post #60

Earlier quoted context omitted.

I think the relevant quotes are these: "Dedekind quickly replied that...he’d worked out a proof that the algebraic numbers (the numbers you get as solutions to algebra problems) could be counted. [...] Weierstrass had been most excited about the proof that algebraic numbers are countable. (He would later use that result to prove a theorem of his own.) So Cantor chose a misleading title [for his paper] that only menti…

> he’d worked out a proof that the algebraic numbers (the numbers you get as solutions to algebra problems) could be counted I can't say I'm fully comfortable with that characterization of the algebraic numbers. The definition itself does suggest a proof that they are countable: 1. The number of symbols that can appear in a well-defined algebra problem is finite. (For example, if we define algebra problems as being p…

Yeah, that was a weird way to describe the algebraic numbers. The formal definition (those numbers for which a finite polynomial p exists with p(x)=0) is not that complicated, is it?
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