Daniel Williams

Mathématicien

[David Louapre @dlouapre]

Mathematicians are angry at AI solving their problems, and it's easy to dismiss this as gatekeeping.

"A proof is a proof. They just want to keep their jobs / prestige / fun !"

But I do think mathematicians have a point, and I'll try to explain using old Civilization-style maps

[Daniel Williams @BarbChapeau]

@jogothy @KevinSVanHorn @52_65_64 https://t.co/Z2DWxf8HWE

Is it clearer after watching this video ?

[Daniel Williams @BarbChapeau]

@jogothy @52_65_64 Is it more clear that way ? (click to see the full image) The ... means we go on like this for every integer, so 4, 5, 6 etc to infinity

The Σ symbol is the greek equivalent of the letter S, because it's the first letter of the word "sum", because as you can see it's a sum

[Daniel Williams @BarbChapeau]

@aslanoffff @RadishHarmers @Lewiswbt1 The sum is infinite if we define it to be the limit of the partial sums. Read his Quora post.

[Daniel Williams @BarbChapeau]

@lorel_sg Je le pense aussi, tant qu'on en connaît les limites c'est plutôt fécond je trouve

[Daniel Williams @BarbChapeau]

@numbertheory0 @ml8_lmn C'est pour ça que je fais des maths sur mon temps libre 😁

[Daniel Williams @BarbChapeau]

@numbertheory0 Félicitations ! Tu le mérites tellement, et ta passion se ressent toujours autant quand on te lit !!

[james rahner 🇻🇦 🇵🇸 @infinirien]

@Tabletop_James @littmath let's say Alice goes (1,2,3...) and Bob does the same, but starts with box 100 (100,1,2...). Who has the advantage here? Alice clearly does: the only way Bob can ever win is if there is a present in box 100, and with 26 presents over 100 boxes, the odds are against this.

[Daniel Williams @BarbChapeau]

@pagu_pagu_pg @miniapeur Why would it be undefined ? There is no reason to not define it as 1.

[Daniel Williams @BarbChapeau]

@WillFindU @miniapeur Why would it be undefined ? It's 1

[Daniel Williams @BarbChapeau]

@BudvarCat @martinmbauer LaTeX notes allowed me to rearrange them later to make it clearer, and to share it to absent people. I just had a notebook for the diagrams that I later draw with TikZ

[Daniel Williams @BarbChapeau]

@Graymanland @dagwieers @adriendecroy @MiguelMadeira11 @Anthony_Bonato There is no need of infinite amount of time to define infinity.

Do you know about set theory ?

[Daniel Williams @BarbChapeau]

@Graymanland @dagwieers @adriendecroy @MiguelMadeira11 @Anthony_Bonato I don't understand the argument with sinus. What does it proves about infinity that inverse doesn't prove about 0 ?

sin isn't defined on infinity.
inverse isn't defined on zero.

[Daniel Williams @BarbChapeau]

@Graymanland @dagwieers @adriendecroy @MiguelMadeira11 @Anthony_Bonato It just means infinity isn't in the domain of sin.

y = 1/x
x = 0
y = ?

Doesn't mean 0 doesn't exist, just that 0 isn't in the domain of the inverse function.

[Daniel Williams @BarbChapeau]

@Graymanland @dagwieers @adriendecroy @MiguelMadeira11 @Anthony_Bonato But in math we don't count. Counting is a matter of the physical world. In math, we can instead for instance talk about bijections between sets and cardinality. We then doesn't have to do anything endlessly.

[Daniel Williams @BarbChapeau]

@Graymanland @dagwieers @adriendecroy @MiguelMadeira11 @Anthony_Bonato I think I do agree with you, but I personnaly think an invention is just a case of discovery so to me it doesn't really matter so much.

[Daniel Williams @BarbChapeau]

@Graymanland @dagwieers @adriendecroy @MiguelMadeira11 @Anthony_Bonato What do you mean "endless" ?
2 and any 2-1/n that you can think of are perfectly defined.

You can also order the pairs of natural numbers (n,m) by lexicographic number.
Then, represent every natural number m by the pair (0,m).
Then you define infinity by (1,0).

[Daniel Williams @BarbChapeau]

@Graymanland @dagwieers @adriendecroy @MiguelMadeira11 @Anthony_Bonato The other idea behind my question is that there whether you can have n apples or not in your hand isn't relevant to n existing in math.

Math isn't necessarily about thing you can manifest in the physical world.

[Daniel Williams @BarbChapeau]

@Graymanland @dagwieers @adriendecroy @MiguelMadeira11 @Anthony_Bonato That way, you can represent the natural number n by it associated real number 2 - 1/n.

So that way, 2 is bigger than every 2 - 1/n you can make. The idea of infinity is just to represent a situation like that where you have something bigger than any member of an endless sequence

[Daniel Williams @BarbChapeau]

@Graymanland @dagwieers @adriendecroy @MiguelMadeira11 @Anthony_Bonato You can define infinity like that.
Look at the sequence
2 - 1/1
2 - 1/2
2 - 1/3
2 - 1/4
etc
2 - 1/n for each natural number n.
I'm not talking about limit here, just the sequence.
Each member of this sequence is smaller than the number 2.