So what the FUCK did he mean by this?

Is that Lacan?

Math can be read and spoken. Many of those symbols can be translated to English language (and, or, such that, is element of, is congruent of, there exist, etc).

Just buy a copy of Euclid's Elements from Green Lion Press. It contains an axiomatic representation of basic stuff like this written in plain English without mathematical notation.

The dots are just punctuation.

If there is a serious interest in knowing what this means at a symbol by symbol level I'll take the time to write it out.

I would unironically be interested.

Pls

plato.stanford.edu/entries/pm-notation/
why not just leave it as an exercise to the reader?

Please, I'm a based CS major and this has got even me fucking confused. It's something to do with set theory I think

are you referring to propositional (0th order) logic?

>Famously, several hundred pages are required in PM to prove the validity of the proposition 1+1=2

You've russelled my jimmies friend
It's nice to see more of this on Veeky Forums though

The analytic is rising

Not that guy but the item posted by the OP is one of the better known items (and a representative example) from the /Principia Mathematica/, a three-volume work on mathematical logic which was written by (Alfred) Whitehead and (Bertrand) Russell and published from 1910-1913. In the text, Whitehead and Russell attempt to systematize all of known mathematics (or as much of it as they are able to) by grounding it in logic, set theory, etc.

Specifically, the bit posted by the OP occurs in the later half of the first volume. The "54.43" bit which starts it off just lets you know whereabouts you are in the text: okay, we're proving this now, okay fine. As you can clearly see scattered throughout the proof, it refers back to previous statements, "54.26", "51.321", etc, in such-and-such ways. this reference back to previously established self-contained results in a long-form work is exactly parallel with Euclid's elements. The point being that what you're looking at is a proof resting on other stuff already proved, but you have to have the text handy to understand it all. Self-citations in steps of the proof.

The backwards "C" symbol represents /implication/. To actually scan the first line, I will hazard the suggestion that two classes represent "1" in such-and-such a way, while their distinct operation later in the line represents "2" in such-and-such a way. Then a bunch of autism about the classes, or whatever, and then the cute line that this is a justification of 1+1=2.

That's basically what is going on. See also, for the most pertinent point-by-point,

en.wikipedia.org/wiki/Glossary_of_Principia_Mathematica#Symbols_introduced_in_Principia_Mathematica_volume_I

Furthermore, Wittgenstein recycles small bits of this notation in his Tractatus, especially the backwards-C for implication. I didn't quickly scan-it just to read the above link but I suspect that they both use a dot in such-and-such a way to represent the AND logical operator.

The "1,2,3,4" square-dot notation in PM has various semantic meanings as well. Pic related is just part of a glossary-list from the 2nd (and final) edition of the text, for a bit of flavor.

I remember this from maybe my first lecture at the university. This got me curious, so I looked it up. I couldn't find this proof in my book on discrete mathematics, and I couldn't find it in my calculus book either. Anything else would be two advanced to contain anything as profoundly stupid as this, but here is what I came up with:

First you define the successor function S

0 is a natural number.
for every natural number x, x = x
for all natural numbers x and y, if x = y, then y = x
for all natural numbers x, y and z, if x = y and y = z, then x = z

for all a and b, if b is a natural number and a = b, then a is also a natural number
for every natural number n, S(n) is a natural number
for all natural numbers m and n, m = n if and only if S(m) = S(n)
for every natural number n, S(n) = 0 is false.
If n is a natural number then S(n) is a natural number

Most people would now probably define natural numbers recursively using the successor function, but for the sake of this proof, it would actually be fine if the successor function magically skipped over some of them.

We define:
S(0) = 1
a + 0 = a
a + S(b) = S(a + b)

Now comes the proof:

1 + 1 = 1 + S(0) = S(1 + 0) = S(1)

First we define 2 recursively to be S(S(0))

S(S(0)) = S(1),

and thus 1 + 1 = 2

The proposition in the image probably uses Archimedes' axioms and not the Peano axioms which I used in my proof. I remember that as well. It was probably presented in the first lecture of my calculus class. Maybe I'll look into that tomorrow.

>0 is a natural number

Stopped reading there

It is a very common convention, depending upon which math text you're reading. That guy is referring to Peano, who IIRC did have 0 as the starting point in his treatment - this would inform later authors who also have 0 natural.

But it is not an overwheming consensus. I personally prefer to think of the naturals as starting with 1 (and other authors do as well) but the real point is that you have a set starting with some least element which then continues. And then whether you want to treat of 0 or not for whatever reason is the sticking point.

Stupidly, mathematicians can't seem to settle this simplest of notational conventions when they got a good bead on the rationals, reals, complexes and more. It is perfectly clear to me what ought to be done.

The set {1,2,3,...} is once-and-for-all denoted by something. doesn't matter what it is, just something.

And the set {0,1,2,3,....} is once-and-for-all denoted by something else. We all agree, they're unequal sets, whoop-dee-doo. sometimes you want zero, sometimes not.

>Anything else would be two advanced to contain anything as profoundly stupid as this, but here is what I came up with:

Why do you think this specific bit of the PM is stupid? Or are you referring to something else

>look how many trees we can kill by using a verbose notation

The proof is unnecessary. When my professor suggested that calculus is just a form of linear algebra (because any point in the definition space of a function can be considered a dimension in an infinite space of elementary functions, and every function value is thus a basis vector in that space, and so any function as a whole is a vector), I let go of mathematics. Proving that 1 + 1 = 2 is just... it derives from the definition of addition.

Thanks anons. Even a brainlet like me could get it.

the project wasn't to prove 1+1=2 it was to ground maths in logic
being able to prove simple arithmetic statements is a requirement, not a goal

The proof is completely trivial. It is much more useful to learn propositional calculus and rebuild the arithmetic operators that way.

Clearly you do not have the severe autism to appreciate the ambitious undertaking that is logicism.

Let's just say that I am more intrigued by actual problems than pedantic formalism.

"A book for those who don't even know that 1 + 1 = 2"
-- Henri Poincare

You must be fun at parties.

Recos for 'philosophy of math' for a layman?

>Russell BTFO Frege
>Gödel BTFO Russell

Who will BTFO Gödel?

>CS major
>based
lad, I hope you are going to grad school, otherwise pajeet who went to a coding bootcamp is going to take your job.