Pasting this from my identical thread on /his/ because it may be relevant:

It might help if I explain why I'm asking. Recently I was introduced to the philosophy of mathematics when a family member asked me if I thought numbers were "real" or not. It seemed like a strange thing to ask and my initial reaction was that it didn't make sense as a question. I can't explain why I feel that way, though, because of the huge gaps in my knowledge. The only way I could try to describe what I felt was by saying something like "2 + 2 = 4 doesn't describe a relationship between actual objects, but it's more like a rule".

Not being able to understand my own intuition was very frustrating as you can imagine, but I wasn't able to find much by googling because I don't know the terms. All I found was that Wittgenstein believed something vaguely similar, but when I tried to read his work, I couldn't understand it. That brings me here.

>>7840298
I know Ray Monk did his thesis on Wittgenstein's mathematics. No clue where to find it though. The Duty of Genius gives a pretty soft and readable introduction anyway.

I think Wittgenstein was not that much clear about what he thought when he was writing those enigmatic lines in the tractatus. My former logic professor was not able to give a lot of clue about how the mathematical formalism used in the tractatus worked. Surely he was not mistaking us when saying that only someone who already had those thoughts could understand them.

>>7840371
same poster as >>7840404
Read the Introduction to Mathematical Philosophy of Bertrand Russell if you are interested in math philosophy. It is a bit dense (for me at least) but gives a very good and intuitive introduction.

Witt math philosophy started with similar views and then departed from it.

>>7840409
If you found it dense, I'm sure I will find it impenetrable, but thank you for the suggestion.

Don't you need to know formal logic in order to get him?

I am very confused by what your asking OP

Wittgenstein thought extensively on pure math with Russell, but I don't see why he would strike your fancy over say Frege.

I recommend the comic Logicomix, it sort of goes over trying to make "contradiction free, and complete mathematics" ending ofc with Godels theorems. Its also a light read, very idea based, not to do with formal logic

I would also add that Wittgensteins major contributions didn't have much to do with "whether numbers are real or not", the Tractatus and his later writings are primarily occupied with the relation between the mind and reality and language and reality. Math to him served primarily as the most formalized mapping, but is not really unique from other languages in anything other than exactitude.

>>7840459
I'm also confused by what I'm asking, so that explains why you are.

The fundamental issue is that I have an intuition about the reality of mathematical objects that I myself don't understand. I have no idea what to search for. Wittgenstein came up while I was looking, and this is all I have to go on so far. I don't particularly care about him.

What I want to know is if this gut feeling has already been felt by someone smarted than I, who turned it into some "-ism" that I can study. It's very frustrating to not be able to properly express my opinion.

>>7840482
Alright, I'm gonna write about on this:

first: It's perfectly fine that you're incredibly vague (apparently, you're even vague in your own head) about this. It just means, that any literature people recommend won't necessarily be topical. As well as I can make out you have some issue with whether numbers are real, this has been an issue confounding mathematicians for as long as mathematics has existed.

Here's the thing, that issue was already settled (basically) by the time Wittgenstein came along. The more interesting question, the question that was all the rage was this: Is Mathematics true? Ofc it depends on what you mean by "Mathematics" and what you mean by "true". This has also been the subject of heated philosophic debate for thousands of years.

So no matter where you exist on the following spectrums:
>What you mean by numbers
>what you mean by mathematics
>what you mean by real

there has probably been a philosopher or mathematician that's felt it too.

Euclid, Pythagoras, Descartes, Spinoza, Kant, Frege, Russell, and Wittgenstein are just a few of the names that dealt with this issue in its most direct form.

second, where to go from here:

Well, I guess its time to start pinning down your thought huh? Seriously, the fact you have a weird unnamed feeling inside you should feel pretty cool. Most of literature is filled with academics feeling around their innards and minds trying (and often failing!) to find something original. (By the time you're done reading philosophy, you might also feel this way, be forewarned)

I'm just not sure if its Wittgenstein you're after, because as I said before Wittgenstein's major works on the subject don't pertain directly to mathematical philosophy insofar as they stem from implications of mathematical philosophy.

I hope this has been clarifying and not mystifying, again, I'm sorry I couldn't be more specific, but until you can pin down these issues with more clarity I'll be unable to do more

http://plato.stanford.edu/entries/wittgenstein-mathematics/

You are all so fucking bad at this, just stop.

>>7840584

Wow, I appreciate the time you took to write this out anon. I'm going to try to explain what I mean further, but please understand that I'm a high school drop-out and not particularly gifted, so this doesn't come easily to me.

When I was asked "are numbers real", my initial reaction wasn't "yes" or "no", but "the question doesn't make sense". And the only way I can think to explain this is to look at a standard mathematical statement like 2 + 2 = 4.

Initially it seems like a proposition which could be true or false (true in this case). But it isn't, really. "2" doesn't refer to some abstract object that we can perform operations on to find answers. Rather "2 + 2 = 4" is a rule that mathematicians have created. In this way, maths is invented by us and not discovered.

The reason I picked Wittgenstein was because when I googled this idea to try and understand it, pages like this came up >>7840599

I think the following extract explains what I mean, but I can't be sure because it's much too dense and scholarly for me to wrap my head around:

Wittgenstein maintains that “mathematical propositions” are not real propositions and that “mathematical truth” is essentially non-referential and purely syntactical in nature. On Wittgenstein's view, we invent mathematical calculi and we expand mathematics by calculation and proof, and though we learn from a proof that a theorem can be derived from axioms by means of certain rules in a particular way, it is not the case that this proof-path pre-exists our construction of it.

What I want to know is if this theory is an already established "-ism" because I want to study it, so that in future when I'm asked about my position I can actually explain it and not make myself look like an idiot (despite clearly being one)

>>7840666
That's just Wittgenstein's philosophy of Mathematics. The "-ism" is "Wittgenstein's philosophy of mathematics"

Depending on your palate for formal logic (I think you demonstrated some understanding in your post, assuming you know what you mean when you invoke a word like proof) you can start reading books on set theory or discrete mathematics.

Otherwise, read Wittgenstein's work and if its too opaque, then read Ray Monk's supporting work (which is very good, and amazingly readable)

There are also a litany of books written in response to Wittgenstein's work which might interest you once you've worked through Wittgenstein.

wittgenstein has no "philosophy of mathematics". he used the learning process in mathematics as an illustration of his anthropology, or as he calls it his grammar.

of course philosophers will force a philosophy of mathematics into it but that is all nonsense.

>>7840749

This is wrong.

>>7840666

As is stated in the article, Wittgenstein ascribes to a type of constructivism.

>>7840666
>When I was asked "are numbers real", my initial reaction wasn't "yes" or "no", but "the question doesn't make sense"

The question makes sense m8. From a philosophical perspective the question is usually taken to mean, "do numbers enjoy an ontologically independent existance, or does their existence depend on other factors (e.g. only in the human mind, etc)?"

The standard theories:
Realism/Platonism: numbers are real things that exist, in a non-spatio-temporal manner, as Platonic forms. Read about Platonic forms if you want to get to grips with this.

Nominalism: numbers exist as concepts/generalisations/abstractions from our interaction with certain features of empirical reality. Upshot is that without humans numbers (or other mathematical objects) don't exist.

Formalism/Logicism: Different but related systems that try to establish the foundations of mathematics via formal systems or logic respectively.

There are lots of other theories of interest. Kant tried to establish arithmetic and geometry as results of the pure intuitions of time and space as part of his metaphysics.

This is all to do with the foundations of mathematics so extends beyond numbers to the objects of geometry, etc too. I believe Russell and Witte did stuff on meaningless statements too but I'm not sure how that links to this. Apologies for any errors this was all a sketch off the top of my head. Check out Stewart Shapiro's Thinking About Mathematics if you want a good intro book to the subject.

>>7840666
Sorry m8, I read the rest of your post. The -ism you want is logicism. It tried to establish mathematical truths as purely syntactic manipulation of a system of logic. It was the project of Russel and Whitehead's Principia Mathematica. Gödel tore that shit up with his incompleteness theorems. I think Witte had some scathing things to say about the Principia if I'm not mistaken.

But, yea I think it's logicism or formalism in the philosophy of mathematics thah you need to read more on. I think it's largely considered a defeated project though.