I'm not sure if I should post this here, but here goes:

I'm trying to determine whether this proposition:

is itself uncertain, or it doesn't refer to reality.

Here are my thoughts:
if we let
p := "the laws of mathematics refer to reality"
q := "they are certain"

Then the proposition becomes:

if p then ~q and if q then ~p

But this is a mathematical proposition (if we include first order logic to mathematics)

By its own standard, if it refers to reality then it is not certain
And if it is certain, then it does not refer to reality.

Am I correct in this or is there a mistake in the argument? I'm really not sure whether it is a mathematical proposition in itself, but from the discrete math I've taken, first order logic is usually thought part of math, even a foundation of math. Or maybe because it does not convert to a math proposition the way I've done so.

They are not mathematical statements, what does "mathematics refer to reality" mean mathematically?

But you can treat them as logical statements. You have p --> not q, and q --> not p, and these statements are equivalent. Basically it says that (p and q) is false, that is, mathematics cannot refer to reality and be certain.

For the record, mathematical statements do not refer to reality!

Your reply made me think that I may have a misconception of what a mathematical statement is. So I tried to look it up, but I couldn't find a reliable source. So please if you have any, share it with me!

But my first thoughts about your reply where that, although "mathematics refer to reality" may mean nothing mathematically, it may not be so for the whole proposition, i.e the whole of the quote. For example when I say:

Ax,Ey :x + y = 3

(Ax,Ey == for every x there exists a y) this is definitely a mathematical statement. I could go on to substitute x = 2 and y = 1. And so I get

2 + 1 = 3

which is also a mathematical statement. But what does 2 mean mathematically? Or 1? I'm not sure, but I don't think it means anything mathematically. Its just a number, which might refer to apples or oranges or whatever, or nothing at all (in this last case, 2+1=3 would simply be a meaningless well formed sentence) . But it doesn't have a mathematical meaning and of course 1, 2 or any number can't be said to be "true" or "false"

So I still believe p ->~q & q -> ~p is a mathematical statement, whatever you substitute p and q with, even if p and q mean nothing mathematically.

As to whether mathematics refer to reality, yes I too believe that it doesn't and that it is just a human construct. But no one can be certain of that! Perhaps reality IS mathematical, and we don't (or can't) know it. The fact that I believe that math is not real is more or less a philosophical position (just like I believe that there is an external reality or that there are other minds)

And thus this is another reason I think that AEs quote, although it refers to reality, it is uncertain. I wouldn't mind really since science is all about falsifiable beliefs, but he talks about certainty and said that if math refers to reality then it is uncertain. But what is his point? Is science certain? Physics? Is his claim certain?

Please go to the official Albert Einstein archives site and show where this quote appears. I've seen it spread around the internet, but cannot find it in his official archives.

Indeed, I read a few pages of the book, and it comes down to this:

There are two important facts about mathematics

- mathematics is always correct. There are no experiments that refute mathematics, unlike all other sciences.
- mathematics is used in science to accurately predict certain phenomena.

So Einstein asks himself the question why mathematics (who's theorems do not follow from experience, but from logical inferences) can so accurately describe reality.
The question to this answer is our famous quote. He claims that the mathematical laws which describe the universe are not certain. That is, it is impossible to axiomatize physics.

To me, the quote is common sense, really. Let's say that we have an axiomatic system that describes reality. How can we ever be certain that these axioms describe reality? We can't.

Thanks for the link! It was a very interesting read.

I believe I made 2 mistakes above (if not correct me).

One was that I took the quote out of context. I always thought this was an answer to an interview question on whether mathematics is real or not. I don't know how that stuck in my mind.

The second mistake is that the quote doesn't refer to reality, but it refers to referring to reality through the "laws of mathematics" (I believe by this he means mathematical axioms after reading through) So judging it by its own standard makes no point.

I think I have a better idea what he meant by it, so thanks for your replies!

Also, forgive me for not replying sooner, but right now I have an examination month. So I didn't want to reply without reading the link first but with all the workload, it had to be a significant time investment.

If you look at them as propositional statements, you can look at the truth assignments which would make the total statement true. Commonly understood as the underlying model.

A bit easier is to rewrite the proposition.

(p => ~q) /\ (q => ~p)
<=> (p|q) /\ (q|p) {definition of '|', the NAND or Sheffer stroke}
<=> (p|q) /\ (p|q) {associativity of '|'}
<=> (p|q) {idempotency of '/\'}
<=> ~(p /\ q) {definition of '|'}

In short, p and q can't both be true at the same time, any other truth assignment to p and q satisfies the original proposition. Moreover, the original proposition was a bit too lengthy, just claiming 'p => ~q' would be enough, the second statement is superfluous.

If you assume q to be true, 'the laws of math are certain,' it follows that p cannot hold, therefor 'the laws of math don't refer to reality'.

Sounds to me like old Al was no linguist. English was his second language, so is this must be a translation, either by someone else, or by Einstein himself.