> Without a mind to understand, interpret, and define mathematics, does it exist?

Unquestioningly so. For instance, there is a number 2 and a number pi which are not caused by thinking. Beings which evolve on separate planets (or whatever) in separate universes have to to come to the conclusion that the ratio between the diameter and circumference of a circle is a certain number. Those numbers will be found to agree, though there is no causal link between the two. You can define what constitutes a circle, and define the question of the ratio of its parts, but you don't get to define pi.

Or, if you define "composite" and "prime", you don't get to decide which integers are one or the other, or facts like that two is the smallest prime and the only even one.

The question is: what is the difference between your existence and the existence of pi? Maybe there isn't any.

> facts like that two is the smallest prime and the only even one

I see the fact that two is the only even prime brought up from time to time as if it's inherently interesting. Why is it more interesting than the identical observation that 37 is the only prime which is a multiple of 37?

I guess this bothers me because 2 being the only even prime isn't a consequence of the definition of "prime"... it's part of the definition.

You're right in that evenness is divisibility by two by definition. For any P which is prime, P is the smallest divisible by P.

It is probably that divisibility by two (evenness) is interesting.

For example, it has the property that if we know the evenness of two integers, then we know the evenness of their sum or product.

Division of cases by even versus odd occurs regularly; in few circumstances do you have to separately reason about cases corresponding somehow to the elements of the congruence modulo 37.

As regards your third line, I feel compelled to note that if we know the equivalence class of two integers (mod 37), we also know the equivalence class (mod 37) of their sum and product. ;)

Indeed, it just the mod 37 congruence doesn't correspond to nice Boolean:

  odd(x + y) = odd(x) XOR odd(y)
  odd(x * y) = odd(x) AND odd(y)

There.

With regards to 1), I remember reading a paper I found on HN not too long ago trying to argue that no, it does not exist. To generalize, any piece of knowledge only exists after it is discovered. Can't remember what it is at the moment, maybe someone else knows what I'm talking about.

With regards to 3), I would assume Tegmark means laws of physics which we have not yet discovered, but nevertheless govern what occurs inside a black hole or at the beginning of the big bang. It is commonly believe that with a complete theory of quantum gravity, we will find that the singularities in these situations disappear and the laws of physics don't break down.

EDIT: Here's the HN submission from a few months back:

https://news.ycombinator.com/item?id=10068676 arxiv link: http://arxiv.org/abs/1506.03733

So before pi was discovered, circles didn't have a ratio of circumference to radius? Or did have one, but a value other than 3.14159...?

My mistake, I was remembering it incorrectly. Edited my previous reply to include a link to this:

http://arxiv.org/abs/1506.03733

Lee Smolin discusses a class of facts that are evoked:

"I would like to propose that there is a class of facts about the world, which concerns structures and objects which come to exist at specific moments, which, nevertheless, have rigid properties once they exist. Let us call this possibility evoked."

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He provides a table of how a fact and it's existence can be described:

Has rigid properties and existed prior? The fact was discovered

Has rigid properties and did not exist prior? The fact was evoked

Has no rigid properties but did exist prior? The fact was fantasized (Smolin does not elaborate on this in the paper)

Has no rigid properties and did not exist prior? The fact was invented

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Roberto Mangabeira Unger and Smolin hypothesize two principles to describe Smolin's view, temporal naturalism:

The singlular universe, all that exists is part of one singular universe

The reality of time, as in reality is not timeless

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With all this in mind, yes circles always did have a ratio of circumference to radius of pi. This is a property of the singular universe, and is a fact that was thus discovered.

The universe of mathematical possibilities that does not describe the universe was not discovered, it was evoked.

P.S. Please forgive the formatting of this response!

Thanks for the reference. From the essay:

1.In the real universe it is always some present moment, which is one of a succession of moments. Properties of mathematical objects, once evoked, are true independent of time.

2. The universe exists apart from being evoked by the human imagination, while mathematical objects do not exist before and apart from being evoked by human imagination.

> In the real universe it is always some present moment

That is rather naive. The entire succession of moments can be described as one object, in which time is a dimension. The unfolding through time is just the subjective experience of the human consciousness.

Some H.264 video also appears to unfold, presenting a depiction of events frame by frame. Yet, at the same time, it's also just a 1.2 gigabyte file: a giant integer.

A ratio doesn't exist outside of the human mind. The same goes for circles.

I think rather than saying "The universe behaves according to laws of physics...", it would be more appropriate/correct to say that our "laws" of physics somewhat consistently describe our observations of the universe.