Applicability of Maths to the Universe

[polymath257]

FTR, I don't see how platonism answers my question about pi better than non-platonist views. It's just pure curiosity for me. And yes, this is a bit of a deviation from the OP topic, but since this is still about maths, whatever. I will get to reading your response soon, polymath, but I do want to try and see if I can answer Rahn's questions first just in the hopes that maybe I can make more clear my state of mind regarding this.

Do you find 1/3 to be messy ?

If you cut a pie into 3 equal pieces, does each person have 1 piece of pie or do they have .3333333333333333333333333 pieces of the whole pie ?

My answer to this is "yes and no". While there are certainly infinite digits to 1/3, I can sort of see a clear enough pattern here and appreciate that, by splitting the set of real numbers between 0 and 1 into three equal subsets, the cutting points are going to have to be values with infinite digits to allow for that, but there is nevertheless an "orderliness" to it anyway, since it's the same value occupying the decimal values and when you look at 2/3, it's 0.666666... and 3/3 = 0.999999 (which is also 1), pattern here being that the decimal value is jumping by 3.

Perhaps for you this understanding should easily be applied to numbers like pi, but I don't see it the way you seem to be seeing it.

You have to have a bit of care here. I'll give you two numbers:

x=.12345678910111213141516171819202122....

What we do is write each integer in decimal, one after the other and produce a new decimal from that.

This number has a very definite order (base 10). But it is irrational (the pattern does not cycle) and (much harder) transcendental. It also has the property that every sequence of digits occurs with the 'probability' you would expect: 1 happens 1/10 of the time, 23 happens 1/100 of the time, etc in the limit.

We say that x is a normal number because these ratios work out. Notice, though, that the digits are far from 'random'.

Another:
y=.110001000000000000000001000000....

where we put a 1 in the n! places: so in decimal place 1!=1, 2!=2, 3!=6, 4!=24, 5!=120 (not shown), etc.

Once again, this clearly has a pattern. But it is *still* an irrational number (the pattern doesn't cycle) and it is also transcendental (in fact, it was one of the first numbers to be proved to be transcendental). But, the digit 2 never appears in its decimal expansion.

Now, with pi and e we know that both are transcendental: they are not the root of any polynomial with integer coefficients.

What we do NOT know is if either of these numbers is normal: we simply don't know if there is, in the long run, just as many appearances of the digit 4 as there are for the digit 1 or 8.

Are the digits of pi and e 'random'? No, of course not. We can write computer programs to determine their digits. They are fixed and determined (by all of our definitions). We can even write programs that can determine (rather quickly) what the 5 quadrilllionth hexadecimal digit of pi is.

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Good videos here:

Khan Academy -- Does .9999 repeating equal 1?

I'd actually rate it as fair. The problem with

.99999.... = 1

is understanding what the left side means. And many people seem to be uncomfortable with the idea that there can be more than one way to write the same number. But they also don't mind saying that 1/2=2/4=3/6, etc

The actual reason for this equality is to look at what the left side means. What it means is the end result of a process.

we look at the sequence of numbers
.9
.99
.999
.9999
.99999
etc

and we ask ourselves the following question: is there some number that these are getting closer and closer to as we go further and further out in the sequence?

If the answer is yes, that *limit* is what the expression means.

And, pretty clearly, as you go further and further out, these numbers get closer and closer to 1. The limit is 1, so the meaning of the left hand side is the number 1.

The same is true for, say .3333333...

We look at
.3
.33
.333
.3333
.33333
etc and ask if there is some number these are getting closer and closer to as we go further and further out in the sequence. And the answer is yes: 1/3. So the infinite decimal

.33333333..

means 1/3.

And yes, the same is true for pi=3.141592653589793.....

As we go further and further out, we get closer and closer to some number. And that number is pi. it is a *finite* number. In fact, 3<pi<4. it just has an infinite decimal expansion.

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Even those mathematicians that are Platonists agree that most real numbers are not computable. What Platonists would say is that these real numbers still exist in some Platonic realm.

Axiomatically, there must be some other definition or description which would be equivalent to the sequence of digits referred to in the halting problem.  It may be the case that we can't get there in the manner described - but there's a there to get to.  Yes.

A Platonist would say that this question has a definite answer.

Axiomatically, just as above, yes.  Every question has a definite answer, regardless of whether we possess it.

And I would say that this is directly contradicted by Godel's results. There are *always* questions that cannot be answered. it isn't simply a matter of not knowing, but that the issue cannot be resolved without making an arbitrary choice in our axioms.

We can construct two models of set theory: one answers the question yes, the other answers the question no.

So, whither Platonism?

A platonist may wonder whether set theory is complete, and suggest that a definite answer which might even damage platonism in some way would still remove this objection in it's entirety.

Well, we *know* that set theory isn't complete in the sense that all meaningful questions have answers that can be proven. And, in any supplemental system, there will ALWAYS be new questions that cannot be answered.

And the *only* real way to resolve such questions is to *arbitrarily* choose which way we want the axioms to go.

Now, we can make such choices based on things like aesthetics, but there is no way to determine the truth or falsity without making additional assumptions.