Applicability of Maths to the Universe

[GrandizerII]

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.

Pi isn't messy in my opinion. It's complex. It's beautiful.
It contains the words to every novel ever written.
(That last one may not be true, but it sounds good)

I didn't say it's not beautiful. I find such numbers very fascinating actually but won't say no to the thinking that it nevertheless appears to contain random sequences of digits that seem to have no clear pattern to them. That's what I mean by "messy".

Also do you find it strange that all four sides of a square are equal in length ? I mean what are the odds that all four sides just happen to be equal ? That sounds like it was designed by some higher power to be that way. What things can you name in nature are exactly the same length on all four sides ?

No, I don't find it strange that all four sides of a square are equal in length. The square, by definition, must have four sides equal in length. The concept of a square, while based initially on observations of approximate squares, is easily defined as such. With pi, it's not like people historically made basic observations and were like "hey, let's equate pi to some complicated value just because it's intuitive". Rather, pi had to be discovered via certain calculations and then over time (with further understanding and technology) refinements continue to be made to the value to make it more and more precise. This is the bit that's fascinating me, that a number discovered in that way and is key to many of the problems solved just happens to be this really "messy" (to me) sort of number.

To be clear, I am not saying there is any hard problem here. I do suspect the answer lies in the definitions being used ultimately. Something about them seem to be leading to such strange (to me) numbers, but I don't know what the specific explanation(s) is/are or whether I am even qualified to ever comprehend the answers.

Ok, time to now read and try to digested what polymath has responded with.

[Jehanne]

Good videos here:

Khan Academy -- Does .9999 repeating equal 1?

[The Grand Nudger]

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.

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.

[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.

---

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.

---

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.

[Rahn127]

This video gives a good example of Pi like fractions.

The sin(1/5555555555...) = very close to Pi 
(if your calculator is in degrees)

[The Grand Nudger]

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.

Are there?  To whom or what?  Godel?  Math?  It doesn't seem like we've exhausted the full list of knowers or knowledge.  This would not be enough to reject platonism as a metaphysical stance.  If we assumed that there were some mathematical objects that demonstrably proved mathematical platonism false (and much more from doing it purely by axiom, as we address below) then a platonist can simply concede that mathematical objects are mental objects (and this isn't the only concession that can be made).  It may be the case that there are not always cognitive answers for mental objects.  This would surprise no one.  Satisfied expressions of taste, like "yum" are not cognitive objects, they cannot be true or false, though we've been known to mistake them as such - on account of being satisfied pattern seekers.  

However, in an effort to maintain the position, it will always be posited first and foremost, that we've got something wrong, perhaps as a consequence of our axioms...and what we've got wrong may be godels results.  

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.

Which is a leveler for platonism.  If it's just a disagreement over axioms, and if the things that are taken to be indicative that platonism might be false are merely products of differing axioms or aesthetics.....

[Jehanne]

Good videos here:

Khan Academy -- Does .9999 repeating equal 1?

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

.99999.... = 1

Here's something more formal:

Wikipedia -- 0.999..

I agree with the formal proof, which is what I think that the author of the Khan video (who, appears to be, at the time at least, a PhD finalist in mathematics) was trying to convey:

[polymath257]

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

.99999.... = 1

Here's something more formal:

Wikipedia -- 0.999..

I agree with the formal proof, which is what I think that the author of the Khan video (who, appears to be, at the time at least, a PhD finalist in mathematics) was trying to convey:

Yes. And the key should *always* be some way that shows the limit exists and is 1.

Just to give a *bad* example,

Suppose we let
x=1+2+2^2 +2^3 +2^4 + 2^5 + ...

Then,
2x=2+2^2 +2^3 +2^4 +2^5 +2^6 +...
=x-1

Which 'shows' that x= -1.

The problem is that the defining series for x doesn't converge, so the algebraic operations are not legitimate.

In the corresponding claim that
1=.9 + .09 + .009 +...

we would start with
x= .9 +.09 +.009 +...
and get
10x = 9 +.9 +.09 +...
=9+x,

leading to 9x=9, and then to x=1.

But this set of operations is ONLY legitimate if the series actually converges. And, yes, in this case, it does,
but that is a separate issue that needs to be addressed.

The Wiki proof is good because it effectively shows that convergence happens and what the convergence is to simultaneously.

[The Grand Nudger]

I wanted to add that we could go through an endless list of maths that might seem to show that this platonist conjecture or that one might be false - but they'll all be as easy to dismiss as the last. It's because we're arguing from a purely axiomatic standpoint, unless we're willing to bring it back to something in the OP conjecture. Unless we want to present some reason to believe that one set of axioms is better than another, then no two positions based on divergent sets of axioms can touch each other.

[polymath257]

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.

Are there?  To whom or what?  Godel?  Math?  It doesn't seem like we've exhausted the full list of knowers or knowledge.  This would not be enough to reject platonism as a metaphysical stance.  If we assumed that there were some mathematical objects that demonstrably proved mathematical platonism false (and much more from doing it purely by axiom, as we address below) then a platonist can simply concede that mathematical objects are mental objects (and this isn't the only concession that can be made).  It may be the case that there are not always cognitive answers for mental objects.  This would surprise no one.  Satisfied expressions of taste, like "yum" are not cognitive objects, they cannot be true or false, though we've been known to mistake them as such - on account of being satisfied pattern seekers.  

However, in an effort to maintain the position, it will always be posited first and foremost, that we've got something wrong, perhaps as a consequence of our axioms...and what we've got wrong may be godels results.  

Godel proved that *any* axiom system strong enough to talk about the natural numbers has sentences that cannot be resolved. So there will *always* be unanswerable questions.

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.

Which is a leveler for platonism.  If it's just a disagreement over axioms, and if the things that are taken to be indicative that platonism might be false are merely products of differing axioms or aesthetics.....

No, that is NOT the point. No matter what axiom system you choose, there will be questions that cannot be answered. The mathematical system is *defined* by the axioms, so the basic definition guarantees there will be unanswerable questions.

Furthermore, you can always *add* to the axioms either way, defining two *new* systems of mathematics.

It's sort of like geometry. Once we found that Euclidean geometry is NOT automatic, that said that the Platonic ideals for geometry simply don't exist in the way that Plato imagined.

---

I wanted to add that we could go through an endless list of maths that might seem to show that this platonist conjecture or that one might be false - but they'll all be as easy to dismiss as the last.  It's because we're arguing from a purely axiomatic standpoint, unless we're willing to bring it back to something in the OP conjecture.  Unless we want to present some reason to believe that one set of axioms is better than another, then no two positions based on divergent sets of axioms can touch each other.

Exactly the point which shows that Platonism isn't correct.

It isn't that one set of axioms is 'correct' and another isn't. It is that *no* axiom system can manage to answer all of the questions and ALL axiom systems produced by adding on undecidable questions are equally legitimate logically.

Platonists like to talk like the notions like 'set' are intuitive and obvious. But, when push came to shove, the intuitive version was self-contradictory.