Dave_Oblad » June 15th, 2016, 9:35 pm wrote:If we go back to Pi (3.14159...etc) we may notice something.. it has sequence. If you were a smart observant Number located in that string of numbers then you might notice that there are other numbers that preceded you and followed you. In those that follow, the string of numbers may extend to an infinite length (the Future) but if looking towards the beginning, it had a definitive beginning as the Number 3. So for that little simple Universe of Pi, the sequence had a beginning and thus Time (from its perspective) had a beginning, even if it has no end in the other direction (future).
That same would apply to any Equation that produces a growing Sequence. It had a first state or beginning.
Time is Sequence.
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IntroHi Dave,
Over in that other thread you invited people to comment on the contents of this post.
I don't have any special insight into the ultimate nature of the universe (I think it's turtles all the way down) but this paragraph fairly jumped out at me as requiring a bit of mathematical perspective so I typed this somewhat long post. It could be shorter. I said the same couple of things in several different ways.
You said you're busy so it's not necessary for you to reply and either agree or disagree. I have no dog in this fight. If you're ultimately right about the universe more power to you. I'll be surprised but as I said in the other thread, Tegmark's all wrong. But he's Max Tegmark and I'm not so there's that.
By the way do you know William Lane Craig's kalam cosmological argument? You're as wrong as he is and for exactly the same reason. You think times flows like the natural numbers in their usual order, one discrete chunk after another with a beginning and no end. There's no evidence for that. It's one of many possible mathematical models of time but not necessarily the right one.
Having said that I'll just give you my mathematical thoughts. If anything I say sounds dogmatic or argumentative that's just how I typed it out. My intent is to document my points of disagreement with your claim that just because we can define a thing called the decimal expansion of pi, that sequences are necessarily the right model of time.
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tl;dr versionMath isn't physics.
The fact that mathematicians use the idea of infnite sequences to represent the decimal expression of a real numbers doesn't tell us anything about time, which is a phenomenon of the real world. Nothing in math necessarily applies to the real world; although many things in math do.
Secondly, it is not logically necessary to define a decimal expression as a sequence. We could reorder the natural numbers and it would make no difference in how we represent pi. For example reverse usual order, ...,-2,-1,0. Now you see "time" never had a beginning, and it arrives at the present moment. And of course you could continue it forever too. Use the evens and odds but reverse the evens:
..., -6, -4, -2, 0, 1, 3, 5, 7, ...
Isn't that cool? Mathematicans know a lot about the concept of order and how to reorder the natural numbers in interesting ways. These kinds of examples are perfectly standard in math and the theory of orderings is extremely well studied.
There's no reason to think any of this has anything to do with the nature of time. And it turns out that decimal representations don't depend on order either. So your thesis is wrong philosophically and also mathematically.
And something about your idea is contradictory. If the world is a discrete sequence of events or instants, what are real numbers? Even if I grant you all the real numbers that can be cranked out by algorithms, most real numbers lie outside the realm of computation. You can't be a discretist and also believe in the real numbers, even if you can sort of believe in pi.
Ok here are my comments, and like I say you don't necessarily need to respond if you're busy or agree or disagree. It won't change what I know about math and it won't change the universe.
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Comments on your paragraphMathematically what is true is that the decimal expansion of pi is normally defined as a particular sequence, but it doesn't necessarily have to be defined that way. It's incorrect to say that a decimal expression "has sequence" because that implies that being a sequence is necessary to the nature of a decimal expression. But that is not true.
If x is a real number then it is a theorem that x possesses something called a "decimal expansion" that is a sequence of digits. Now a "sequence" is defined as simply a function from the counting numbers 1, 2, 3, ... to the set of digits 0 throgh 9.
For simplicity let's just talk about what's to the right of the decimal point. So let's talk about pi - 3 = .14159...
If the function for pi-3 is called f, then f(1) = 1, f(2) = 4, f(3) = 1, f(4) = 5, and so forth.
Point being that
a decimal expression is DEFINED as a particular sequence. So it doesn't "have" sequence, it's just a mathematical object that happens to be defined as a sequence.
But we could define it differently and it would make no difference.
Consider the reverse order ω*, sometimes denoted *ω, where that's the Greek lower case omega. This is the reverse of the usual order of the naturals. It goes "..., -4, -3, -2, -1, 0". It's the
exact same set of numbers, though ordered differently. Whenever "n < m" in the usual order, we just write "m <* n" in our reverse order.
Now we could just as well define decimal expressions based on the reverse order. We'd probably have to adjust a few things but no fundamental math would change.
What would happen if someone asks us for, say, the digit corresponding to 1 million? We just look up f(1 million) and that's the millionth digit!
The digits of pi don't depend on being a sequence. If you "mixed up" the digits by putting them in a completely different order, it woudn't matter a bit as long as we have the function f.
Conclusion: A decimal expression does happen to be a sequence, but it doesn't have to be. We could define it differently and nothing in math would change.
Dave_Oblad » June 15th, 2016, 9:35 pm wrote:If you were a smart observant Number located in that string of numbers then you might notice that there are other numbers that preceded you and followed you.
Of course you are anthropomorphising numbers. Which perhaps is a bit of a mystical intuitionistic point of view, even if unintended. You mean to make a joke or a light-hearted metaphor, but I think perhaps you should question your hidden assumptions. Such as, that numbers possess a subjective experience of the numbers nearby in some order. Or do you believe they do? You said somewhere that every gene has a copy of the universal quantum computer program, and if you believe that then I have no way to know what you might think about numbers.
Psychology aside, we already debunked that point. If I tell the schoolkids to line up by height today and by last name tomorrow, it's still the same set of kids. What if we reorder the natural numbers by the rule: "Evens before Odds!" Then we get the order
0, 2, 4, 6, ..., 1, 3, 5, 7, ...
This order type is called ω + ω. It's a copy of the natural numbers followed by another copy of the natural numbers. Now we've split up those adjoining kids who were always one-upping each other with "I'm one more than yoooou"
Yet the decimal expression is exactly the same as long as I have the function f. A decimal expression is completely determined by any function from the natural numbers to the digits and is independent of the order we place on the natural numbers.
Can you tell me, by the way, why you happen to believe that the way decimal representations are defined, is necessarily a model of time in the real world? Have you evidence? It's true that we have a mathematical function from the natural numbers to the digits. But that's only because we have a set of natural numbers. And THAT is only because we assume something called the axiom of infinity.
https://en.wikipedia.org/wiki/Axiom_of_infinityThe axiom of infinity is an ARBITRARY ASSUMPTION that gets modern set theory and the entire theory of the real numbers off the ground. And it does happen to be logically consistent to deny it. But as far as what we know of the physical world, it's doubtful. You can wave your arms about infinite multiverses or whatever but no physicist can demonstrate an instance of an actual infinity in the world. In math, sure we use it every day. As a purely abstract game. We assume there are infinite sets and derive the consequences, one of which is the fact that a real number has an infinite decimal expression.
But the fact that we can create a
formal, symbolic set of rules and then define decimal expressions, while being very cool math, has
nothing at all to do with time. Math isn't physics. The fact that we can do a particular formal derivation in math tells us nothing about the world we live in.
You have too much real-world faith in math and not enough appreciation for the nature of its abstraction.
Again we've debunked that line of thinking. In ω* there is no beginning and we could describe the exact same decimal expansions as functions on ω*. It would result in the exact same theory, we'd just write decimals right-to-left.
In math 3 doesn't exist "before 4" or something. Everything is a set. All the elements of a set exist at once. That's a point of confusion in our modern age of computers. When a programmer writes a For loop, they think of it as iterating in time. And of course it is. It's a description of a computational process that takes place in time and space, on physical hardward that someone made in a factory. When you add 1 + 1 in a computer a chip engineer can tell you how exactly much heat was produced.
But in math, the terms of a summation exist
all at once. It's not a process or the steps of an algorithm. All the terms exist at once, and so does their sum.
There is no time in math, although people can certainly use math to model abstract time. What actual time is, nobody knows. Do you think time itself is correctly modeled by the natural numbers in their usual order? As a discrete sequence of instants with a beginning and no end?
What is the evidence for that claim?
You really just have no argument here. In fact a contradictory one since I gather you thnk the world is a quantum computer. Now it's been proven that although quantum computers give you huge speedups in some special cases (integer factoring being the most striking example to date), it is still true that a quantum computer can NOT compute anything that a conventional computer can't. Quantum TMs and conventional TMs compute exactly the same set of functions.
You may well believe in pi, because pi is computable. We can program a computer to crank out as many digits as we want, limited only by computational resources. But most real numbers are not computable, and your real number line is shot full of holes and is in no way a sensible model of the continuum.
In short if you believe the world is a computer, you can't possibly believe that time is modeled by the standard real numbers (*).
These are mutually inconsistent beliefs.
(*) I mean standard reals as opposed to the intuitionist reals or the constructivist reals or the hyperreals or one of the many other possible conceptual models of the continuum.
[NOTE -- Are you arguing that time is modeled by the usual sequence of natural numbers 1, 2, 3, ...? Or that there's a real number line with pi on it somewhere? Are the real numbers a model of time too? I'm unclear on your position here. If the world is a discrete sequence of instants, what is pi? And even if pi is an algorithm, what about all the other real numbers that
aren't algorithms?]
Dave_Oblad » June 15th, 2016, 9:35 pm wrote:So for that little simple Universe of Pi, the sequence had a beginning and thus Time (from its perspective) had a beginning, even if it has no end in the other direction (future).
Not only aren't the standard real numbers OR the usual sequence of naturals necessarily correct models of time; but as we note, the digits of pi could be reordered any way we like and it wouldn't matter. Thinking of them as a sequence is convenience. And of course "infinite sequences" are just mathematical fictions based on the manifestly absurd claim that there is an infinite set of natural numbers. Of course there is
in our minds, but you sure can't point to one in the real world. And till you do, please keep infinitary math separate from any philosophizing about what might or might not be true of the world.
That right there is a conflation of two different things. There are sequences, and there are equations. Now by equations I'm sure you mean formulas. Algorithms that crank out numbers. All very cool. But there are only countably many Turing machines, and uncountably many sequences of digits. That's Cantor's diagonal argument. Or the Halting problem if you had the misfortone to study CS. (Haha just kidding. Or not ...). So you are confusing the real numbers with the computable real numbers, and that is a philosophical error, because those two models of the reals are very very different.
Of course that does not logically follow at all from the fact that mathematicians can write down the rules of formal set theory and then logically define something called the "decimal representation of the real number pi". It's purely a formal exercise. It depends on the axiom of infinity, without which it can't even be expressed(*), let alone proved.
(*) I'm not sure about that. Finitists (people who don't believe in infinite sets) do attempt to create a theory of the real numbers, but I don't know anything about it. They can't have been too successful else I'd have heard about it.
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ConclusionThe math of the real numbers says nothing at all about the true nature of time or the universe. And the order properties of the natural numbers are not necessary to the definition of a decimal expansion.
Well I hope some of this is helpful to your thinking.