I like this better than your first statement. Stick with Frylock, he’ll show you the way. His points of argument may seem dimensionless, but in reality I think they may have some substance. Who can say for sure either way?
I really do have to take an extended break from this discussion. It’s been fun, enlightening, and entertaining.
Thanks and Cheers all.
I’d love to know more, myself! I remember that Martin Gardner, when he introduced John Horton Conway’s “Game of Life” to the greater world (and tied up umpteen per cent of the world’s computer power in Life games!) suggested that, at the very smallest level, the cosmos might be “cellular.” It would entail some sort of digital (or even binary) modality. A cell either contains a unit of some sort, or is empty. Units cause other, nearby cells to become filled…or empty…and so, at the smallest scale, “movement” is really “jumping” or a kind of micro-teleportation.
Buckminster Fuller fans might fancy to suggest that, at this smallest scale, the cells are not a cubical grid, like a 3-D checkerboard, but a space-filling mesh of tetrahedrons.
(“Kites and Darts” are right out!)
The Higgs boson gives them that.
Deleted.
erik150x writes:
> In “my” mind and I am sure not the only one who thinks this way, the ultimate
> goal of math is to describe the fundamental language of nature, of which if we
> were all knowing there would be some perfect description of.
If you’re asserting that once we find a perfect description of nature, we would know that the mathematics that we used for that description must be the “real” mathematics, it’s not at all clear that that’s true. It’s possible for several different types of mathematics to completely and exactly describe a single physical situation. The example that comes to my mind is that given by Noether’s Theorem:
What this says is that it’s possible to describe certain physical facts either by symmetry laws or by conservation laws. So the fact that the physical world is space-invariant implies and is implied by the conservation of linear momentum and the fact that the physical world is time-invariant implies and is implied by the conservation of energy. There are many examples like this within mathematical models for physical systems, where the world fits one mathematical description completely and exactly and another mathematical description completely and exactly.
Now, admittedly, these are using two different subfields of one mathematical system, not completely different mathematical systems. But it’s possible to imagine that something like this might be true of different mathematical systems. It’s possible to imagine that we might someday have two different mathematical systems, each built up from completely different axioms but each consistent within themselves. It’s possible to imagine that each of them might completely and exactly describe the physical world. There would then be no way to say which of them is the “real” mathematics.
On the other side of the coin is Gödel’s incompleteness theorems which show us that no single axiomatic system can ever be complete and therefore will ever be capable of fully describing nature.
The idea of “real mathematics” vs. “made-up mathematics” is nonsense. It’s a misunderstanding of what mathematics is.
Now, if you want to ask whether there are infinitesimal distance ratios in the world, or infinitesimal timespan ratios, or infinitesimal energy deviation ratios, or what have you (and note that these are three completely different questions), you can go ahead and ask that physical, empirical question.
But earlier, erik150x, you had repeatedly accepted that infinitesimal such differences were of no real-world importance. Thus, the theory ignoring infinitesimal differences is, at the very least, a useful, accurate approximation, if not ultimately precise, in the same way as so many other physical theories. And if we are to care about infinitesimal physical differences, well… what sort of experiment do you have in mind which would discern their existence or lack thereof?
It’s that physical, empirical question, then, with which you are to be concerned. It’s not an issue of mathematics. Mathematics studies all games (or at least, all the ones anyone is interested in studying).
As for Gödel’s incompleteness theorem, it does not in any practical sense rule out a complete theory of physical law. You could perfectly well discover a series of rules which determine the result of every experiment. You could imagine a set of calculation rules such that, if you put to them the question “What will the world look like at date and location so-and-so?”, and began chugging away, they would spit out the answer. What more could you want?
I could imagine a physical universe which followed the rules of Mario physics quite precisely. I could sit down, write up those rules as code, and even simulate that universe. Indeed, it’s already been done, to my great childhood joy. Would anyone deny that there is a small set of formal rules which describes that universe’s physical laws completely? Yet that particular universe is just as subject to Gödel’s incompleteness theorem as anyone else. The Gödelian incompleteness just isn’t a relevant kind of incompleteness; it’s not the sort of thing we would ordinarily consider a failure to completely enumerate physical law.
Here is the only problem Gödel’s incompleteness theorem presents:
Suppose John sits down with some formal physical theory and gets to drawing out all the predictions it produces, with a particular devious plan in mind. He commits himself to ringing a bell whenever, and only whenever, he finds the theory producing the conclusion “John will never ring that bell”.
Then, if the theory ever says “John will never ring that bell”, John will follow it up by immediately disproving the theory. Accordingly, to the extent that the theory is sound, the theory will never make that particular prediction, and thus John will never ring that bell, and thus that the theory’s failure to produce the conclusion “John will never ring that bell” will be failure to draw a true conclusion.
Fine. That is all as it is. But so what? Would we admonish a physical theory as incomplete if it only told us, in painstaking detail, how to calculate whether John would or would not ring the bell by any particular time, without specifically addressing the matter of whether John ever rings the bell? This is weak criticism; what role do statements like “X will never happen” play above and beyond the combined effect of statements like “X will not happen by the year 2000”, “X will not happen by the year 3000”, “X will not happen by the year 4000”, etc.?
The question “Does John ever ring the bell?” plays no role in physics, beyond that played by sharper questions of a sort Gödel puts no proscription on our ability to handle. If someone were to make the mere prediction “John will at some point, eventually, ring that bell”, there’s no way to directly falsify it, since they can always hold “Sure, not yet, but just wait…”. We might well consider such propositions scientifically meaningless. (I would even go so far as to not consider them to have some status as either true or false; I don’t see the value in pretending every such thing automatically has some externally assigned and potentially inaccessible truth status). At any rate, they needn’t be the sort of thing we mean when seeking the complete laws of physics.
Theories of physical law tend to be empirical rather than axiomatic, or, like quantum mechanics, a hybrid of the two.
I just ran across this book. It’s new and I don’t know anything about it beyond its description, but it certainly looks relevant to this topic: Which Numbers are Real? by Michael Henle.
Another new book I saw while browsing is The Irrationals: A Story of the Numbers You Can’t Count On by Julian Havil.
One the reviewers says:
I’ll bet an irrational amount of money that’s correct.
Pi dollars? Why not just bet with imaginary money?
There’s some large men who will have words with your kneecaps if you try that.
Mmmm. Pi dollars.
There is an interesting website that has an in depth discussion of this topic at:
http://www.youwantedanswers.net
Suffice it to say that the answer to the question of whether or not .999… = 1 can be seen
to be more profound than a simple math topic.
At a quick glance, that page is a lot of crankery (which isn’t to deny that sense could be made of that crankery).
Can you say a little bit about what you mean by “crankery”?
(:p;) (but that’s not to say I wouldn’t read an answer…))
It’s sheer crankery. As soon as you read “at infinity,” which he repeats about a thousand times, you know that it’s crank math. He also repeatedly mentions 1/infinity as an infinitesimal that balances out “AT INFINITY the infinitely precise sequence .9999…,”
I like this bit:
You can, as all the real mathematicians here have said, design a mathematics that uses infinitesimals and defines 1/infinity. That’s not what he says. And the way infinity is normally defined and used accords perfectly with the real world and doesn’t use infinitesimals, no matter what he says. It’s almost exactly the same lack of understanding as erik150x, merely expressed differently.
It’s not an interesting website and the discussion is not in depth; however, the question really is a simply math topic. He leaves out all the math.
I was amused by this false claim: