OK, I’ve been thinking about this for a bit and, while it could be complete BS, I was wondering if it had any validity to it whatsoever.
Are numbers set in stone? Is it a universal constant that whole numbers go 1, 2, 3, 4, etc.? How was it decided that 1+1 = 2? Who decided that 1 would be a whole number?
OK, imagine an intelligent civilization on another planet that has never visited Earth. Whatever they decided to call their numbers, would they have numbers like us? Would they be able to count to ten using the same ten numbers we did? Or is it that our numbers are our own creation and our understanding of the universal conditions is directly related to the numbers we’ve created?
I’m sorry if this doesn’t seem to make sense. If you’re not understanding I’ll try to elaborate further later.
Yeah, math is more or less completely arbitrary. You can invent any kind of logical system you want, pick off anything you like as numbers, and it’s just as valid in a certain sense.
However, the math we have is pretty good for modelling physical phenomena, and it has some interesting properties, so I’m gonna stick with it.
There’s something called topos theory that I think relates to this, but it’s a bit advanced…I’ll see if I can find a decent link.
Even here on Earth, we have other numerical systems with varying properties, but basic operations (+ - / *) are likely to be the standard across the universe.
Remember that here we have the decimal, hexadecimal, octagonal, binary, and other numerical variations of a counting system.
What ultrafilter said. I think mathematics is a complete mental exersize, albeit one that has had some amazing effects on how we perceive the world.
If I pick up on apple and place it next to another one, then say “two” then do the same thing with paper, then sheep, then cups… etc, etc… you would see the concept of the number “two.”
That concept, though instilled in you by the presence of material objects, in no way depends on those objects. That is, I can eliminate all apples from the world and still you would have the number two in your head.
So, was “two” in the apples or in your head? I think it is obvious that our numbers are nowhere but in our heads, though many people seem to disagree for reasons I’ve never quite grasped. Damn nominalists, the whole bunch of them. Or is that materialists? Can a materialist be a nominalist? It almost seems they must be…
That is an interesting thread ultrafilter, I don’t get into MPSIMS much and I definitely missed that one. Shame it died…
There is a school of thought that says mathematics exists without any need of recognition by humans. That it is a feature of reality and not reliant upon human perception and predates our own discovery of it. When you closely examine the fabulous nature of numbers and calculation it’s hard not to feel that way sometimes.
Well when discussing the philosophy of mathematics, you can forget everything except for the whole numbers. All else is derived from those according to arbitrary rules that lead to interesting results.
The fundamental axiom is this: the whole numbers exist and they go on for ever.
“Addition”, Ender, is simply the name for the mapping that takes one whole number to another one. “1+1 = 2” merely defines what “+” means. Similarly multiplication or any other arithmetic function you wish to name. So forget 'em. It’s the axiom above you want to concentrate on.
Assuming it is based on the whole numbers, I would expect an alien mathematics to be very similar to our own. They would have come up with some clever stuff that we haven’t thought of yet and vice versa, but it would all be compatible.
So in the end we come to erl’s quandry: does “2” exist outside the framework of material objects? And is there something else we could base a mathematic on? Sadly there is no proof either way for the former question, since it exists without the system we would seek to prove it with. As for the latter question - well, it could be argued that some areas of maths do indeed attempt to do without number. But whether it is still all related in the end or not, I couldn’t tell you.
But (unless I’m completely misinterpreting it) kabbes is saying that the mathematical signs are tools used for the numbers while JepSnertRF says that it’s the signs themselves that are the constants. It seems as if you both have opposite cause and effects.
Well my point of view comes from building up the arithmetic from first principles in algebra lectures at university. It is the pure mathematical approach.
JepSnert seems to be coming at it from an applied maths point of view and is no less correct in essence, though once you start to plug the gaps with more rigour you’ll run into trouble.
I have always thought that basic mathematic knowledge is a priori and that the concepts existed in the universe for humans to discover. As for advanced mathematics, I believe that it becomes much more of an excerise in logic. It, as it has been explained to me, is sort of a game or language experiment and complicated proofs usually involve some sort of breakthrough in how to manipulate the symbols correctly.
As for me, I always struggled with math, but my cousin has a Ph.D. in mathematics from Pitt and freakin amazes me. She said that one day a divine revelation occured and she just understod.
Well, I have asked this before of a Maths Professor when I was at Uni and he came up with a very long and complicated theory, none of which I understood!! Essentially, the gist of what he said was that existence IS maths, never mind whether maths exists!! What we know as maths is just the way we describe it……
I don’t want to get too bogged down in deriving arithmetic from first principles, but I will say that it’s all based on group theory.
Basically (IIRC), “+” is the mapping from the ordered pair (n,1) to m, where we define m as n+1. Having defined an operation, we can then use it to create identities, other operations, closed groups, rings, fields and eventually the whole of arithmetic. Once you have arithmetic you can go on to develop polynomial groups and vector spaces. Then you’re really motoring.
It all begins with defining “+” as being the mapping that takes (1,1) to 2.
Surely Integers simply are; one rock, one planet, one molecule, one atom, one electron, one person, one tentacled monstrosity, one stalk-eyeballed slime-being; it is difficult to imagine a scenario in which intelligent life wouldn’t have stumbled across the idea of integers.
Differential calculus, on the other hand, is a huge conspiratorial joke to which I alone have not been initiated.
Hmm, and I found differential calculus to be infinitely more intuitive than even regular old algebra. Strange how different people think.
I agree that the case for an intuitive grasp of what we call integers is going to be felt across the universe for any beings which need to do stuff. Hell, even birds can count! Ever hear about those animal counting experiments?
That said, “one” is still a pretty abstract idea when divorced from particulars. And it is amusing to me that all operations that I know of are derived from addition.
For the record, I do not believe that mathematical concepts have any existence outside of our own minds. I also think that this is one of those questions where the evidence falls squarely on the side of whatever you happen to believe already. From a working mathematician’s viewpoint, there isn’t any difference (although it will influence whether they accept certain statements as true, but that’s a story for another post).
What we normally consider math is the consequences of set theory, and every theory can be phrased in terms of sets. There are those who are researching alternative foundations, but I don’t think that any of those have met with wide acceptance.
A very similar debate was hashed out a few months ago here, but not very conclusively.
Regarding this OP, as several people have pointed out the concepts behind the integers follow pretty directly from observations of the universe around us. (Try to imagine an atomic theory where one electron plus one electron equals, say, three electrons. Molecular chemistry would suddenly be a whole lot different!) But on the other hand, keep in mind that there are an infinite number of equally valid “theories of arithmetic”, thanks to Godel. These theories all agree on the simple stuff, like 1+1=2, but disagree on certain huge complicated axioms. So it’s possible that an alien species could have a different notion of arithmetic than we do. It’s just that no one would notice the difference, except for number theorists.
I think that you have it backwards. Are you saying that “+” is defined such that “n+1” is equal to one more than n? But how do we know what comes after n? The way I look at it, “1” is defined as the multiplicative identity. “+” is simply a basic concept that is not defined in terms of anything else. It is simply the additive operation. Everything else is defined in terms of that. We know that 1+1=2 because 2 is defined as the number that is obtained from adding 1 to 1. All fields, and most nonfields, have subfields isomorphic to the rationals. So no matter what number system these aliens have, there would be a portion of their system that we recognize. For example, suppose the aliens consider five by five matrices to be the fundamental numbers, and build everything else out of them. Then if we say 1=
10000
01000
00100
00010
00001
All of the other rational numbers can be constructed just from the identity matrix. You might say “but those aren’t rational numbers, those are matrices”. My response to that is “they behave in exactly the same way as rational numbers; just how are they different?”
There might be some extra part that we ask “why did you put that in?” (in this example, there’s the extra part of the 24 degrees of freedom that aren’t included in the rational number system) but the main part will be the same as ours. Unless the aliens never develop fields, which seems incredibly unlikely.
No, mathematics doesn’t use make-up. You’ve got to accept mathematics as she is, warts and all.
::dodges brickbats::
In any world with discrete but more or less equivalent objects (oranges, cattle) that intelligent beings want to collect, those beings - in my opinion, anyway - are overwhelmingly likely to develop the counting numbers, and addition.
From there, it’s only a matter of time until they derive the complex number field, as kabbes indicated.
So if mathematics is ‘made up’, I’d argue that there are powerful forces that would push most intelligent races to develop a great deal of the same basic mathematics.
My head is already spinning. I guess you can say 1 + 1 = 2 IF the object that 1 is representing is perfectly equal to what the OTHER 1 is representing. Hence, mathematics is philosphical until you find two objects in the universe that are 100% equal.
As for aliens coming up with a different system, it’s kind of hard to take 1, subtract 0 and not get zero. But they may have a more detailed way of doing it, I guess.
