Are mathematical theories and such solved in the same ways throughout each culture?
If another language was using a different numbering system, could it be “translated” per se the way a spoken language is?
–
It seems mathematics is a connection we all can truly share.
Um, I’m not sure what you mean but mathematical theorems and their proofs still have to be expressed in some sort of language. Sure, if you use standard notations and diagrams it’ll be a little more understandable, but I doubt I could make heads or tails of say, Gödel’s Incompleteness Theorem proof in Hindi or Mandarin. I wouldn’t probably be even able to identify it as such.
Mandarin’s a bit of an odd case, but I could probably spot a Hindi proof of GIT, just from knowing the notation. I’ve got Dr. Landau’s Grundlagen der Analysis, and while I don’t speak any German at all, I can generally puzzle out the meanings of paragraphs with symbols in them. I don’t think I’m the only one, either.
I took those as extreme cases because in countries with utterly different languages the notation seems to morph and change. Maybe I exaggerated a little, but I have trouble reading unfamiliar math papers in Russian even though I speak russian because I stumble over terms I can’t translate immediately. Sure, things like “group”, “ring” or “recursive function” are pretty simple, but I’ve seen some completely bizarre Russian math paragraphs that I would understand were they in English/standard notation, but the notation and terms completely baffled me.
I have to add that I would make a completely terrible mathematician. If you took something I’m familiar with intimately, say formal languages, I don’t think a language barrier would stop me. Slow me down, yes, but not stop me.
I think the real question here is whether or not any cultures use mathematics that are not based on Base-10. As long as all cultures use Base-10, then math should be constant, it seems to me, although the numbers themselves may be written differently.
Real discrepancies would occur if other counting systems than Base-10 are taken as the norm. The hoopla that prime numbers are somehow “universal” (allowing us to use them to commicate in some way with alien cultures), for example, is ridiculous. Our prime numbers (numbers that are divisible only by themselves and 1) are only “prime” in a base ten system. If an alien culture were to use base 12, base 15, or some other counting system, their “prime numbers” would be completely different.
If you include geometric measurements–especially angles–in the “equation”, things are even more fluid. The size of a “degree” is completely arbitrary in our way of measuring things. It would make a lot more sense to define degrees in terms of units of 100 degrees (e.g. straight angle = 100 degrees or right angles = 100 degrees or a full circle angle = 100 degrees), but the fact that it is based on a completely different and arbitary system means that there is even less chance of it being “universal”. As far as I know, though, all Earth cultures use the base that a circle represents 360 degrees, and if this is considered a constant, then all geometric theorems should work in every culture.
Only if you think math is limited to arithmetic.
This is not true.
Not true “Seventeen” is prine in any (integer) base system. It is true that 17 in base nine which is “sixteen” is not prime but the number seventeen is.
While this is true, pure math almost always works in radians as measure of angles. Radians are the natural unitl to use since one radian is the angle subtended at a distance of one unit by an arc one unit in length. When you see math formulas like sin(x) = x - (x^3)/6 + … they are alwasy expressed in radians.
I was checking out cheap mathematics books at Xidan Bookshop in Beijing. They were using very familar notation and I could piece together a fair bit.
Everything in here is actually rather good evidence that “mathematics” is language and culture independent. Kiminy is so dead-set on specific notations and notions of scale that he fails to realize that none of that is actually mathematics.
Number theory, for instance, is about the natural number system, which I can define using a specific set of properties. It is the only structure that has these properties (up to isomorphism, of course). That is, no matter what other culture used for scales or bases or notations, they would still come up with (something essentially the same as) the natural numbers.
I agree with your conclusion, but I don’t think the fact that the Germans use the same notation as us Americans is good evidence for it.
What I mean is that a close reading of Kiminy’s post and an understanding of why it’s flawed points one towards the proper philosophical argument. Mathematics is formulated independently of cultural referents. Culture can well influence the ways in which mathematicians discover new mathematics, and thus the “feel” of the mathematics, but it cannot change whether a given derivation is valid or not.
Suppose I were to meet a Martian mathematician. The symbols he would use for various numbers would be completely different, but specific numbers actually show up a lot less than you’d think in pure mathematics. He’d have different symbols that he’d use for names of variables, but there’s no actual mathematical content in those: A theorem can be expressed with any symbols at all used for the variables (though there are some which are conventionally used). And he’d use different notation for the various operators, relations, and the like. This would be a barrier to communication, but it would be fairly easily overcome, and once it was, exact translations could be made (far more exact than any translations of verbal languages).
Now, if we’re just talking about us humans here on Earth, I think that all cultures now use the Arabic numeral system, though I won’t swear to it. And I’m even more certain that all cultures use the same notation for operators and the like (everyone uses the same integral sign, for instance). However, the alphabet typically used for variable names does vary. Russian mathematicians, for instance, will generally express their equations using Cyrillic letters.
Strictly speaking, that’s not true, at least not if your “specific set of properties” is finite. If your specific set is P, then, by Gödel’s Incompleteness Theorem, there is a statement G whose truth is not decided by the properties in P. There is one (nonunique) structure N that satisfies the properties in P∪{G} and another structure N’ that satisfies the properties in P∪{not-G}. Both N and N’ have the properties in your original set P, but they are clearly not isomorphic structures.
In principle, one could imagine two cultures such that, for one, the concept of number naturally includes the property G, but for the other culture, not-G naturally applies to numbers.
Welcome to the world of universal properties, Tyrell. Us category theorists run the show here.
Now, think back to the Peano axioms. Remember there’s that one about induction? Specifically, if a set contains “0” and if a set contains n it contains the successor of n, then the set contains the natural numbers. That means that the Natural Numbers Object in sets is the limit of a certain diagram in the topos of sets. As such, there is a unique map from the natural numbers object to any other object that can fill in the diagram, and so for any two natural numbers objects there’s an isomorphism between them. See, for instance, Lambek & Scott or MacLane & Moerdijk.
The problem with your argument is one of scope. You’re mistaken about how Incompleteness applies. That is, if there is a formal system (which is a kind of structure, but not the other way around) that is sufficiently powerful to encode arithmetic, then Incompleteness applies. First of all, I never claimed to be talking about formal systems. Second of all, the natural numbers are the subject of arithmetic, but they are not themselves arithmetic. That is, they’re the subject matter, but not the system itself. Incompleteness is a totally moot point here.
I’m not entirely clear on how you’re working from axioms but avoiding a formal system. Care to elaborate?
** Mathochist**, all of your reasoning, category-theoretic as it may be, can be encoded in a formal system. You seem to have confused your ability to prove uniqueness theorems within this system as the ability to prove the uniqueness of the objects of thought that satisfy the axioms of the formal system that encodes your reasoning. This is an error.
A structure is just a list of properties which must be satisfied. “Group” is a non-unique structure, while “Natural Numbers” is a unique one. More classically, “Euclidean Geometry” is a structure.
Now, there is a formal system of theorems about each structure, the structure itself is not a formal system. That is, while there certainly are true-but-unprovable statements about the natural numbers – which leads to multiple distinct systems all starting from the same Peano axioms – there is still only one natural numbers object in the topos of sets – up to isomorphism, of course.
More down to earth, Peano can be reformulated as “the universal set with property N”. There may be many sets with property N, but we choose one – N – with the further property that if S has property N then there exists a unique function f from N to S “preserving” the property N (think homomorphism). This is essentially unique, because if S and S’ are both universal then there are f from S to S’ and f’ from S’ to S, and further the identities are the unique maps from S and S’ to themselves that universality requires. Thus the composition S -f-> S’ -f’-> S – being a map from S to itself preserving property N – is the identity on S (uniqueness), and so f and f’ are isomorphisms.
Similar arguments apply to why we can talk about “the” product of two sets or “the” disjoint union. Any two instances of the “Natural Numbers” structure are isomorphic, just like any two sets which behave as the product of A and B should are isomorphic.
My response is essentially in the latest post in re ultrafilter. You’re still confusing the map (the formal system of arithmetic) with the terrain (the structure of the Natural Numbers).
My wife is from Taiwan and she says she was taught a method of subtraction that I was never taught in school here in the USA.
026
My wife was taught the following method to solve the problem above.
The 6 is bigger than the 9 so you borrow a one from the tens column. The one becomes a “ten” in the ones column. Subtract the 9 from this ten and you get 1. One plus 6 equals 7.
Although this method may look slower and more complicated, it is used because its easier and faster to figure out subtraction problems using 10 minus a number.