Is mathematics synthetic?

That's just fine (I do not know the answers)! But you surprised me by saying "Every cube has six sides" is contingent... Also, the stick statement --- I thought you'd say it's like a math statement. The reason is this: this particular statement can be known by chimp as well, although chimps have no language and therefore no logic. ???

 

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I recently took courses on the foundation and philosophy of mathematics. The main book we read from was "Philosophy of Mathematics," edited by Paul Benacerraf and Hilary Putnam. You could also check out "Principles of Mathematics" or other stuff by Betrand Russell. Depending on your tastes - whether you like math itself or the logic behind it - either of these books should suit you well.

 

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From my point of view, considerate mathemathics as sintethics a priori is one of worst errors of Kant. Of course they are a priori in spite of Lakatos who considere mathemathics as quasi empirical sentences (but nobody have never seen "sinus" walking down the street!). But they're necesarialy analithic because its negation implicates a contradiction. Since the non contradiction principle (Leibnitz) is the only criterium to validate a mathemathical sentence, we must recognise, I think, that aritmethics, algebra, geometry... are analithical a priori.

 

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Their negation necessarily implies contradiction?? And what about non-Euclidean geometries? You get them by denying the axiom of parallels (the 5th postulate in the Elements). Its negation does not imply any contradiction, contrary to what you're claiming!

 

Says proteus:

"You must be in a lousy mood these days, gendanken."

If by lousy you imply a bitter old hag, no; whereas lousy in terms of possibly being riddled with lice…maybe

"you know, gendanken, you can also know that logicism faliled even if you hadn't read Principia Mathematica"

I have found two basics in quantificational logic: one, that it takes years and a day of university to grasp the damn thing and two, that I’m among the basest of asses when it comes down to it because, with due honesty, its all mud to me.

Number theory, analytical philosophy, constructivism, all those gadgets like Hegel’s of thesis in flux……..are in my opinion a big old maggoty cheese with too many irregular verbs! But that is not say the quest to see what had to be said fizzled out empty-handed. My understanding of math logic, so far, is that there’s always been a hunt for veracity in math. Frege, from my take of it, did not believe that math was an extended branch of human logic- but that was *not* to say it couldn’t be proved deductively using logic systems already extant. Its absurd to think that these things would not be had they not been analyzed and proven by man. Don't you think?

So it was, as I see it, that he gagged at psychologism in logic, specifically those ideas trying to sneak in with math that say numbers are subjective and lifeless without us.
Allright, numbers maybe, but quanitifiable properties? Blashphemy!

And that's where I jumped in and grabbed him- I too believe that value ranges, number entities, true ,false, quantity, mass and property exist in a world of their own, objectively, and separate from the mental world.

What I feel is something akin to what wemorris here stated about 5 posts up:

” Will the concept of 1 arise regardless of the nature of the consciousness that searches for it? One can only guess really, but I'd guess yes.

Its incredibly enticing to imagine a topaz planet whizzing about somewhere in Andromeda and its beings a civilized bunch, what would their 1’s amd 2’s look like? Would they too know of geometry and algebra? I wish I lived long enough to find out.

The quantifiable is free and beautiful. Say math to me and I think ordered chaos. Numbers remind me of noble savages.

And no, proteus, I don’t think I’ve mixed up anything. Had I plodded down and dogged over all that blasted terminology and wrung my brains over Zermelo-Fraenkel Set Theories and isomorphic classic math as you have I probably would have. Likewise, no, I wasn’t trying to test you on whether you actually read Russell + Whitehead’s work; I’m glad to hear no one really has read the damn thing because I was already strongly convinced no one had. I only remember the threat of seizure when first seeing it and this has happened only twice in my life. The other was on coming cross "Atlas Shrugged". No-thank-you.


(((((Ok, fine...I was testing you.

 

Says charles:

Holy mother, where do I start? Consider fractals, the idea of reiterative patterns in nature, that no matter how you tweak it, chaos shows up as sexy, fierce, simple, and beautiful.

The sieving out of prime numbers, take the first number (conventionally 2) and going down the line scratch out all its multiples, then take the next number and do the same; then again then again. The numbers you're left with are primes.

The golden ratio, the Grecian idea of proportial beauty in quanta- not to mention that research has been done showing that polygons consisting of this very ratio are seen as considerably more pleasing than those otherwise.

The Fabonocci (sp?) sequence and its stamp in nature, the conch the most beautiful example of all.

Fermat's last thereom teasing a young college graduate like a beautiful harlott.....nothing drives men like beauty, and nothing drives learned men like math.
Force, what is it? a mystical current only explainable in books the size of small dogs? or simply mass times acceleration?
Look at it:
F= ma

that's what its is. What's mass? a ratio of said force to that of its movement, acceleration of the same to its mass.

So so so so many- prime numbers, polygons, holy god functions ( I remeber in grade school being struck by how obediently a line sways to a function; give me any function, all I had to do was pucnh in a number and the thing did my bidding without fail). Polynomials and Pascal discoveirng his triangle......................WHERE DO I STOP?

BEAUTIFUL

 

I'd like to. But I'm not sure it's not wishful thinking on my part.

What is it about math statements that can't be explained in the fashion of the empirical sciences? What comes to mind at once is, it's their degree of certainty. Although natural laws can be imagined to be different, the truths of arithmetics cannot. Or a "rectangular circle" seems more impossible than the law of gravity being different from what it really is. But, as some say, there's only one basic kind of certainty, that of the laws of nature, and all other certainties are only parasitic on it. Deduction, they say, is just an extremely reliable type of induction. If you break a stick, you'll get two shorter sticks. This statement has a strong feeling of necessity about it. But is it a law of nature or is it rather a consequence of Eucledian geometry? It can hardly be denied that some primates are aware of the fact that breaking a stick will shorten it. It's like an inductive generalization, still it's about as certain as any theorem of Euclid. Maybe when we want to say that mathematics is in a different sphere of existence than physical reality, we're just being dogmatic. But, to be frank, I don't believe this tale myself. I think the laws of logic and math are really different in their existence from those of nature.

Their numbers couldn't be different from ours --- otherwise they wouldn't be numbers. Now, this is the rigidity that makes me think that numbers are not physical in any sense.
I can imagine little green Martian mathematicians with 5 legs swimming in lava but I can't imagine 5 being even. Or any number being different from itself. They are absolutely immutable. (Of course, I'm not talking about the names of numbers that surely could be changed and by such a trick you could say that "5 is even", denoting some other (even) number by the sign "5". But that would only be a nominal trick with names.) And if somebody likes evolution theory and believes it is only evolution that has shaped our thinking, then I'd really like to ask him or her to explain why is it possible for us to imagine wildly different worlds but impossible to imagine a number that has been changed just a tiny little bit?

 

Nice argument (I mean the one of the non euclidian geometry). But, if its negation does'nt implicates contradiction, can we aseverate they are true? I think there are some abuse of the reasonament in any parts of mathemathics.

 

I think I understand your point. If the negation of a mathematical statement is not a contradiction, then the statement itself cannot be logically necessary, only contingent. And this seems to be problematic because math statements are thought to be necessary. This means that the certain axioms of geometry are not self-evident, that is, they are not logical axioms. You can use this argument to prove that mathematics is not a part of logic because certain mathematical axioms are less necessary than those of logic.

If you equate "true" with "logically necessary", then they are not true. If you equate "true" with something like "describes a particular type of space adequately", then they are true with respect to the corresponding space but false with respect to the others.

 

I see you understand completely what I said. Imre Lakatos thought that mathemathical sentences were a almost empirical ones. I'm not agree with him, what do you think?

 

Mathematics is like a spectrum: arithmetics is pretty close to logic (though not identical to it) and geometry is pretty close to an emprical physical theory of space. Maybe geometry is after all nothing but a very simple physical theory. I don't have the same feeling about set theory or arithmetics because they don't seem empirical theories at all. The rigidity of their concepts pushes one toward a Platonic theory, I think. So, I accept Platonism in set theory and arithmetics and think that their concepts correspond to Platonic entities. (Incidentally, I think many concepts of natural languages like English also correspond to Platonic entities, pace Chomsky and his buddies.)

 

I was just wondering Wes Morris; when you work maths puzzles do you see the face of God?

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Why does no language mean no logic? Logic is a cognitive process and just because chimps may not be able to express it, doesn't mean that it doesn't exist.

Well I figured that a cube itself is not a neccesary truth, however it is contingent: Collins Gem and Thesaurus; 'a. depending (on) -n. group part of larger group'. The fact that a cube, by definition, has to have six sides means that a cube is dependent on sides; sides are not dependent on the cube, i.e., six sides do not neccessarily make a cube, but a cube is always made of six sides! Maybe in mathemtaical terms contingent has a slightly different meaning, however if this is true then you should have elaborated and explained in your post

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Surely it is possible to test mathematics itself, to find the truth, by creating an equation that should work, if mathematics is fact! I'm sure this must have been done for years but something like

sqrt/10 = 3.1622776601683793319988935444327,

and so sqrt/1 (should) = 1/10th of 3.1622776601683793319988935444327. This probably doesn't even work, because of ratios, or something, but there must be some mathematical equation that can test maths itself.

 

Why does no language mean no logic? Logic is a cognitive process and just because chimps may not be able to express it, doesn't mean that it doesn't exist.

(I don't have much time to write, so just a short remark on this.)

You claimed that the statement "If I break a part off a stick, I'll get a shorter stick" is a statement of logic. Okay, what does this mean? One way we can approach logic is saying that logic is a means to explicate implicit relations between concepts. Remember the worn-out example, "All bachelors are unmarried"? Here we use logic to see that being a bachelor and being unmarried are related in such a way as the sentence comes out true.

Now if you accept this "definition" of logic, then your claim commits you to saying that the concepts "stick" and "break" are such that simply by understanding what they mean you can come to the conclusion that if you break off part of a stick you'll get a shorter stick... Well, I think I have to admit that it's very well possible that I was mistaken and you were right. As far as I can see the concept of "breaking a longish thing" does imply that the broken thing will be shorter than before. Which means that it is a truth of logic after all, as you claimed.

 

Yes, every time his face appears before me in a mystical cloud. Then he's all "Guess what? I'm damning the heck right out of Charles Fleming", and I'm all "but that's what you said last time!" and then we laugh and laugh and laugh. *sigh* Math is cool.

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Dontcha just hate it when you assert a universal law and you get it WRONG!

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If you meant "1+ 1= 2", then it's true precisely because of the way "1", "2", and "+" are defined. Mathematics is "verbal" knowledge and therefore "synthetic" (in the sense of Kant).

To respond to some of the other points made:

Mathematics is peculiar in that it is BOTH created and discovered!

Every theorem in mathematics resides in some "axiom system".
We create the theorem when we create the axioms. We discover the theorem when we prove it.

As to why mathematics (if it is "synthetic") applies to the universe: it is simply that mathematical systems (what I just called "axiom systems" above) are templates.

A mathematical system (or axiom system) consists of
1) Undefined terms
2) Terms defined using the undefined terms
3) Axioms that apply to both undefined and defined terms
4) Theorems proved from the axioms

These are entirely arbitrary (as long as they are consistent).

To APPLY mathematics, we have to give meaning (from our application) to the undefined terms, in such a way that the axioms are still true. That's why calculus, for example, can be applied to economics, or biology, as well as physics without having to be re-created. As long as we know the axioms are true (in our application) we know that the theorems (and methods of solving problems derived from those theorems) are true.

Of course, applications to the real world always involve measurements and measurements are always approximate so we can never say that the axioms are exactly correct. We have to choose from a large number of mathematical systems that which best fits our need.

 

Yes it is! I have an interest in computing at the moment and understanding the processors etc. is mind-blowing stuff. Not because it's really complicated (though some parts are) but because when you can see how the system actually works it opens up whole new avenues.

It does that to me anyway, but maybe that is just me.

 

You're making the mystery seem less mysterious by saying "they are simply templates". But why is the universe such that it obeys those templates so readily and, to make things more strange, necessarily? There's nothing in the concept of a template itself that excludes the possibility that it can only be applied sometimes. What's interesting about "math templates" is their universality.

 

because the universe follow certain laws. why? that's a physics question. we don't know the answer yet. why does math map to the universe so well? be cause we mapped it to match what we saw in the universe! Circular logic to think that Math is anything more then invented. Math is a symbolic language which defines, on paper, real-world events. however, it only works when certain assumptions are made first. If *any* of those assumptions is taken out of the real world example, math falls apart.

Math is not inherent to an object like Mass or Density. it is an applied attrebute to a certain set of phsycial events, assuming certain things.

1+1!=2 if the begining objects do not match in the properties we are trying to add. How many oranges do you get when adding an orange and an apple? How much speed do you get when adding a photon's velocity to a photon's velocity? How much mass do you get when adding a proton to an anti-proton? none of the answers to the above is 2.

You can make 1+1=2, but only if the defined situation fits the limited scope of what addition of integers was designed to represent. You have to make the problem fit within the boundries that were defined by those humans with invented math.

As I did in the last 1+1=2 thread, I seriously urge people to learn Abstract Algebra. It will open up dorrs which have been closed by the classical math training in the public school system, where to say that 1+1!=2 is to be suspended for disrupting class.


edit: oops, meant to post this in the "why does 1+1=2" thread. oh, well, still aplies here.

 

Gendanken (whoever you are) you're a rembrant! I don't mean this in a derogatory way of any kind, but your posts so far have been pure truth.

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Not always (in my opinion)! Nature is only the result of pure physics (and logic) and this forms us. logic can be found in nature.

Wes Morris, I apologize, I didn't mean any disrespect! It's good to see you on here!

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I think mathematics has a real place in the world, and only a few of us are allowed to see it. It is fortunate for us few, that this allows us to see the truth, and this truth, allows us to see everything. I have to say, as a general statement, everything can be viewed by anyone, at anytime. This is the nature of truth: everyone knows everything. It is with good nature and good will that I greet you.

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I have nothing to add but thanks for all the post in this thread. Im very interested in all these arguments.