User talk:Faye Kane
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Help counter systemic bias by creating new articles on important women.
Help improve popular pages, especially those of low quality.
Good luck. --Daniel()Folsom T|C|U 17:32, 14 January 2007 (UTC)
ttyl = talk to you later
[edit]pretty standard abbreviation: ttyl = talk to you later. --moof 05:13, 30 January 2007 (UTC)
Some answers to stuff on your webpage
[edit]I am familiar with math, so I could give some answers. I post them here because they are long and I couldnt be bothered to open my email. Forgive the nonexistent copyediting.
Math
[edit]Given P(A union B) = 1/4.2, P(A) = 1/4, then P(B) =? First, notice that P(A union B) < P(A), which doesnt make sense since the latter event is contained in the former. So we can't use your numbers. In general, though, P(A union B) = P(A) + P(B) - P(A intersect B)
We know the values only 2 of the terms, so we cannot solve for P(B). Thats why you never got an answer!
Philosophy
[edit]- This statement is false
You are right of course. In a formal language , a statement cannot directly refer to itself because a statement is merely string of symbols. There are rules for what strings can be written in the language, but the strings have no innate meaning. Logic and all other axiomatic systems are just formal languages (with additional "rules of inference" that allow one string to be transformed into another). Since a typographical string of symbols cannot "refer" to itself, there seems to be no way to write the liar paradox as a well-formed statement.
Yet we cannot dismiss this paradox so quickyly. Bertrand Russel was the first to give the paradox a clear mathematical statement. Russel's paradox asks us to consider the set containing all sets that do not contain themselves. (A set that does contain itself would be like a picture that depicts itself) Let us call this collection A. Is A contained in A? This devilish question is in many ways similar to the liar's paradox, but manages to skirt its most obvious problem. Naive set theory allows Russel's paradox to be written as a well-formed statement. Russel famously sent his paradox to Frege just as the latter was about to publish the second volume of his vast mathematical enterprise, and thus destroyed the whole work by showing it to be inconsistent. Various proposals were offered for correcting the basic axioms to make the paradox impossible. Zermelo–Fraenkel set theory disallowed the construction of a set of all sets, whereas NBG allowed the construction but redefined it as a new type of object called a "class". Thus, once again, mathematics was saved by invalidating the liar's paradox.
Things took a drastic, catastrophic turn in 1931 with the publication by Kurt Godel of his famous incompleteness theorems. Godel found an ingenious, devious trick to eliminate the self-referance of the liar's paradox by making it into a perfectly legitimate mathematical statement about numbers. How did he do this? Well, as I said above, a statement in an axiomatic system is merely a string of symbols (e.g. parenthesis, plus signs, variables). Godel assigns a number to each symbol and combines the numbers in such a way that an entire statement or even a whole proof becomes a single integer. Thus instead of saying "this sentence" he can refer to an actual number, so that the paradox reduces to the prosaic question of whether one number divides another. Since the question cannot be answered either way, Godel concludes that in every axiomatic system that allows numbers, there must be statements that cannot be decided to be true or false.
Therefore, the liar's paradox is an essential feature of even very simple systems. It cannot be defined away or somehow fixed.
- this statement is true
well, if it is a true statement, then there is no contradiction. If it is false, we get a contradiction. In formal logic this would merely be a proof that the statement is true.
- is mathematics God?
I will break down your argument point by point
>it has been and always will be infinitely complex
Complexity can be defined in many ways, so I am not sure what you mean. I assume you view math in the Platonic sense: you believe there exists some sort of ideal world where the pure idealized objects of math live. This world is perfect and eternal, and we are merely explorers discovering its secrets one bit at a time with the tool of formal proofs. There are many reasons to reject this belief.
- It has very little to do with how math develops as a field. Mathematics is a human enterprise subject to the biases of those who engage in it. Axioms are chosen merely because they look like they could lead to useful or interesting results, or because they model reality in some way. Why did the Greeks develop Euclidean geometry? Because it (falsely) seemed to them like it described the space around them. Why were imaginary numbers created? Because they were useful aids in solving equations. Why was the strange, unintuitive axiom of choice invented? Because it gave mathematicians the power to prove things they couldn't prove before.
- There is nothing inevitable about our mathematics. There is no reason to believe that aliens on other planets developed the same math as us. In fact, in his book "A New Kind of Science" Wolfram pursuasively argues that the very axioms of logic are in fact incredibly arbitrary! We developed it because it suits how our brain works, seems to model many things around us (but who says it really does? Newton and the Greeks both had wrong models), and leads to things we find interesting.
I will present more objections to the platonic conception below. Now if we reject the Platonic idea, then math is merely a function of the different axioms we may use! If we use only very simple axioms, (such as first order logic) then our math is not very complex. In fact even if we use all the axioms we have, our math is still not complex at all! Why? (this idea follows from Chaitin) Because our axioms are finite. I or anybody else can design a computer program that starts with the axioms (which are just strings) and recursively applies all possible legal transformations to them. This program can thus derive every possible theorem given enough time! Thus there is nothing complex about a static, unchanging formal system with a fixed set of axioms.
> [It] is fixed and unchanging forever
Obviously, math (as a human enterprise) is not the simple proof-outputting machine I described above. The main reason for this is that, as a discipline, it is NOT fixed and unchanging. Mathematicians constantly invent new axioms and perspectives. After thousands of years, people tired of the boring axioms of Euclidean geometry, so they thought to change them (originally to show that a certain axiom could be derived from the others) This led to new mathematics in the form of non-Euclidean geometry. Klein then came along, and realized that different geometries are really just applications of group theory. Cantor meanwhile developed set theory and showed that it implied the existence of different sizes of infinity. Each of these changes entailed a revolution in thought and perspective that created with it whole new fields of mathematics. What does all this show? That mathematics is a dynamic, constantly changing discipline whose fronteirs are ever expanding.
New perspectives and axioms are crucial, since without it mathematics would stagnate as it would devolve into building better proof-making machines.
>it will always exist, even if there is no universe and no space or time.
I made the case above that math is essentially a HUMAN activity. It is our basic thoughts put into rigorous form and the logical consequences of them. Since it is a human activity, it cannot exist without humans (and obviously it can't exist outside the universe, whatever that means)
>It controls every single thing that happens in the universe,
No, you are confusing the model with the object it models. Math doesn't control anything. When we construct a model, we choose axioms that seem to describe the world (e.g. F=ma) and then use math to see what they imply mathematically. We then interpret these results to be statements about physical objects, and we test to see if they hold. If they do hold we feel confident that the axioms we chose will give true results again in the future. But really, there is no reason to believe that math will always give true facts about the universe even if the axioms are true (i.e. that our system of logic actually models the unverse). And in any case, how the hell do we know what assumptions to pick for our model except guess and check (which no mathematician would accept).
>And though it can't NOT be, it doesn't actually exist anywhere,
Math is ideas written on paper or in our mind. When humans don't think to invent these ideas (i.e. they don't think to invent real numbers because they view the world as countable, or they dont think to invent non-Euclidean geometry because they can only consider flat spacetime) then they dont actually exist in any tangible way. When math does exist, it is in our minds and in books.
>It permeates everything everywhere.
OK, I admit it. Math we have developed so far has been surprisingly effective in predicting many observable things once we feed it well-chosen assumptions. But it can't predict everything! In fact, if you accept the assumption of quantum mechanics that the universe contains random events, then obviously there are a huuuge number of things (most, really) that cannot be predicted, or else they would not be random (using Chaitin's defn. of random). These things (if I understand the physics) can have macroscopic effects, as when stray radiation causes a mutation in the genome that unexpectedly causes humans to evolve. (Unrelatedly, biologists agree that randomness has a huge impact on development and creation of species. Natural selection eliminates unfit animals, but it doesnt decide that humans would evolve. We are lucky that the dinosaurs died out and we replaced them). So really, math has surprisingly little to say about the universe if there is randomness! (Wolfram recovers from this dilemma by arguing that the universe has no randomness, only pseudo-random phenomena that looks that way because we do not know from what basic principles they emerge.)
So math really cannot tell us all that much, even if we feed it "true" assumptions. True randomness = complete unexplainability and uselessness of math in predicting it behavior. (See Chaitin's Metamathematics for more on this). Even if you had all the math knowledge possible at the time of the big bang, you would not be able to predict that humanity would evolve.
> It certainly explains why God doesn't seem to give a damn about us!
So as I said above, math has no innate power over anything. It is merely ideas we find interesting. It certainly has little to do with God. Personally, I like the idea of there being a God, so I decided to believe in his existence. I don't see how math or physics could have much bearing on that, but that's a separate matter.
- Did we evolve so that mathematics can have feelings? Do we exist so the gods can harvest our emotions? Is earth someone's science project (or garden)?
I like these. They are whimsical questions. I wonder too...
- What does it really mean for two objects to be orthogonal to each other? Ultimately, what does it mean for two events not to be the same event?
In math orthoganality obviously has a precise defn. I assume you mean the following instead: The physical phenomenon that we model using the mathematical concept of orthoganality: can it be explained using a more compact or more predictive model, or else in terms of other more familiar concepts? I am not a physicist or a philosopher, so I don't know. nadav 08:19, 14 April 2007 (UTC)