ntkwild, thank you for accepting the parameters (it would be interesting to discuss all this in light of the 2 'truths' you have stated), but royalchicken does not accept the math part so perhaps you and he can work out how you want things to be for the math. presumably we are all agreed for the physics.
royalchicken, very glad to see you again and of course i can 'stand' you. i am very appreciative of the assistance you gave me regarding the P->~P stuff. please work out with ntkwild what you would like as the math parameters. i am ok with either and would be willing to continue separate discussions with each of you should the two of you not agree.
Originally posted by pradtfYes, thought I'd drop back in. I think nktwild and I should come to some consensus here; I just think the Aristotelian definition would lead to some erroneous conclusions.
royalchicken, very glad to see you again and of course i can 'stand' you. i am very appreciative of the assistance you gave me regarding the P->~P stuff. please work out with ntkwild what you would like as the math parameters. i am ok ...[text shortened]... rate discussions with each of you should the two of you not agree.
The bit about being able to stand me was just satirizing my repeated proffering of the same definition of math(s).
Originally posted by pradtfI think it's even more extreme than that. I seem to remember having to work out at school what would happen if you put ice in a kettle and left it on. It would spend about 10 minutes melting the ice, under 2 minutes bringing it to the boil, and then over an hour to boil dry!
the primary reason, jearl believes as do the authors, is the evaporation. less water will freeze faster than more water (notice the loss of 16% going from 100 to 0 celsius degrees). the loss of heat as a result of change of state (water evaporating into gas) is pretty hefty as well - to raise 1 g of water from 0 to 100 requires 100 cal of energy, but then to turn that 1g of water into steam requires 540 cal of energy!
Ice is actually a very good way of keeping your drink cool, though you don't need that much of it. If you have a drink with ice in it, then no matter how hot the surroundings, the drink will stay at 0C/32F (ignoring time taken for convection etc) until all the ice has melted! If, when you finish the drink, there's ice in the bottom, you put more ice in it than you needed; restaurants are notorious for this.
Originally posted by nktwildThat is certainly its history, but maths has evolved to much more than that, and I think we should consider it as it is now.
hmmm, i prefer to use pradf's version, maths really is just numbers. the maths we know today is IMHO just the results of complexe numerical maths. in a way its a whole new subject written in the language of the old subject.
Originally posted by royalchickenbut much that is taught in modern maths isnt maths at all. In maths at the mo, we are being taught mechanics (phyisics) and discrete which is most definatly basic computer programing and NOT maths. any one who knows/is learning C++ will recognise discrete maths.
That is certainly its history, but maths has evolved to much more than that, and I think we should consider it as it is now.
Originally posted by nktwildI would agree that mechanics is not maths. What do you mean by 'discrete maths'. Many things are called that, and if you just mean the mathematics of discrete phenomena, then you are wrong. Huge parts of number theory, most of abstract algebra, probability etc. are 'discrete'. Furthermore, remember that computers have their origins in maths, and were largely developed by mathematicians (Blaise Pascal, Charles Babbidge, John von Neumann, Vannevar Bush, Alan Turing...). In fact, one of the first conceptions of a digital computer was the Turing machine, which was really just a concrete (althouh not actually built) illustration of the logical Entscheidunsproblem. I would say all but the physics you mentioned is maths.
but much that is taught in modern maths isnt maths at all. In maths at the mo, we are being taught mechanics (phyisics) and discrete which is most definatly basic computer programing and NOT maths. any one who knows/is learning C++ will recognise discrete maths.
it's been several days since you last posted and since ntkwild has not replied to you, perhaps you would like to proceed using your definition of math. i saw from another post that you are in the middle of exams, so if you would prefer to wait till they are over, that's fine - just let me know.
Originally posted by royalchickeni think that definition is a little arbitrary itself.
Mathematics is the study of the logical consequences of an arbitrary set of self-consistent axioms.
An instance of mathematics is any statement derived from the axioms. You take it from here...
for instance, here are 2 fundamental axioms of social psychology:
people construct their own reality
social influences are pervasive
the 'study of the logical consequences of this 'srbitrary set of self-consistent axioms' is surely not maathematics.
or how about from the declaration of independence:
We hold these truths to be self-evident[axiomatic], that all men are created equal; that they are endowed by their Creator with certain unalienable rights; that among these are life, liberty, and the pursuit of happiness.
these are pretty self-consistent, but the behaviour that follows from them is not to be considered a mathematical study.
i think your statement
" ... is the study of the logical consequences of an arbitrary set of self-consistent axioms."
can apply and is used in many fields of human knowledge.
what you have here is a pretty good definition of deductive reasoning (which of course is utilized in mathematics as well as many other fields). compare unabridged random house dictionary definition of deductive reasoning: a logical process in which a conclusion [consequences] is drawn from a set of premises [axioms].
however, if you really want to use this definition because your idea is that all mathematics is really deductive reasoning (something that may be debated though not here), i can accept it.
Very good post, first. I think both of those things you mentioned are mathematics, but I'd like to compromise. Clearly, for practical purposes my definition is too broad. So we can also stipulate that the arbitrary axioms must have to do with a certain large but well-defined subject matter. However, virtually everything in what we call mathematics is, in principle, reducible to fooling around with sets, and axiomatic set theory really does encompass all of mathe,atics. So maybe we should say that mathematics is that which deals with numbers, sets, functions, operators, manifolds, sequences, groups, fields, vectors, relations, and Paul Erdos. Or, we can use the nice definition due to someone: "Mathematics is the science that knows nothing of (empirical-ed.) observation."
Basically, all I want to get across is that math(s) is a form of deductive reasoning in which the premises are assumed and which is not inherently based on empirical science. Further, it is the only science in which a statement being 'true' has any real meaning.