Any argument carried out with sufficient precision.All of these answers are unsatisfactory.
Using arguments with more than two steps.
Mathematics is what mathematicians do.
Mathematics is the branch of natural philosophy that concerns itself with only making true statements.
Mathematics ought to be considered as a set of precise, symbolic, languages that serves as a lingua franca for the physical sciences.
Mathematics is knowledge obtained by logical proofs.
Saying that mathematics is a language is like saying music or philosophy is a language. Sure they use language to communicate, but so does everyone else.
Using phrases like "sufficient precision" ignores the fact that some arguments are proofs, and some are not. Math demands proofs.
Saying "true statements" comes the closest to describing math, but of course many other fields claim to be finding truth. Only math finds it with logical proofs.
Math still IS a language, much as you say just like music or philosophy. It either communicates a clearly, or it doesn't. If you can say something very complicated and esoteric in a language that no one else speaks or understands, you aren't really saying anything, and stamping your foot about it because others fail to understand you is really your problem (not theirs) in failing to communicate.
Let me give it a try:ReplyDelete
``The science of inventing methods for defining and finding relations among quantities.''
Sizes exists metaphysically, as attributes of physical objects.
Physics investigates the physical world in order to isolate the existing ``size-wise'' (i.e. quantitative) relations among physical objects, including their attributes, characteristics, etc. (Relations among objects also include physical interactions between them.)
Maths supplies the methods for making the physical concepts and statements involving quantitative relations precise.
Does heavier object fall faster? Yes, in relatively denser fluids (hence Aristotleans' answer). No, in more rarified fluids like the air, esp. over small distances. (Hence Galileo.) Etc.
Answering such questions involves, in turn, finding answers for how big or how small the objects and their effects are, including the context in which the observations about the relative big-ness or small-ness were made. To make such statements precise, numbers are used. Numbers are the invented concepts. Sizes exist with physical objects, numbers don't. Numbers is where maths begins. Numbers are also useful in other contexts which, with growth of knowledge, we don't regard as physics proper. For instance, measuring areas and volumes (for planning agriculture and trade). Physical objects and their sizes are involved, but it's too elementary a physics. Numbers are involved for measuring sizes, but even when it uses calculus, economics isn't the killer app for maths; physics is.
Maths has ``unreasonable'' effectiveness in physics---and rather only in physics---because in other sciences, their subject matter is too complex for anyone to find concepts that are both *quantitative* and *fundamentally* useful in deriving or organizing knowledge about its domain. No quantitative units of sufficient fundamentality or of general usefulness can be found for measuring joy, pain, boredom, or ecstasy.
Not all logical relations are mathematical. Lawyers (including judges), biologists discovering new sub-sepecies, forensic scientists, and failure analysis engineers aren't mathematicians, even if their most crucial arguments often involve ``more than two steps.''
Only mathematicians can be as professionally stupid as claiming that ``all logic is `math','' that ``all `math' is nothing but logic,'' et cetera.
(I should deal with the backlog of moving my comments from here to my blog. It will make for at least 3--4 posts straight!)
Math is a subset of logic. Mathematicians hate this fact, but any computer science person knows there is a good reason they call the very foundation for electronic digital computation a 'logic gate' and not a 'math' anything.
Many famous mathematicians from history also have a very long and illustrious reputation of bad behavior, often as charlatans and con artists duping the wealthy and powerful, and frequently as card cheats. Not much has changed.
On the face of it, this is an issue more of terminology. However, on second thoughts, I realized, there are some finer points too.
All valid maths *is* a kind of some valid thought. Inasmuch as logic is the art of conducting correct thinking (or reaching right conclusions), it therefore may be thought that maths itself is a subset of logic. ... Seems valid, until you look at it closely.
Maths isn't "just" logic. After all, the most basic laws of logic, namely, the law of identity, of non-contradiction, and of the excluded middle, are validated without reference to numbers. In contrast, maths cannot get going without the concept of quantities, of numbers.
It's true that laws of logic continue to apply also to numbers. But they are rather "only" taken for granted, because explicitly invoking them does not help solve the specifically mathematical problems.
Logic is concerned with the right methods, and in this broad sense, *all* knowledge (of all domains) must conform to the principles of logic.
But precisely because logic is a science of (broad) *methods*, you can't say that the specific items of validated knowledge in some (or every) specific area is a *subset* of logic. That would be confusing between the method and the content (or products of rightful thinking).
Aristotle, precisely for this reason, didn't teach the procedures of classifying biological species (e.g., details like the gaits of horses) in his course on logic.
Albeit maths *is* a science of methods (the methods of measurements, as Ayn Rand put it), it still remains a science of only *one* particular type of methods, viz., those with which we can define *quantities* (numbers) and their inter-relationships.
*All* special sciences have some component of methods. They don't thereby become subsets of logic.
Inasmuch as maths has a definite *content* (as apart from only the methods specific to the domain of its (valid) concern), it too should be seen as a science in its own right, i.e., as being separate from the broadest and most basic science of logic.
So, all in all, I think that the basic issue here is one of recognizing the distinction between *methods* of thinking and *products* of thinking, and further, between methods appropriate to some special domain (quantities) vs methods applicable to all thought.
So, to conclude: Inasumuch as maths has its own content which is separately identifiable from that of logic, it's not a subset of logic. However, like all sciences, it's a product of applying logic.
Polemics: Both the following statements are wrong: ``all logic is X'' and ``all X is nothing but logic,'' where you can substitute any science in place of X, including philosophy.
Thanks for a thoughtful comment, though! (I first agree with you, started writing accordingly, and only then realized the content-method distinction.)
PS: As to your assessment of mathematicians (as in contrast to mathematics), I think that yours is an understatement. :-)