And according to the Tooth Fairy, Herodetus is wrong. (Isn't argument from authority fun?) To say that the river is never the same is exactly as wrong as saying it's always the same.
Hmmm... Well damned if you do, damned if you don’t. That’s a safe, but ultimately a dead-end view. It may be the most correct, but it certainly doesn’t leave much to say... Perhaps you’re right in that I’m just being silly drKitten, hence the anonymous forum
Regardless of how “silly” it may be, debate over the nature of numbers and mathematics has been a fiery subject form the time of the Greek and continues to rage (in fact, Santa’s opinion on the issue is quite enlightening
).
Below are just a sampling of philosophical school that focus on the nature of mathematics. So while this conversation may seem laughable there are some people out there who get paid to ponder on this stuff, and thus it is at least as good as porter potty cleaning, if not better.
From the Wiki page on Philosophy of Mathematics:
Mathematical realism, like realism in general, holds that mathematical entities exist independently of the human mind.
Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the naive view most people have of numbers.
Logicism is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic (Carnap 1931/1883, 41). Logicists hold that mathematics can be known a priori, but suggest that our knowledge of mathematics is just part of our knowledge of logic in general, and is thus analytic, not requiring any special faculty of mathematical intuition.
Empiricism is a form of realism that denies that mathematics can be known a priori at all. It says that we discovered mathematical facts by empirical research, just like facts in any of the other sciences
Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. For example, in the "game" of Euclidean geometry (which is seen as consisting of some strings called "axioms", and some "rules of inference" to generate new strings from given ones), one can prove that the Pythagorean theorem holds (that is, you can generate the string corresponding to the Pythagorean theorem). Mathematical truths are not about numbers and sets and triangles and the like — in fact, they aren't "about" anything at all!
Mathematical intuitionism: In mathematics, intuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" (L.E.J. Brouwer).
Constructivism: Like intuitionism, constructivism involves the regulative principle that only mathematical entities which can be explicitly constructed in a certain sense should be admitted to mathematical discourse. In this view, mathematics is an exercise of the human intuition, not a game played with meaningless symbols. Instead, it is about entities that we can create directly through mental activity. In addition, some adherents of these schools reject non-constructive proofs, such as a proof by contradiction.
Fictionalism was introduced in 1980 when Hartry Field published Science Without Numbers, which rejected and in fact reversed Quine's indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as a body of truths talking about independently existing entities, Field suggested that mathematics was dispensable, and therefore should be considered as a body of falsehoods not talking about anything real.
***
Embodied mind theories hold that mathematical thought is a natural outgrowth of the human cognitive apparatus which finds itself in our physical universe. For example, the abstract concept of number springs from the experience of counting discrete objects. It is held that mathematics is not universal and does not exist in any real sense, other than in human brains. Humans construct, but do not discover, mathematics.
*** this is the one I gather most people here ascribe to.
Social constructivism or social realism theories see mathematics primarily as a social construct, as a product of culture, subject to correction and change.
There were also “non-traditional schools listed but I though it best to keep this post in the category of “obnoxiously long” as opposed to “not worth the effort”.
