"There are no such things as numbers"

The most exciting essay I've read so far this year: Paul Benacerraf, "What numbers could not be," Philosophical Review 74 (1965), reprinted in Benacerraf and Putnam, eds., Philosophy of Mathematics: Selected Readings (Cambridge UP, 1983). This was the essay that spurred the development of the first new position in the philosophy of mathematics - structuralism - since the debates on the foundation of mathematics in the early 20th century. Some stand-up-and-cheer goodness:

...that any recursive sequence whatever would do suggests that what is important is not the individuality of each element but the structure which they jointly exhibit..."Objects" do not do the job of numbers singly; the whole system performs the job or nothing does. I therefore argue, extending the argument that led to the conclusion that numbers could not be sets, that numbers could not be objects at all; for there is no more reason to identify any individual number with any one particular object than with any other (not already known to be a number). ...
[N]umbers are not objects at all, because in giving the properties (that is, necessary and sufficient) of numbers you merely characterize an abstract structure - and the distinction lies in the fact that the "elements" of the structure have no properties other than those relating them to other "elements" of the same structure....Any object can play the role of 3; that is, any object can be the third element in some progression. What is peculiar to 3 is that it defines that role - not by being a paradigm of any object which plays it, but by representing the relation that any third member of a progression bears to the rest of the progression. Arithmetic is therefore the science that elaborates the abstract structure that all progressions have in common merely in virtue of being progressions.

Benacerraf is positioned against not only Fregean logicism, but also classical platonism; and he claims at the end to explain "what the ordinary formalist apparently cannot - why these axioms were chosen [in the first place]". I will go read Stewart Shapiro's book on structuralism and let you know what happens.

robert thefamilyghost wrote:

Herr Tim wrote:It would be a vast understatement to say that numbers are important.

you raise a valid point, yet somehow i can't help but feel that i'm angry at numbers...

hart wrote:What if it's slightly off? What if 1 + 1 = 2.000000000000000000000000000000000...(etc, you don't need to mess up the screen display to pove your point)

kerble wrote:

Andrew L. wrote:Trivia question: When did the number zero first enter western/European mathematics?

1979, dude.

robert thefamilyghost wrote:

hart wrote:What if it's slightly off? What if 1 + 1 = 2.000000000000000000000000000000000...(etc, you don't need to mess up the screen display to pove your point)

numbers are an abstraction...to assume there's a "measurement" error in their defined integer states is to assume they actually exist in a manner that can be deviated from...by definition the addition function, when applied between two integers, cannot yeild a non-integer...

when you define things that do not exist in nature (such as numbers...both 1 and 2 are concepts, not physical entities) you cannot make assumptions about them! there's nothing to assume, they haven't existed yet, you're generating an IDEA! therefore there is nothing "assumed" about "1 + 1 = 2"...your assumption otherwise is what's irrational and emotional...

numbers are basically adjectives...they're descriptions of the things that actually exist in the universe...mathematics describe how these things interact and function...

how can you claim that such a simple equation as 1 + 1 is equal to a more complex expression such as 2 + (739468 / 100000...) ?

and how do you make that exact measurement on an idea that has no physical existence?

hart wrote:I suppose the idea would be this. If 1 + 1 really equalled... slightly more than 2... then you could gather a whole bunch of apples, and after a WHOLE lot of apple gathering, you would actually have one more apple than you would have expected. Apply this notion to all of its implications (1 + 2 = 3 ... 2 x 3 = 6.... etc...) and you may find that previously supposed problems, such as entropy, don't really exist if you work the numbers right. Perceived inevitable inefficiencies- loss of energy to heat, or even "the aging process"- were merely byproducts of our faulty ability to count.

hart wrote:If one can start to fathom that some endless quanitities are greater than others, than why can't 1 + 1 = 2 be ever so slightly off?