Case for math antirealism

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What's the title of the book? I seem to remember searching for it on Amazon a few years back before giving up.

Can't the whole realism/anti-realism discussion be construed as a discussion about the (un)reality of mathematical entities (e.g. those ostensibly referred to by such terms as "2" and "triangle")?

(Or more specifically, about the (un)reality of numbers? Or sets? (Hasn't it been accepted that numbers are the fundamental entities of all mathematics including geometry; and hasn't it been accepted that numbers are a special kind of set? I might be a bit off on this.))

Mathematical anti-realism - that is, if I'm right, the view that there are no such things as mathematical entities (sets, numbers, figures, what-have-you) - sounds preposterous at first, but once you try to specify what mathematical entities are, then it's easier to appreciate the controversy of the question regarding their reality.

What is 2? It isn't identical to this or that physical object-pair (two apples, two hands, etc.); nor, perhaps, is it just an idea in the mind (though some metaphysicians might disagree) - more generally, it has no spatiotemporal properties (i.e. it doesn't occupy space and time), and it doesn't have any effects in the way that physical events do; therefore (so it's been said), it's eternal and immaterial, and we know it only through a special faculty of rational intuition, or some such.

Now if you're suspicious of what's written in the previous paragraph, then you might actually have some anti-realist sympathies! One of the challenges for anti-realism is to account for what it is that practitioners of mathematics, as such, actually do. (Language-games?)

Perhaps you mean this book ...

"Fictionalism in mathematics was brought to fame in 1980 when Hartry Field published Science Without Numbers ..."
http://www.amazon.com/Science-Without-N ... 0691072604
http://en.wikipedia.org/wiki/Philosophy ... ctionalism




Yes.

This happens often in my field of work where say a "tripod" may be defined to have two legs, not three

Math is great since it objective and not judging for how outgoing and conformist you are. Wish I could have been a math major in college.

Well, shoot. I hope that book gets reissued sometime! Currently it seems that it's only available for a few hundred dollars.

I do, however, have in my wish-list a much less expensive book called A Subject With No Object, by Burgess and Rosen. Here's the summary:

I'm not sure how much this (nominalism) overlaps with fictionalism, though. My guess is that while both deny the existence of abstract objects, fictionalism emphasizes the usefulness of thinking and communicating as if there are abstract objects, despite the unreality of such objects; whereas (perhaps - again, I'm guessing) according to nominalism, we don't have anything like abstract objects in mind at all when we talk about (e.g.) the number 2, justice, or redness, and to think otherwise is to misinterpret convenient figures of speech.

Analogously, consider the following proposition, (p), and the following sentence, (S):

(p) Geographic boundaries are real independently of anyone's say-so.
(S) "If all human beings simultaneously lapsed into comas tomorrow, there would still be a boundary separating the US from Canada."

A fictionalist about geographic boundaries might say that (1) (S) is a false utterance because its speaker falsely presupposes that (p) is true; and (2) for lack of a more useful alternative, we'd do well in ordinary circumstances to pretend that (p) and (S) are true anyway.

A nominalist about geographic boundaries might say that (1) (p) is indeed false; but (2) hardly anyone has (p) in mind when uttering (S), which happens to be a true sentence.

(Again, I don't know for sure. The Wikipedia article you linked doesn't include nominalism at all, so maybe it's just a matter of preferring one word or another for the same thing.)

It seems that math makes things nice and structured, as well as nicely organized. Ahh two of everything all in a row in Noah's Ark. Math to help make buildings so nice and wonderful to look as well as make buildings structurally sound and maybe withstand an earthquake or two.

Good point. I think you are right. Fictionalists appear to be OK with using math as a tool, while nominalists appear to reject math entirely. However, I need to read site below, do a compare and contrast, and then come back to your analogy

I wonder if nominalists view fictionalists as hypocrites (i.e., fictionalists recgonize that math is fiction, yet use it anyway as a "tool").

Nominalism in the Philosophy of Mathematics
http://plato.stanford.edu/entries/nomin ... thematics/

ikr, I'm pretty sure the numbers were invented in the first place just cause humans coincidentally have 10 fingers. That was the first method for counting stuff at least. The majority of intelligent aliens from other planets probably don't have the same number of fingers that humans do if any at all.

If this were the case, we would only have one finger. The unary system was probaly the earliest (i.e. /=1; //=2; ///=3; ////=4).

A base 10 system came about long after number systems did (generally assumed to be around the 5th century, although many say it was influenced by the earlier base 5 system). Zero in the 6th century was a pretty big breakthrough for actual calculations, though. The idea of nothing was tricky to represent (and is still difficult for some people to visualize). A lot of the Anglo cultures were a base 20 system (i.e. scores like four score and seven). Then you have the computer based numerical systems (binary, binay-octal, the binary based hexidecimal system). You also have circular numeral systems (base pi) that we translate into base 10 systems as best we can. There are somewhat arbitrary ternary systems (like the old English current American measurement systems (inch-foot, teaspoon-tablespoon, etc.)), septenary systems (based on days of the week), the tridecimal system of the Mayans, NAF (works well with binary systems), various fractal based systems, golden ratio based, seasonal based, and my personal favorite the senary system (base 6 based on the prime factors of 2 and 3 instead of 2 and 5 like base 10), which makes so much more sense in my head than more widely used systems for some reason.

The fact that we can translate between all these number system serves to reinforce that math is quite real, and it is only the way we physically represent it that is variable.

Great thread!

much food for thought

Nonsense.

The very word "digit" (meaning "number" in English) comes from the Greek word for "finger".

Counting systems LONG predate the Fifth centurey AD. And most of the first independently invented systems around the world were based on five, ten, or 20 (the Incans counted on both their fingers AND their toes and used a base 20 system). Roman Numerals, for example, are based on five and ten.

But even the ancients experimented with other base numbers. The ancient Babylonians had an odd habit of counting by fives,AND by 12's.And by fives in groups of 12 (sixties). Thats why today a circle has 360 degrees.