I'm inclined towards Brent's point of view. IMHO, "real" means something that can be *experimentally* verified -- or more importantly -- something that can be experimentally *refuted*. Mathematics is completely disconnected from the "real world"; things are true mathematically *no matter what the "real world" looks like*. You can do experiments from now til kingdom come, and none of them will change any mathematical truths. In mathematics, you set up a set of axioms/axiom schema, and you then look for "models" of those axioms. Sets of axioms that have models are consistent; if the axioms are incomplete, there will be *undecidable* statements which can be either true or false, depending upon which of multiple different models you choose. ---- There is another closely related notion which really bothers me: the *legal* notion of "facts". Legal "facts" are determined by "fact-finders" which are judges and juries; this legal "fact" system is completely disconnected from both mathematical truths and scientific truths. Moynihan's old saw, "Everyone is entitled to their own opinions, but they are not entitled to their own facts", is actually a really, really cynical lawyer joke, since the population at large doesn't usually realize that the legal meaning of "facts" has nothing to do with "truth"! At 12:27 PM 4/2/2018, Brent Meeker wrote:
In my view they instantiate quite different meanings of "truth" and "real".
In mathematics and logic "2+2=4 is true." means something like "Given the Peano axioms and rules of inference '2+2=4' follows."
"True" is an attribute that is conserved by the rules of inference; so if the axioms are true the theorems are true.
But "true" is just a marker like "t" in a programming language.
In science "true" is almost never used.
Theories are "accepted" when they entail the observations and nothing to the contrary.
Then they are provisionally thought to correspond to facts.
This is always hypothetical and provisional because one can never be sure of the facts and because other theories might entail the same observations.
So my personal view is that mathematics exists in a different domain.
The declarative part of all possible language = Platonia.
Observations, perceptions, exist and from them we formulate theories of physical reality, which are (so far) always provisoinal and incomplete.
Brent
As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.
-- Albert Einstein
On 4/2/2018 9:42 AM, Dan Asimov wrote:
If this common discussion subject among mathematicians has come up in math-fun previously, it must have been a while ago:
Is mathematical truth real? I was reminded of this question when I happened to sit next to an esteemed molecular biologist who has strong opinions on the matter: He thinks math is "all in the brain" of humans who think about it. I could not convince him that math has any kind of independent existence -- though I certainly believe it mmyself.
What do other think about this? I would say that mathematical truth is *at least* as real of a thing as physical truth.
--Dan