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

My email is capricious, so I often miss current threads, but I found this one interesting, so I offer my belated opinions. I believe that there exists an objective physical reality (I am a physicalist). I believe that my body and mind supervene on that reality (I am a materialist). Reality includes other beings with minds and experiences similar to my own (I am not a solipsist). "Real" describes entities or events that exist within reality. "Truth" describes statements that correctly describe entities or events within reality (I believe in correspondence theory of truth). However, I recognize that due to failings in perception, understanding, and expression, we cannot attain truth with certainty. We must settle for confidence based on observation, experiment and external confirmation (I am an evidentialist). Mental constructs, including concepts and abstractions, are not real in any sense. Specifically, mathematical concepts such as objects, sets, shapes, numbers are not real (I am not a Platonist). IMHO, the proper subject of pure mathematics is not objects, numbers, shapes, patterns, collections, or any of the popularly ascribed subjects. Indeed, our eyes and brains are wired to interpret the world in terms of objects, numbers, shapes, patterns, collections, &c, even if our mathematical talents are modest. These concepts are no more inherently mathematical than are colors, tastes or smells. It is just that these concepts are more amenable to reasoning, and were therefore the earliest subjects of mathematics. But since George Boole (bless his heart) unified mathematics and logic, it has become clear (at least to me) that the proper subject of mathematics is logical systems. By logical system I mean a set of strings of symbols manipulated via formal rules (logic). Applying pure mathematics to a problem domain essentially devolves to: - Translate relevant aspects of the problem domain into a logical system (applied mathematics) - Use the formal rules of the logical system to generate new strings of symbols (pure mathematics) - Interpret and test the new strings of symbols within the problem domain (more applied mathematics) The process of translating a problem domain to a logical system effectively strips it of meaning: - The truths of the problem domain become meaningless strings of symbols in a logical system. - The strings in a logical system can be applied with different meanings to different problem domains. - The strings generable in a logical system are determined by the rules of the system. A computer with infinite time and memory could, given the rules of an arbitrary formal system, apply the logic to generate all and only the strings of an arbitrary logical theory. Such a computer, given the axioms and logic of ZFC, could eventually spit out almost all of known mathematics with no inkling of what it means. This is not to denigrate pure mathematicians. Even if we could create a computer that could spit out all of mathematics, interesting results would be painfully few and far between. There is great deal of art to pure mathematics, in choosing axioms and applying clever logic leading to new proofs and ideas. To do so effectively, mathematicians must need invest their strings of symbols with meaning, it's our human failing to see meaning where there is none. I don't believe in "mathematical truth". Mathematics doesn't magically generate truth, rather, if you feed it truth it poops out more truth. Unfortunately, we don't have any truth to feed it, so we have to accept beauty instead. I also take issue with the Mathematical Universe Hypothesis. I can buy that the universe in mathematical in the sense that its behavior is determined by laws that might possibly be translated faithfully into the strings of a logical system. I can't, however, buy the idea that the universe is mathematics. What else could this mean but that the universe IS a formal system. If so, where are its axioms and theorems? Are they the strings of string theory? What are the atomic symbols used by the universe to write its theorems? And then Tegmark postulates "self-aware substructures" of his mathematical universe. Good luck writing a logical system with one of those. But Tegmark has already jumped from one grand conceit (MUH) to another (CUH), claiming that all computable mathematical structures are real. It all sounds farfetched bordering on religious.