QFT isn’t inconsistent, just it is not known yet if it is consistent

When physicists reason they often do so using techniques that aren’t generally valid. They may work in one particular situation but not in another, without a good explanation as to why. For example, in the absence of proper foundations to QFT, physicists use heuristic arguments about limits and infinities that really are not valid mathematics. I guess you could view this not as an inconsistent theory, but as a theory with a very complicated set of deduction rules about when you can use this and that rule. So complicated, in fact, that the rules haven’t been codified and you have to learn them by reading between the lines of your lecture notes.

This is quite different from mathematics where the moment you smell a whiff of inconsistency somewhere it’s imperative that you rush towards it as fast as you can.

Not always, and not everybody. I for example have no problems with (some level of) sloppyness and heuristic arguments, as long as at least the objects in question are more-or-less clearly defined. The problem is, when you talk with physicists, or read physics books, they are often not…

(btw, I don’t think “inconsistent” is the good word here. QFT isn’t inconsistent, just it is not known yet if it is consistent)

“Don’t knock masturbation. It’s sex with somebody I love.”

Physicists work under the extra constraint that theories should not only be consistent, but true (or in accordance with nature) as well.

These are more like ideals towards which physicsists strive rather than actually being constraints. Even elementary quantum field theory is inconsistent (eg. AFAIK path integrals in Lorentzian spacetimes still have no rigorous foundation). Part of learning how to do physics is learning when it is and isn’t appropriate to use various kinds of argument so you don’t actually hit those inconsistencies. This is quite different from mathematics where the moment you smell a whiff of inconsistency somewhere it’s imperative that you rush towards it as fast as you can.

Physicists work under the extra constraint that theories should not only be consistent, but true (or in accordance with nature) as well.

Physical intuition can convince you certain mathematical statements are true, even without the comfort of a mathematical proof.

It’s worth noting that Feynman also said

Physics is like sex: sure, it may give some practical results, but that’s not why we do it.

and in a letter,

Dear Mrs. Chown, Ignore your son’s attempts to teach you physics. Physics isn’t the most important thing. Love is. Best wishes, Richard Feynman.

Having tried my hand at both math and physics, I understand why Feynman made this remark. There are lots of nice mathematical structures. For mysterious reasons, only certain special ones are the laws of our universe (unless
Max Tegmark) is right. So, with math you can go anywhere, limited only by your own imagination and skill. With physics you are seeking intimacy with a mysterious other called Nature, who may or may not respond to your advances.

But, Feynman’s remark underestimates the way in which deep mathematics is not at all arbitrary – discovered rather than invented.

He also said:

Mathematics is not real, but it feels real. Where is this place?

To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature … If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.

This shows a certain appreciation for mathematics but also perhaps too much of a sense that mathematics is just a tool, not something real.