What would you say the meaning is? Just curious. — jgill
A quarter counterclockwise turn in the plane. That's the simple meaning — fishfry
That's what happens when multiplying a+bi by i. — jgill
I play in the complex plane all the time, and I have always visualized figures and imagery and motion. Even created what might be considered art in the process. — jgill
"There exists a unique x such that x^2 = 2." — GrandMinnow
E!x x^ — GrandMinnow
Why would you think that someone who has not studied existence would know as much about existence as someone who has studied existence? — Metaphysician Undercover
Philosophers don't necessarily lead better lives than others, nor are they more moral, and they most definitely don't know any more about existence than the general public.
In particular, a philosopher who knows hardly anything about mathematics is in no position whatsoever to comment on mathematical existence. Many philosophers of mathematics are in this position. They simply don't know enough math to comment intelligently on the subject of mathematical existence. — fishfry
"There exists an object that has the property that its square is equal to 2" is perfectly fine English. — GrandMinnow
Existence is the same. If someone's been existing for a few decades they know as much about existence than a philosopher. The philosopher knows the history of what great thinkers have written about the subject. But philosophy does not confer actual knowledge of its subject; only knowledge of what others have said. — fishfry
In particular, a philosopher who knows hardly anything about mathematics is in no position whatsoever to comment on mathematical existence. Many philosophers of mathematics are in this position. They simply don't know enough math to comment intelligently on the subject of mathematical existence. — fishfry
I didn't say it isn't perfectly fine English. I said you haven't properly identified the subject signified with "there", to which "exists an object" is predicated. — Metaphysician Undercover
[Many philosophers of mathematics] simply don't know enough math to comment intelligently on the subject of mathematical existence. — fishfry
The problem with this perspective is that "mathematical existence" means something completely different than "existence" in the philosophical sense. — Metaphysician Undercover
The op does not ask about "mathematical existence", it asks about "existence". — Metaphysician Undercover
If it asked about the mathematical existence of irrational numbers there would be nothing to discuss. Clearly irrational numbers are used by mathematicians therefore they have mathematical existence. — Metaphysician Undercover
The op is asking a philosophical question about the existence of certain mathematical objects, not whether those mathematical objects have mathematical existence. — Metaphysician Undercover
That would be self-evident. So mathematicians who hardly know anything about existence, yet think they do because they know something about mathematical existence really seem to have very little to say about the philosophical question of whether certain mathematical objects which obviously have mathematical existence, have existence. — Metaphysician Undercover
My rough impression is that professionals in the field of philosophy of mathematics usually do know about mathematics. Which philosophers in, say, the last 85 years do you have in mind? — GrandMinnow
today’s generation of mathematics undergraduates who are studying mathematics using theorem provers from the outset. — sime
My rough impression is that professionals in the field of philosophy of mathematics usually do know about mathematics. Which philosophers in, say, the last 85 years do you have in mind? — GrandMinnow
What's a "theorem prover"? Computer program? A tutor? — jgill
But why the square root of 2? How about the number 3? That has no more claim on existence than sqrt(2). — fishfry
Clearly numbers don't have the same claim to existence as rocks or fish. — fishfry
This thread hasn't even begun to touch on the subtleties of that subject. I've seen no decent arguments one way or the other. And if that's what the OP really cared about, they'd have asked if 3 exists. Once you bring in sqrt(2) you are talking about mathematical existence. — fishfry
It's "above their pay grade" as Obama would have said. So make an argument. Do you think 3 exists? Of course the positive integers have a pretty good claim on existence because we can so easily instantiate the smaller ones. So how about 2googolplex2googolplex? That's a finite positive integer that could in theory be instantiated with rocks or atoms, but there aren't that many distinct physical objects in the multiverse. So make an argument, say something interesting about this. Forget sqrt(2). Do you think that extremely large finite positive integers exist? — fishfry
More generally, it is a dependently typed logical programming language, with clause resolution and other rules of logical inference, together with SAT solvers, methods of analytic tableaux and heuristics for automated or interactive theorem proving — sime
the square root operation is closed over real numbers — TheMadFool
The op is asking a philosophical question about the existence of certain mathematical objects, not whether those mathematical objects have mathematical existence. — Metaphysician Undercover
I can accept that the square root operation is closed over the 'reals', but that doesn't mean it's closed over the real numbers. — Ryan O'Connor
Cauchy Sequences — Ryan O'Connor
why do we even need to assume that irrational 'numbers' exist? Why not assume that irrationals are the algorithms that we actually work with? — Ryan O'Connor
algorithm (described completely with finite characters) which if executed to completion — Ryan O'Connor
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