I will remind you, that Pythagoras demonstrated the irrational nature of the square. The relation between two perpendicular sides of a square produces the infinite, which as I argued above is bad. This makes the square a truly impossible, or irrational figure. And, all "powers" are fundamentally derived from the square. Therefore any exponentiation is fundamentally unsound in relation to a spatial representation.. — Metaphysician Undercover
I will remind you, that Pythagoras demonstrated the irrational nature of the square. — Metaphysician Undercover
The relation between two perpendicular sides of a square produces the infinite ... — Metaphysician Undercover
which as I argued above is bad. — Metaphysician Undercover
This makes the square a truly impossible ... — Metaphysician Undercover
, or irrational figure. — Metaphysician Undercover
And, all "powers" are fundamentally derived from the square. — Metaphysician Undercover
Therefore any exponentiation is fundamentally unsound in relation to a spatial representation.. — Metaphysician Undercover
I've read through much if not most of your writings in this thread. I think I understand where you're coming from. I found two remarks I can get some traction on. — fishfry
So ok. ZF[C] is unsound. This is a commonplace observation. Might you be elevating it to a status it doesn't deserve? Are you thinking of it as an endpoint of thinking? What if it's only the beginning of philosophical inquiry? — fishfry
That, I submit, is one of the key issues of the philosophy of math. We agree that math isn't true, but it nevertheless seems intimately related to the real or the actual.
This is what I mean when I say that "ZCF is unsound" is the starting point of philosophical inquiry into the nature of mathematics; not the end of it. I get the feeling you think it's the last word. It's only the first. — fishfry
That's just silly. There's nothing fundamentally unsound. The diagonal of a unit square can't be expressed as a ratio of integers. It's just a fact. It doesn't invalidate math. Even the Pythagoreans threw someone overboard then accepted the truth. — fishfry
That it comes up so naturally, as the diagonal of a unit square, shows that (at least some) irrational numbers are inevitable. Even if we're not Platonists we suspect that a Martian mathematician would discover the irrationality of 2–√2. There's a certain universality to math. Euclid's proof is as compelling today as it was millenia ago. The philosopher has to acknowledge and account for the beauty, simplicity, and power of Euclid's proof, undiminished for 2400 years; not just deny mathematics because some number doesn't happen to be rational. — fishfry
Among the irrationals, the very simplest are the quadratic irrationals like 2–√2, meaning that they're roots of quadratic equations. They're so easily cooked up using basic algebra, or Turing machines, or continued fractions. — fishfry
However, there is this problem which the right angle creates, and that is that it allows us to very easily make two points with an immeasurable distance between them. — Metaphysician Undercover
I assume you mean there are many sets of two points in which the distance between the two points in each such set is "measurable." As opposed to the two points "easily made" you mention. You can make me go away to an "immeasurable distance" by making very clear how you might realize your claim. — tim wood
You seem to be lacking in reading skills tim, it's no wonder you're so confused. I was talking about what someone can do with Euclidean geometry, not about what someone can do walking on the ground. So your example is way off track. — Metaphysician Undercover
and that is that it allows us to very easily make two points with an immeasurable distance between them. — Metaphysician Undercover
You claim it is possible to "make two points with an immeasurable distance between them."... You can make me go away to an "immeasurable distance" by making very clear how you might realize your claim. But I think you cannot. — tim wood
On the assumption that there can be two points the distance between them being "measurable," how do you "very easily make two points with an immeasurable distance between them"? — tim wood
It seems to me your entire post is deeply confused, rampant with ambiguity, amphiboly, conflation, undefined terms - or if they'e defined then the definitions are not held to - and faulty argument, all in a toxic mix and mess, that like most messes, is easy to make but labor intensive to clean up . — tim wood
I'm afraid you have your own free will and I can't make you go away. — Metaphysician Undercover
Construct a square according to the rules of Euclidian geometry, the opposing corners are an immeasurable distance apart. — Metaphysician Undercover
Thank you fishfry. I'm vey impressed that you actually took the time to read and try to understand what I was saying. — Metaphysician Undercover
Most just dismiss me as incomprehensible or unreasonable ... — Metaphysician Undercover
I have been reading Meta's post and it's hard to know where to start. @Meta, do you agree that math isn't physics? You complain that we can't measure 2–√2 but we can't measure 1 either. — fishfry
Yes, you've got the diagonal of a square. Why is that distance "immeasurable"? — tim wood
I believe I did. You think math should be physics. You think perfectly accurate distances exist in the real world. You seem to deny abstraction, as in your claim that squares are "impossible," your word. — fishfry
In each case, mathematicians of a given historical age become interested in an equation they can't solve. They say, "Well what if there were a funny gadget that solved the equation?" Then after a few decades the made-up gadget becomes commonplace and people come to believe in it as a first-rate mathematical object, and not just a convenient fiction. Eventually we can construct these gadgets both axiomatically in terms of their desired properties; and as explicit set-theoretic constructions. They do indeed become real. Mathematically real, of course. Nobody ever claims any of this stuff is physical. You are fighting a strawman. — fishfry
So this is where I want to engage. You seem to want math to be what it can never be. You want an abstract, symbolic system to be real, or actual. That can never be. Your hope can never be realized. — fishfry
But abstractions can be real as well. Driving on the left or right is a social convention that varies by country. It's artificial. Made up. It's nothing but a shared agreement. It could easily be different. Yet it can be fatal to violate it. Driving laws are social conventions made real. See Searle, The Construction of Social Reality. — fishfry
I submit that 5 is prime and the square root of 2 both exists and is rational. — fishfry
Yes, you've got the diagonal of a square. Why is that distance "immeasurable"?
— tim wood
It's what we call an "irrational number", implying an immeasurable length. Are you familiar with basic geometry? — Metaphysician Undercover
I submit that 5 is prime and the square root of 2 both exists and is rational. . . . But they are true.
— fishfry
What am I missing here? — jgill
For the side of the square, take a straightedge, lay it along the side and mark the straightedge at the ends of the side. That's the length of the side — tim wood
As pointed out repeatedly by others, the problem with irrationals is not with their number, but with some of their numeric representations. — tim wood
Don Quixote at least had dignity.I do not understand the construction of the real numbers, but I am willing to argue — Metaphysician Undercover
Are you saying there is no square root of two?This is absolutely false. The reason why the "numeric representation" of irrational numbers is a problem is because there is no number to be represented. — Metaphysician Undercover
Are you saying there is no square root of two? — tim wood
There is no rational number which is equivalent what is represented by "square root of two". So the problem is not as you say, one of numerical representation, it is that there is no number for what is represented. There is a numerical representation, commonly used, but no number which is represented by it. — Metaphysician Undercover
You apparently take "rational number" to mean the same thing as "number." And you apparently think that rational numbers possess some quality that those irrational thingies don't and can't have. What quality might that be? What quality can you name, identify, describe that rational numbers share but that the irrationals cannot have - other than, of course, being rational. — tim wood
And if you allow for there being such a thing as the square root of two, never mind what it might be beyond being the square root of two, then you're obliged to acknowledge that it has a lot of brothers and sisters, in fact a very large family. So large that in comparison, the rationals are next to just no size at all. — tim wood
Perhaps it might help if you tell us what you suppose a number, or number itself, to be. No doubt the difficulties here originate in the foundations of your thinking. — tim wood
A number is a definite unit of measurement, a definite quantity. The so-called "irrational number" is deficient in the criteria of definiteness. The idea of an indefinite number is incoherent, irrational and contradictory. How would an indefinite number work, it might be 2, 3; 4, or something around there? — Metaphysician Undercover
Indeed it is. "I do not allow." That says it all.No, I do not allow that there is such a thing as the square root of two, that's the whole point, — Metaphysician Undercover
As I have pointed out in other recent threads, mathematics is the science of drawing necessary inferences about hypothetical states. Consequently, mathematical existence does not entail metaphysical actuality, only logical possibility in accordance with a specified set of definitions and axioms.Do abstractions exist at all? — Relativist
As I have pointed out in other recent threads, mathematics is the science of drawing necessary inferences about hypothetical states. Consequently, mathematical existence does not entail metaphysical actuality, only logical possibility in accordance with a specified set of definitions and axioms. — aletheist
Yes, it corresponds to the difference between metaphysical actuality and logical possibility. Again, mathematical existence refers to the latter, not the former.In that vein, do you recognize that there's a conceptual distinction between an "actual infinity" and a "potential infinity"? — Relativist
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