On 10/8/2013 12:43 AM, Arturo Magidin wrote: > On Monday, October 7, 2013 7:52:32 PM UTC-5, Hetware wrote:
>> Too late, I already have. I now realize I was asserting my >> assumptions >> >> in the wrong order. > > It wasn't a problem of order. The problem was that you were asserting > assumptions without warrant. >
I can assert a function to be continuous, and then give a rule for determining its mapping in some part of the domain of reals. There may be times when such an assertion leads to a contradiction, and thus the definition of the function must be abandoned. In the case under discussion, there is no contradiction if I define f(x)=(x^2-9)/(x-3) where x!=3, and then use the assumption of continuity to determine f(3).
According to Thomas a function f(x) which is continuous in some neighborhood of x = c
1) has a definite finite value at f(c), and
2) limit[f(x),x->c] = f(c).
If I assert that f(x) is a continuous function over the domain of real numbers I am asserting that it has a definite finite value at f(c). If I then provide a rule for determining f(x) for all reals x != c, and limit[f(x),x->c] exists and is finite, I can then use that fact and the definition of continuity to determine the value of f(c). To wit limit[f(x),x->c] = f(c).
Therefore f(x) provides a continuous mapping from the domain of all reals onto a range of reals.
f(x) = (x^2-9)/(x-3) where x!=3 and f(c) = limit[f(x),x->3],
is synonymous with the definition
f(x)=(x^2-9)/(x-3) where x!=3 and f(x) is continuous at x = 3.
>> Axiomatic set theory, though it's been a long time since I went >> through >> >> the entire exercise. Some concepts must be taken /a priori/, such >> as the >> >> dictates of logic. > > And where do you think that axiomatic set theory came from? From some > immortal? You said you did not accept things handed down to you from > mortals... but apparently you do; so it wasn't so much that you were > against accepting things "from mortals", but rather something else. > Which was... what?
I am Neoplatonic in this respect. That is to say, the foundations of mathematics are a-priori, and the essence of mathematics is not invented, it is discovered. Extra terrestrial mathematics is isomorphic to terrestrial mathematics.
One expression of this sentiment is found in the Diary of John Adams:
Drank Tea at the Majors. The Reasoning of Mathematicians is founded on certain and infallible Principles. Every Word they Use, conveys a determinate Idea, and by accurate Definitions they excite the same Ideas in the mind of the Reader that were in the mind of the Writer. When they have defined the Terms they intend to make use of, they premise a few Axioms, or Self evident Principles, that every man must assent to as soon as proposed. They then take for granted certain Postulates, that no one can deny them, such as, that a right Line may be drawn from one given Point to another, and from these plain simple Principles, they have raised most astonishing Speculations, and proved the Extent of the human mind to be more spacious and capable than any other Science."
>>> Again: get off the high horse; because you are being either >>> ignorant, >> >>> or hypocritical. >> >> >> >> In this context /argumentum ad hominem/ is clearly a fallacy. > > Sigh... > > First: saying that you are being ignorant or hypocritical is not, in > and of itself, an "argumentum ad hominem". Contrary to common > misconception, "argumentum ad hominem" is not merely the saying of > bad things about others.
There's something about a "high horse" in that as well.
It is quite legitimate for a mathematician or anybody, for that matter, to appeal to native first principles rather than the authority of others as the first and final arbiter of Truth. Any other position would be unacceptable (mental slavery).
> You wrote: "I don't answer to the authority of mortals." > > Yet you admit that you take as authority axiomatic set theory, which > comes from, surprise, mortals.
> (i) It is defined at a; (ii) The limit of f(t) as t approaches a > exists; and (iii) the value of the function at a equals the value of > the limit of f(t) as t approaches a. > > A real valued function of real variable f(t) is "continuous at all > real numbers" or "continuous everywhere" if and only if it is > continuous at x=a for all real numbers a. > > A real valued function of real variable f(t) is "continuous" if and > only if it is continuous at all points in its domain.
Is limit[(t^2-9)/(t-3), t->c] continuous for all real numbers c?
Saying that f(t) is continuous everywhere is the same as saying that f(t) has a finite limit for all real numbers t, and that f(t) = limit[f(x), x->t].
Is f(u) = limit[(t^2-9)/(t-3), t->u] a continuous function of u over the reals?
Does f(u) = (u^2-9)/(u-3) everywhere u!=3?
If g(t) is continuous and g(t) = f(t) everywhere t!=3, and limit[f(t), t->3] exists and is finite, then what is the value of g(3)? Is it not g(3) = limit[f(t), t->3]? Could it be anything else?
I contend that asserting f(x) is continuous over the domain of real numbers, and f(x) = (x^2-9)/(x-3) everywhere the rhs is meaningful (i.e., x!=3), unambiguously determines a one-to-one mapping of the real numbers onto the real numbers. This description of f(x) thus constitutes a clear definition of a function which is continuous over the real numbers. Furthermore, f(x) <=> x+3.
Quod erat demonstrandum.
> -- End Quote -- > > and which you deleted, and then continued to conflate "continuous" > and "continuous everywhere" as if they meant the same thing? > > What's the point of me giving you citations and quotes, if all you > want to do is stroke your ego saying that *you* discovered your error > (though you do not correctly identify what the error is), or that you > don't answer to "authority of mortals"? > > Simple fact: you screwed up; you screwed up because you ignored the > conventions at play. Then you tried to justify your screw up through > some rather impressive, but ultimately doomed, mental acrobatics. > When this was explained to you clearly and simply, you got off on a > high horse and made grand-elloquent but ultimately pompous and > ignorant statements such as "I don't answer to the authority of > mortals". Eventually, you conceded error, but were not willing or > able to admit the explanations of others, and came up with an > "explanation" of your error which is itself in error and inaccurate. > Given that you have failed to adequately understand what your > multiple errors are, it is unlikely that you will be able to proceed > without committing even more in the future. > > But thanks for playing. >
You quoted yourself and made claims about my responses, but cited none of those responses.