Not much of a looker, is she.
(She's still out of my league though.)
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Scientists on RF: sharing your expertise
- Thread starter Laika
- Start date
(She's still out of my league though.)
I'm an entomologist. Combining, school, brief jobs with state government and a career in industry, I have about 30 years experience in science.
I turned chasing insects as a child into a career and got paid for it.
I turned chasing insects as a child into a career and got paid for it.
It's not??PhD in applied mathematics. Field of specialization in differential geometry and topology.
Help physicists to get they math straight, sometimes. They still call the Dirac delta a function.
Poor losers
Ciao
- viole
PhD in applied mathematics. Field of specialization in differential geometry and topology.
Help physicists to get they math straight, sometimes. They still call the Dirac delta a function.
Poor losers
Ciao
- viole
Do you call it is a measure or a distribution?
It's not??
Well, it is.....but not in the way physicists think.
It's not??
Functions are not zero almost anywhere and infinite at a point so that they read out the values of other functions when used as kernels in integrals. Functions have clear values for any point of their domain.
No. It is a distribution. A generalization of the concept of function. Distributions as defined in functional analysis (not statistics). As such it is also differentiable an arbitrary number of times, which is quite handy.
Ciao
- viole
I had a kernel stuck in me teeth once.
There was calculus build up too.
Please don't report me to staff.
I tend to think of the Dirac delta as a measure. Measures are ways to describe the 'size' of a set and generalize concepts like length, area and volume. They also allow a fairly general integration theory. The Dirac delta at a point gives a set a size of 1 if the point is in the set and 0 otherwise. The integral of a continuous function just gives the value of that function at the point.
The good thing is that measures make sense even if there is no differential structure around. On the other hand, if there is, the derivative of a measure need not be a measure (in fact it usually isn't).
Now, technically both measures and distributions *are* functions. But they are not functions on the space most engineers and physicists think of them as being on. Instead, they are functionals which take other functions and give out numbers.
Well, you can see it like that. But, as you correctly noticed, things like the derivative of a delta is not a measure. Yet, it is still a distribution. I don't like this sort of interruption of properties, and maybe that is why I prefer distributions.
But that is mainly a question of taste.
By the way, what do you think of the axiom of choice?
Ciao
- viole
Absolutely essential for much of what I do. I rely on Tychonoff's theorem, which is equivalent to AC (although the Hausdorff version isn't). Not to mention Hahn-Banach; another absolutely crucial result.
I generally reside in some sort of Banach space, so measures being equivalent to the dual space of the collection of continuous functions on a compact space is more important than the distributions being the dual of the test functions (whichever test functions you choose to select).
I am much more up in the air about the Continuum Hypothesis. Or, for that matter, large cardinals.
Oh, a functional analyst, I see. I also love functional analysis. It is amazing how things like Hilbert spaces map so naturally in physical things like Quantum Mechanics.
Well, I am also pro-choice, so to speak.
But I cannot help feeling a bad taste about it. I cannot get it out of my mind that assuming it as true is mainly justified by its utility. But are the fruits that it produces justification enough that it is true?
Ciao
- viole
Given that it is independent of the other axioms, I would say yes.
The question boils down to how we select our axiom systems. Clearly, any system that doesn't allow basic calculus would be discarded. But even that is based on utility.
The goal of mathematics is to produce beautiful theorems and proofs. We choose the axioms to satisfy this aesthetic. So we adopt AC because the theorems proven from it are more beautiful than those possible without it.
True. But the parallels axiom was also independent from the other Euclidean axioms, and see what happened when we deny it. A completely new world opened up.
Since I am an applied mathematician, I would not indulge in phylosophical or meta-logical questions when it come to the AC. I mean, I am not even sure that the rule of the third excluded is true, even though I do use it all the time when I work.
I have simply no easy way to destroy the constructivist approach. I find it, prima facie, absurd to declare that it is not the case that either the Goldbach conjecture is true or false, until we have found a way to construct a proof or a defeater thereof, but I have no easy way to show that absurdity without being circular.
I am just a wee concerned that we give too much priority to esthetics than to truth. Nobody says that truth must be beautiful ( to human mathematicians).
Unless we reduce truth to tautologies, of course.
Ciao
- viole
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Sapiens
Polymathematician
Seems like we're just setting up an appeal to authority here, arguments on RF should stand or fall on their own and not, tempting as it is, on our credentials. That said: I did Tussman Tutorials at Berkeley followed by three more years in the sciences, completing the lower division required sequence in Math, Chemistry, Physics, Paleontology and finally getting a degree in Zoology. My honor's thesis began with a study of the foraging behavior of Brant's Cormorant and expanded to address the niche separation of the suite of diving birds found in the Monterey Bay area.
My PhD work was in Oceanography with a thesis topic on the performance of Antillean Artisanal Fish Traps.
My PhD work was in Oceanography with a thesis topic on the performance of Antillean Artisanal Fish Traps.
The problem isn't just with constructivism. The problem is that 'truth' only makes sense in the context of a model while provability makes sense with only the axioms. Any statement that is independent in the sense of Godel can neither be proved nor disproved from the axioms. Because of this, adding either that statement or its negation to the axioms will preserve consistency: we get to assume either way!
So, what happens if the Goldbach conjecture is neither provable nor disprovable from the current axioms? Do you still have the same feelings about truth or falsity? Remember, in that case, there would be two different set theories that are equally consistent, one has GC true and the other has it false.
I don't think there is a single a priori model for mathematics. I am certainly NOT a Platonist when it comes to math. In fact, I tend towards formalism, but allow for intuitions to determine axioms. We have certain intuitions and some of those are independent of the other axioms we assume. If that happens, we can assume a new axiom encompassing those intuitions. Yes, other axiom systems are possible. And we may select those axiom systems that work well in formulating what we want to say.
So, the reason I choose the axiom of choice is that it gives good results and is independent of the other axioms. The quality of the math and the generality of the results serve to justify the assumption of AC. I would strongly suspect that you also assume at least the axiom of countable choice in your work. It is used whenever subsequences are chosen without a construction criterion and is the basis of many existence results for ODE and PDEs.
Now, the continuum hypothesis is another statement that is independent of the other axioms (including choice). But, from what I have found, neither it nor its negation gives significantly nice results to lead people to assume it or deny it. That said, the bias among set theorists is to deny it. I tend, on the other hand, to like the symmetry of the generalized continuum hypothesis.
Oh yeah. I could not live without the axiom of choice. I told you, I am pro-choice
It just smells slightly, but, as they say, you get used to smells after a while. A bit like the beauty of Switzerland coming at the small price of smelling cows' left overs.
And let's be honest. Without it, we would not have nice things like the Banach-Tarski paradox. Which would be a huge loss. I would love to apply it to this coin in my hand. Alas, I have not worked out yet, how to cut a coin into not measurable sets.
Ciao
- viole
I did not see this earlier. Sorry. No, I have never worked in Alaska.