Here’s one of Bathory’s videos (only video?) entitled One Road to Asa Bay. Go Vikings!
Let be a linear map between vector spaces. Show that T is injective if and only if .
Suppose first that T is injective. Then implies , so .
Now suppose . Then implies which implies , and . So, T is injective.
Let be a homomorphism of Abelian groups. Show that for each .
Since f is a homomorphism, and since , an Abelian group, then:
(f is a homomorphism)
The last line is true because (but that’s another proof).
Michael Atiyah gives a presidential address on Mind, Matter, and Mathematics (good alliteration). In it he discusses the difference between mathematical philosophy and natural philosophy. It’s an interesting read throughout.
But, near the end he says:
Mathematical physicists believe that there are indeed simple and beautiful mathematical equations that govern the universe, and that the task of the scientist is to search for them. This is an article of faith.
An alternative faith is to believe in a God who created the universe and kindly provided us with laws or equations that we would be able to understand.
He touts that these are compatible philosophies. As faiths, they are similar (I take issue with the first idea: Mathematical equations do not “govern” the universe, they are just really good at representing it.)
But, more importantly, I disagree with the idea that belief in God is always compatible with science (an implication I think he was making). In physics, it’s an easier sell. There is nothing alive in physics.
A harder sell is in biology. Belief in God is one thing, but belief in a soul is problematic. If one believes in a soul, that every human (homo sapien) is singled out from among God’s creatures as different (better), then all of biological evolution (and what it can tell us about who we are) falls apart.
If it is true that humans ARE totally and fundamentally different than all of the other creatures on earth (and potentially on other planets), and if this is due to our having a soul, then we MUST abandon much of what we believe to be true in biology.
If biology is right, then we are not different in any fundamental way than other species. Unique, sure. (So is the norwhal.) But, not totally different.
I am hesitant to say that a belief in a God-given soul is compatible with biological science, and from there science generally.
(HAT TIP: Noncommutative Geo)
Well, my class-load this term is not as large as Felisis’s. I’m only taking 3: Abstract Algebra, Topology, and Multi-linear Algebra (with Erdman, for those who know what that means). I’m also still working (as a private weightlifting coach).
I’m gonna try and post a few times a week, including POD’s and other tidbits.
That is what I have to contend with this quarter. One class, Calculus II, I’m teaching; the other three: Stochastic Processes and Probability Theory; Graph Theory; and Partial Differential Equations; I’m taking.
I had hoped to be posting something weekly about all of my classes, but after getting started on this post last weekend and not taking it any further, I’ve decided to just make one math-related post a week. Later today or tomorrow I am going to put up an explanation of l’Hospital’s rule, which gives us a way to calculate limits that would otherwise give rise to what are called ‘indeterminate forms’. Sometime over this week I’ll throw in a note about where Bessel functions come into play, and next weekend I’ll catch up with some measure theory from the Stochastic Processes class.
ciao for now!