PLS Science I tutorial - Spring 2008

Something I Found to be Interesting

mitchbpls — Tue, 29 Apr 2008 00:16:25 +0000

I found Socrates’s answer concerning the source of mathematics’ success to be one of the most interesting points within the dialogue. Socrates claims that he believes the “source of the success of mathematicians lies in their methods, the high standards of their logic, their striving without the least compromise to full truth…and avoiding self-contradictions.” (24) He furthers these claims by suggesting that he utilizes these same logical underpinnings in his own political and philosophical discourse. In this manner, the principles of mathematics underlie politics and philosophy.

Question: Are these principles necessary for pure discussion? pure art? or skill? For example, is their a “successful” discipline that discourages avoiding self-contradictions and/or striving for full truth?

Socrates determines that this form of questioning has made him an enemy of almost everybody. (25) In spite of this, he refuses to compromise truth. Throughout time, philosophers and mathematicians who’ve advocated similar ideas have frequently been met with strife. Why were/are these very basic ideas dangerous to middle society?

a priori or experience

jared33 — Mon, 28 Apr 2008 04:26:19 +0000

A point that did not make much sense for me in the reading of Space, Ether, and Field in Physics (1930) was the reason why “an all-pervading medium was needed for the light waves, which penetrate even empty space” (175). I am sure that the reason why I am confused is just my unfamiliarity with the science of light waves and whatnot, but I am just hoping that someone will explain this to me in lay-man’s terms.

On the dialogue, at first it seemed as if the author had a somewhat Kantian view of geometry. He seemed to believe that mathematics was something pure in our minds that we apply to the physical world, in a way that reminded me of Kant. But then he spoke of mathematical knowledge only coming from experience, like the child counting pebbles to reach a notion of 2+3=5. I was wondering how these two ideas can be reconciled? Does it have to be just one or can it be a combination of the both, as Einstein seems to assert? How can it be both pre-existing in the mind and from experience?

Smilies concerning math

cbh59 — Mon, 28 Apr 2008 02:30:04 +0000

I really liked the idea of math as a map. It helps us to understand our world from a removed point of view, and it explains things that otherwise would be hard to see or grasp. We could never know what a perfect circle was if there was no concept of such outside of this world.

I didn’t like the concept of math as a reflection of the world–it seems our world is more a reflection of math than vice versa. Reflections in the water are flawed and wrinkled, just as the embodiment of perfect mathematical principles in this world aren’t up to par. We can never hope to find a perfect circle on earth, or a perfect form of any geometrical shape. Even if we did, as humans our instruments would never be fine-tuned enough to measure and be sure that this was indeed the case.

Thoughts?

Hyperspace links

Robert Goulding — Mon, 07 Apr 2008 16:13:13 +0000

In response to a couple of queries, some websites on four-dimensional space.

This page is a step-by-step tutorial to visualizing higher spaces, using different approaches (dimensional analogies like Flatland, perspectives of 4-d objects etc.).

This is a movie of a rotating hypercube (or tesseract). What are we watching here? A two-dimensional representation of the three-dimensional shadow of a hypercube rotating “about a plane” in four-dimensional space. (I am reminded of Plato’s divided line: pictorial representations are only shadows of shadows).

Here is a more technical page. In the first Java applet, you can move the hypercube in various directions and see all the aspects its 3-dimensional shadow can take.

Some of the best movies of rotating hypersolids are found here. Warning: these movies will make you feel ill if you watch them too long…

The Mind’s role in all this

jared33 — Sun, 06 Apr 2008 18:59:59 +0000

I may be preempting some of our future Kant reading, but since I just read some of him in my Seminar IV class I was wondering a little bit about the mind’s role in all of these considerations about the fourth dimension. Like Kant, Aristotle also believed that our minds imprint an order upon the outside world, and that is why we see our external space to be Euclidean and 3-dimensional. I believe we said that our minds are like a ‘software program’ in a computer that interprets and translates for us the external data and then presents it to us in an orderly fashion. Thus, we imagined Euclidean geometry to be the geometry of our space because that is how our ‘software program’ interprets the data. If our minds were purely receiving input from the outside world without this interpretation, I imagine we would have probably considered space to be spherical, since that is the kind of world we live on for the most part (like sphereland). But instead we came up with Euclidean geometry. So what is the implication for this imposition of our minds upon the outside world for the idea of a 4^th dimension? Does this mean that these dimensions are not real but instead just ‘figments of our imaginations’? And if that is the case, then aren’t we really just talking about extending our minds into another dimension, and thus that dimension may not really exist in actual space? I am of the opinion that our notions of dimensions do exist in real space, but how do we know that all the talk we’ve had about the fourth dimension is not just a thought experiment that we have completely made up within our minds and not a true extension of space? Maybe I am just not smart enough to know what all these philosophers are saying, but I do know that I’m confused.

Corpus Hypercubous

katieshircliff — Wed, 02 Apr 2008 06:59:00 +0000

An interesting 2D representation of the hypercube (an unfolded one) by Dali, and a work of brilliant religious symbolism. Check out the plane of 2D squares below, the 3D space it’s set in, then the frame, I guess, of the hypercube of the cross!

Practicality of Geometry?

jared33 — Tue, 25 Mar 2008 22:26:38 +0000

Davis’s discussion of sphere-land and spherical geometry brought up some serious questions for me about the practicality of geometry in the real world. It would seem that Euclidean geometry was actually an incorrect representation of the world we live in. For Davis says when talking about people that live on a sphere that: “They would have no parallel lines, although they would probably think they did. The lines they think of as parallel and everywhere equidistant would actually not be equidistant” (127). We were these people! My understanding was that most people believed the world was flat for a long time, although perhaps not some of the Greek philosophers. So was Euclid operating from the belief that his system of geometry was an accurate representation of the world we live in, or was his geometry entirely a mind-experiment? And if it was merely a mind experiment, what is the practical application to our lives and why should we do it at all?

Hyperbolic Art

roosterhoff — Sat, 22 Mar 2008 15:28:06 +0000

M.C. Escher, as it happens, was hooked on non-Euclidean geometry.

Parallel Bogus

sjohns12 — Wed, 19 Mar 2008 01:15:57 +0000

I’m not sure what everyone else is thinking, but I know what I know, and there is no way there is more than two parallel lines to a given line from a single point not on that line. I think it all comes back to the “straight” lines idea because even hyperbolic geometry’s own diagrams obviously show that the other “parallel” is going to cross the given line because it is pointed right at it!!!! ( see Figure 2.23 and almost all the others)

It’s like this: I think just about everybody would assert that a wall could be made to infinite length and if you kept putting less and less bricks on top of it would get shorter and shorter until the top and bottom intersected each other. It’s just plain ridiculous to say they wouldn’t. I agree with Euclid. It’s just obvious. This multiple parallel stuff is crazy. (I cite railroad tracks and fences as proof).

Brains, Numbers, and Learning

roosterhoff — Wed, 27 Feb 2008 20:40:21 +0000

A fun article in The New Yorker tells of a brilliant Frenchman who thinks that all children learn mathematics alike, and that they shouldn’t learn to do long division.