Comments on: On Abstraction

By: Sasank

Sasank — Fri, 16 May 2008 03:30:10 +0000

Great explanation of abstraction. Thank you.

By: kane

kane — Wed, 26 Sep 2007 17:23:53 +0000

I do like that raindrop analogy.

By: The Narrow Road » Blog Archive » A Transfinite Landscape

The Narrow Road » Blog Archive » A Transfinite Landscape — Wed, 04 Jul 2007 05:24:01 +0000

[...] Going all the way back to the first entry, On Abstraction, things start to get a little clearer however. As long as we view mathematics as a matter of making effective and powerful abstractions from the real world, rather than describing some platonic universe, having a choice of abstraction doesn’t seem so bad. We can choose how to interpret the continuum to suit our needs — indeed, we can even reject transfinite arithmetic and opt for the intuitionist conception of the continuum if we wish; we choose the abstraction that best suits our purposes for the moment. You could view it as little different than choosing to work at the genetic level as a molecular biologist instead of the considering subatomic particles as a physicist would: the level and manner of abstraction matters only with regard to the level and manner of detail you wish to obtain in the way of results. The more layers of abstraction we apply, the greater the chances of running into quandaries and choices; by abstracting away more and more detail, and by piling abstractions upon abstractions, we push further and further into the realm of pure possibility. This has the potential to lead us to strange and confusing trails, but it also gives us the power to see beyond our own limited horizons. In broadening our minds to embrace worlds of possibility we conceive of realities that transcend our conceptions, and probe our own reality in ways far beyond the limits evolution has shackled our perceptions with. [...]

By: Greg

Greg — Tue, 29 May 2007 23:47:08 +0000

I am delighted to see you undertaking this project. I’ve wished for such an introduction to Mathematics most of my life. As a computer scientist and someone generally interested in science and mathematics I am always trying to expand my understanding. Most mathematics texts read to me like the work of old-fashioned programmers who delight in low-level coding. I prefer a modular and incremental presentation of complex ideas.

I find your words and imagery both clear and beautiful. I find Russell’s convoluted and deadening. I would encourage you to replace the Russell quote with something of your own.

What I get from Russell’s torturous expression is that he is trying to say that counting presupposes distinctness, and then he generates a basis for that with a deliberately non-intuitive approach. Trying to kill intuition may have been desirable for his Principia Mathematica program, but it is very off-putting to a beginner! I would prefer something more like this:

(1) Let “x differs from y” mean some property p exists which holds for x and does not hold for y.

(2) Let “property p is a unit property” mean that
(2a) an entity a exists satisfying property p, and
(2b) for all entities x where “x differs from a”, x does not satisfy property p.

and so on. However, even after disentangling Russell’s expression, I’m not at all convinced that this is a good basis for understanding counting. I would prefer an approach involving one-to-one correspondence, as that is closer to how humans (and some other animals!) learn counting. I think also we draw on our geometrical notions of magnitude. Counting sits on a pleasant peak which can be reached by several easy (and some quite challenging) paths. The views from that peak are illuminating and many paths from there lead onwards and upwards!

By: angelina

angelina — Fri, 18 May 2007 09:35:30 +0000

hello. very nice. good luck

By: bruce

bruce — Sun, 13 May 2007 09:24:24 +0000

Hello, nice site. Good luck

By: Zachariah

Zachariah — Wed, 02 May 2007 22:14:25 +0000

This is beautifully thought out and eloquently written. Thank you so much for an intelligent and thought-provoking insight into a staggeringly sumptuous world of wonder and delight.

I wish you the best of luck in this adventure, and sincerely hope that you can impart theories of calculus in an accordingly dazzling manner; it deserves an exquisite treatment.

By: carlos

carlos — Tue, 24 Apr 2007 19:22:50 +0000

please keep this up! this is the way math should’ve been taught to me in school!!! i totally get this..

By: nice article

nice article — Sat, 21 Apr 2007 16:41:44 +0000

this guy is good…

lol amazing…

By: hfamk

hfamk — Sat, 21 Apr 2007 09:19:10 +0000

Hello, you have a nice site, good LUCK!