Learning Project: Symbolic Logic

Some of the things I learn will take more than a day. For these topics, I’ll post an overview like this one, and then occasionally use a fragment of what I’ve learned as a daily post.

Something I’m very keen on learning is symbolic logic, along with number theory, set theory, and proof writing. These topics bring us to the most basic foundations of mathematics.

I’ve always been bad at math. In elementary school, I had the unsettling realization that if you subtract 2 from 6, you get 4, but if you are told to read pages 2 to 6, you wind up reading 5 pages. I’ve found math to be slippery and untrustworthy ever since.

This is one reason for my desire to return to the absolute basics. What is a negative number, really, and why does multiplying two of them yield a positive? What is going on when you divide a fraction by a fraction? Why are we able to tell the length of one side of a triangle given two angles? What underpins the way numbers change as you count- i.e. representing the number “10” in base 10 versus “1010” in binary.

Another reason I want to study logic is that it was the only unit in math class that was fun and made sense. Everything about (p v ~q ) and truth tables made perfect sense, and doing it made my brain flow in a way that felt so right! It wasn’t until years later that I found out the p and q exercises were in fact the gateway to understanding all of mathematics.

Logic is the language in which proofs are written (well, some of them, anyway,) and proofs give exactly the explanation to all these confusing math equations I seek. I was simply taught the “plug and chug” method- memorize an equation, memorize a process, and just calculate the numbers. There was no explanation for why it worked. Proofs demonstrate the inner workings of all mathematical entities. There are proofs for the Pythagorean Theorem, calculus, and even things as fundamental as the fact 2 is an even number. I have a lot of hope that being able to read proofs will not only solidify the basics, but also provide a backwards approach for grasping higher math.

Moving forward, why care about math? Someone I’m not going to bother to look up to quote properly said, “Math is the language of the universe.” I’m interested in learning about all sorts of things that require math to truly understand, such as physics, computer science, and evolution. Also, math itself can be fascinating. I read a book that explained great math concepts to laymen, and it made math in and of itself interesting. Such topics were visited as infinity, the harmonic series, and transcendent numbers.

For example, take the harmonic series, studied by the Bernoulli brothers.

Consider this progression of fractions: 1/1, 1/2, 1/3, 1/4, 1/5, 1/6…

Adding 1/1 + 1/2 + 1/3 + 1/4 will get you to the number 2. But, these fractions are getting smaller and smaller, so you need more of them to reach the next whole number– in fact, you need to add them all the way up to 1/11 to reach 3 and 1/31 to reach 4. Let’s skip a few steps and simply double the 4… in order to reach 8, we need to add the fractions through 1/1674. Reaching 19 requires more than 100 million fractions, and that’s about where the calculator stopped working.

The breathtaking realization here is that these fractions will continue to get infinitely smaller, and infinitely more of them will need to be added together to reach the next whole number, and this will go on to support an infinitely large quantity of whole numbers.

This inspired Jacob Bernoulli to write a poem, which I will immodestly boast I typed from memory:

As the finite encloses an infinite series
And in the unlimited, limits appear
So the soul of immensity dwells in minutia
And in narrowest limits, no limits inhere.

What joy to discern the minute in infinity!
The vast to perceive in the small, what divinity!

So yeah, I’m interested in learning math.

 

Sources:

Harmonic series: Journey Through Genius: The Great Theorems of Mathematics by William Dunham

Harmonic series calculator: http://www.math.utah.edu/~carlson/teaching/calculus/harmonic.html

My main source for this Learning Project is Sets, Logic, and Numbers by Clayton W Dodge.

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