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Tag Archives: mathematics

In the *Rational vs Real* post I mentioned that real numbers were each *“an infinitely tiny island separated from direct contact with all other numbers.”* The metaphor of each real number as an island comes from how, given any real number, it’s not possible to name the next (or previous) real number.

It’s easy enough to name a *particular* real number. For instance **1.0** are **3.14159…** real numbers. There are infinitely many more we can name, but given any one of them, there is no way to get to any other number other than by explicitly naming it, too.

This applies to a variety of numeric spaces.

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6 Comments | tags: complex numbers, continuum, integers, math theory, mathematics, natural numbers, numbers, rational numbers, real numbers, Theory of Mathematics | posted in Math

One of the great philosophical conundrums involves the origin of numbers and mathematics. I first learned of it as Platonic vs Aristotelian views, but these days it’s generally called **Platonism** vs **Nominalism**. I usually think of it as the question of whether numbers are *invented* or *discovered*.

Whatever it’s called, there is something transcendental about numbers and math. It’s hard *not* to discover (or invent) the natural numbers. Even from a theory standpoint, the natural numbers are very simply defined. Yet they directly invoke infinity — which doesn’t exist in the physical world.

There is also the “unreasonable effectiveness” of numbers in describing our world.

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6 Comments | tags: math theory, mathematics, natural numbers, nominalism, numbers, Plato, Platonic, Platonism, rational numbers, real numbers, Theory of Mathematics | posted in Math, Philosophy

Musicians practice; actors rehearse; athletes work out; and mathematicians play with numbers. Some of the games they play may seem as silly or pointless as musicians playing scales, but there is a point to it all. That old saying defining insanity as doing the same thing over and over and expecting different results was never really correct (or intended to be used as it often is).

An old joke is more on point: *“How do you get to Carnegie Hall?”* (Asked the first-time visitors to New York.) — *“Practice, practice, practice!”* (Replied the street musician they asked.) The point of mathematical play can be sheer exercise for the mind, sometimes can uncover unexpected insights, and once in a while can be sheer fun.

As when finally solving a 65-year-old puzzle involving the number **42**!

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8 Comments | tags: 42, Andrew Booker, Diophantine equation, discrete mathematics, mathematics, Numberphile | posted in Math

Mandelbrot Antennae

*[click for big]*

I realized that, if I’m going to do the **Mandelbrot** in May, I’d better get a move on it. This ties to the main theme of Mind in May only in being about *computation* — but not about computation**alism** or consciousness. (Other than in the *subjective appreciation* of its sheer beauty.)
I’ve heard it called “the most complex” mathematical object, but that’s a hard title to earn, let alone hold. It’s complexity does have attractive and fascinating aspects, though. For most, its visceral visual beauty puts it miles ahead of the cool intellectual poetry of Euler’s Identity (both beauties live on the same block, though).

For me, the cool thing about the Mandelbrot is that it’s a computation that can never be fully computed.

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9 Comments | tags: computation, computer model, computer program, fractals, Mandelbrot, Mandelbrot fractal, mathematics, Turing Halting Problem | posted in Computers, Math

An old saying has it that *“March comes in like a lion and goes out like a lamb.”* That was certainly the case for us this year. February and early March were full-on old-fashioned winter, yet when baseball season started (in the USA) this past Thursday, the snow was mostly gone, and temps were in the 50s. (That’s the thing about winter: spring is pretty sweet.)

The end of March means the official end of the Mathness, but it’s not exactly the end of the math. The whole point of the rotation study was trying to understand 4D rotation, and I haven’t explored that, yet. I plan to, and soon.

But today, as an exit March, I want to talk about math phobia.

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9 Comments | tags: math anxiety, math phobia, mathematics | posted in Math

Last week we celebrated Albert Einstein’s birthday (he turned 140). Now we need another cake so we can celebrate the *other* March major mathematician’s birthday — **Emmy Noether** turns 137 today.

To my regret, despite that I frequently invoke her name (she co-starred with Albert in the Special Relativity series), her work in mathematics is pretty far above my head, and I’m simply not qualified to write about it. I can say that her work connects mathematical symmetry with physical conservation laws. She also made significant contributions to abstract algebra.

Just recently, I’ve begun to nibble at the edges of the latter in the form of group theory as a part of studying rotation.

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20 Comments | tags: algebra, cube, Emmy Noether, group theory, mathematics, rotation, square, tesseract | posted in Math

Back at the start of March Mathness I promised the math would be “fun” (really!), but anyone would be forgiven for thinking the previous two posts about Special Relativity weren’t all that much “fun.” (I really enjoy stuff like that, so it’s fun for *me*, but there’s no question it’s not everyone’s cup of tea.)

Trying to reach for something a bit lighter and potentially more appealing as the promised “fun,” I present, for your dining and dancing pleasure, a trio of number games that anyone can play and which might just tug at the corners of your enjoyment.

We can start with **277777788888899** (and why it’s special).

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2 Comments | tags: 277777788888899, algorithm, calculation, Collatz conjecture, computation, Mandelbrot, mathematics, multiplicative persistence, Numberphile | posted in Math

*Time for math!*

I have a special fondness for the month of March. For one thing, it contains the Vernal Equinox — one of my favorite days, because it heralds six months of light. (As a Minnesotan, Spring has much more impact than it did when I lived in Los Angeles.)

March is when the weather elves begin preparing for the April Showers that create May Flowers. It’s when baseball Spring Training is in full swing with the regular season looming (lately, even at the end of the month; this year on the 28th).

It also contains some important birthdays: Albert Einstein (3/14) and Emmy Noether (3/23), to name two, and in their honor I have myriad math posts planned!

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8 Comments | tags: 3Blue1Brown, Albert Einstein, complex numbers, Emmy Noether, mathematics, Mathologer, Numberphile, tesseract | posted in Math

Folded into the mixed baklava of my 2018, was a special mathematical bit of honey. With the help of some excellent YouTube videos, the light bulb finally went on for me, and I could see **quaternions**. Judging by online comments I’ve read, I wasn’t alone in the dark.

There does seem a conceptual stumbling block (I tripped, anyway), but once that’s cleared up, quaternions turn out to be pretty easy to use. Which is cool, because they are very useful if you want to rotate some points in 3D space (a need I’m sure many of have experienced over the years).

The stumbling block has to do with quaternions having not one, not two, but three *distinct* “imaginary” numbers.

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16 Comments | tags: complex numbers, math theory, mathematics, natural numbers, number systems, numbers, octonion, quaternion, real numbers | posted in Math, Sideband

I seem to be doing a lot of reblogging lately (a lot for me, anyway). But I’m on kind of a math kick right now, and this ties in nicely with all that.

4 gravitons

You’ve probably seen it somewhere on your facebook feed, likely shared by a particularly wide-eyed friend: pi found hidden in the hydrogen atom!

From the headlines, this sounds like some sort of kabbalistic nonsense, like finding the golden ratio in random pictures.

Read the actual articles, and the story is a bit more reasonable. The last two I linked above seem to be decent takes on it, they’re just saddled with ridiculous headlines. As usual, I blame the editors. This time, they’ve obscured an interesting point about the link between physics and mathematics.

So what does “pi found hidden in the hydrogen atom” actually mean?

It doesn’t mean that there’s some deep importance to the number pi in nature, beyond its relevance in mathematics in general. The reason that pi is showing up here isn’t especially deep.

It isn’t trivial either, though. I’ve seen a few people…

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2 Comments | tags: math theory, mathematics, pi, quantum physics | posted in Math