Friday, July 6, 2018

Curved Perspective - The Great Switcheroo

I'm mainly blogging today to share a video I recently watched which I think is well worth sharing. It concerns a topic I've touched upon on here before, namely the idea that our vision is curved. I last posted about it well over a year ago when I reviewed and shared some information from the book Space-Perception and the Philosophy of Science by Patrick A. Heelan.

The links for those articles;

Hyperbolic Perspective: The Visual Dome You Experience

Hyperbolic Perspective: Bedroom in Arles

Anyway, the video in question is tilted The Great Switcheroo and was uploaded by the YouTuber Vortexpuppy (aka Gav from Beyond the Imaginary Curve). The video is a discussion/presentation where Gav explains some of his thoughts on perspective and mathematics. I can't vouch entirely for the mathematics and views contained within the video as it would require me to pretend that I entirely understand everything, when in fact I'm still in the process of learning about it all at the moment. Everything in the video does seem to tally with my own previous thinking on the topic though, especially with regard to perspective.

In fact, as an aside, I think this is often a problem with things of this nature. People who don't entirely understand something like this tend to fall into two categories - people who pretend they understand it all, and people who happily admit they don't, but who because of that lack of understanding feel they can't ever understand it, and that it's simply beyond them. Fortunately I have the honesty to admit that I don't understand things, but also the arrogance to believe I will at some point in the future :)

(Flat Earth - The Great Switcheroo)

Hopefully the PDF files used in the presentation will also become publicly available at some point, in which case I'll provide a link for those as well.

Sharing this video also gives me a chance to talk about mathematics in general, as I said I would in my last post.

Often when the average person sees a blackboard full of complex equations it just looks like gobbledygook to them. It looks like another language, and in many ways that's exactly what it is. People see it and assume that the maths and the concepts underlying the maths are simply beyond them. That they're not intelligent enough to ever grasp such an understanding. However, it's not so much the maths, or the concepts, it's more that they simply do not understand the language.

It's like seeing a blackboard full of Chinese - that too looks like gobbledygook to people not familiar with that language. However, Chinese children understand it with perfect ease, just like you understand the English words you're now reading with equal ease. You're perfectly capable of learning Chinese, after all you've managed to learn English without trouble. It's just that it takes time. You can't expect to look at something and instantly understand it straight away.

(Mathematical equations)

("It's all Chinese to me")

Anyhow, I often wonder if mathematics really needs to be so foreign and confusing. Would it not be possible to write maths in a way that is less opaque? In a way that removes some of the language barriers and that makes it much more intuitive for people.

Just to give a simple example;

If you show people the following equation most will find it fairly easy to understand;

2 + ? = 5

However, if you show people the following equation they often run in fear;

2 + x = 5

Now both these equations are exactly the same, however, one makes perfect sense in plain English and the other seems a little bizarre when read in plain English.

When we read "2 + ? = 5" we read "two plus question mark equals five". We all know that the question mark symbol signifies a question, an unknown. So it's easy for us to understand that 2 plus "an unknown number" equals 5, and that the answer is therefore 3.

However, when we read "2 + x = 5" we read "two plus x equals five". In English this makes no sense, and it's what leads people to find it so confusing. People are used to seeing the "x" symbol represent a letter in a word. Now all of a sudden it means something else entirely - in this case an unknown number.

People who can get their head around this substitution ("ah! okay, so in this case "x" represents an unknown number, right I see, I get it now") will be able to cope and make progress. However, people who get stuck on this will have a hard time going any further with algebra. In fact, even people who have a natural aptitude for this type of thing often have to take a breather and remind themselves what everything means. A "question mark" is simpler for everyone, not just the lesser able.

I actually have quite vivid recollections of trying to explain this substitution to other children at school. Fortunately I was always quite comfortable with maths, however I remember some of my friends really struggling. I can remember saying to them;
"Listen, "x" is just a question mark really - it just symbolises a number we don't know."
And they'd always reply;
"Yeah, but why is it an "x" , I don't get why it's an "x" ?"
They'd always assume that there was some profound reason why an "x" was chosen (or a "y", or an "a", or whatever other symbol, etc). Again I'd try to demystify things and say;
"It's just because that's what the first people doing these equations decided to use, and now that's what everyone else uses too - they could have used anything, it doesn't really matter."
But they'd dismiss me, I was just a child with no authority, obviously these really intelligent mathematicians must have chosen these symbols for some reason. It can't just have been a random selection.

Of course, the teachers would never help on this matter. They'd ask the teacher the same question;
"But why is it an "x" ?"
And instead of saying;
"No real reason really - any symbol would do the job, we just use "x" because that's what somebody randomly chose a long time ago."
They'd just go off into some funk about the x axis on a graph and its relation to the y and z axes, which would just baffle the child even more. Missing the real gist of their question.

In reflection I think it's probably anathema for mathematicians and teachers to speak about maths as blithely as I do. I think they like to treat our maths forefathers with a kind of worshipful reverence. Of course, these great minds of the past are absolutely worthy of our respect and appreciation, however at the same time simplicity is beautiful, and it's always worth reminding ourselves that all these great minds were just men no different to you or I.

Returning to our beautifully simple and literal "question mark" the obvious problem is what to do when there are multiple unknown values in an equation.

It's okay for 2 + ? = 5, but what if we have;

a + b = c

In that case ? + ? = ? doesn't really do the job. We'd need three different question mark symbols. So we've hit a bit of brick wall in our attempts to make our equations more easy to understand. Though I don't see why it should be beyond our ingenuity to have a range of question mark symbols specifically for multiple unknowns.

It's things such as this that I'd like to look at and discuss in the future. Obviously it's quite an ambitious, and perhaps completely fruitless task. I definitely feel it's an interesting avenue of thought to follow though :)

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