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Lockie

The next big break through in our understanding of the Universe

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10 minutes ago, JamesF said:

Perhaps you don't even need to go as far as "String Theory".  Consider for example the square root of -1.  In the physical world it has no obvious meaning, but it's still a very useful concept in maths (and indeed in using maths to describe the physical world).  Those numbers involving the square root of -1 are perhaps more usually known as "complex" numbers, but are also sometimes called "imaginary" numbers.

James

This is good example of how math was discovered rather than invented.

If we invented certain imaginary unit, let's call it i with property that i^2 = -1 (i squared is equal to -1) - then we would also invent its properties, however, we found out that complex numbers - which are numbers that consist out of two parts - real part and imaginary part - follow the same set of rules as other numbers do. How come? Well rather simple really - real numbers that we discovered before are in fact a subset of complex numbers. Any real number is just complex number with imaginary part set to 0.

If we were inventing things - we would only invent things that a) useful - which this one is not - because as you put it - in physical world it has no obvious meaning or b) have certain traits that suit us, but what are the odds that something that suits us perfectly fits with things that we already "invented", on such a deep level?

What are the odds for example that this imaginary unit that we "invented" fits perfectly within this identity:

image.png.0b16aa8481415b8b83ca6f2675a33c5c.png

That joins together 5 of what we could call fundamental constants - 0 which represents absence of things, 1 which represents singular (special state of existence - there are many plurals but only one singular :D ), base of natural logarithm, constant of circle and imaginary unit.

If we invented i - we would need to invent above equation at the same time, at the time we were describing i and its properties (here is i, it's square is -1 and it fits above equation like so, and has all these other properties ....). Otherwise we are discovering that i that we just invented (and did not exist prior) fits something that we later discovered - but if we invented it just now - how could it possibly be part of something that we discovered later - it must have existed and been part of that equation all along - we did not invent it - we merely discovered it.

I know this sounds confusing when I write it like that - but think of it - above equation was discovered much after i was defined to be what it is.

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@vlaiv That's the best equation I have seen this decade! Does it have a name?

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3 minutes ago, Ags said:

@vlaiv That's the best equation I have seen this decade! Does it have a name?

Euler's (pronounced Oiler)  equation .

Jim

 

Edited by saac
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2 minutes ago, Ags said:

@vlaiv That's the best equation I have seen this decade! Does it have a name?

It's called Euler's Identity, if I recall correctly.

James

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Oddly, when I read the email of vlaiv's post (which doesn't contain the graphic when sent), at this point:

15 minutes ago, vlaiv said:

What are the odds for example that this imaginary unit that we "invented" fits perfectly within this identity:

I was certain that it was going to be the equation he posted :D

James

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17 minutes ago, vlaiv said:

Sure, it is very famous and often quoted as one of the most beautiful if not the most beautiful math equations:

https://en.wikipedia.org/wiki/Euler's_identity

 

I'd hazard a guess it has been much exercised by epidemiologists in the battle of Covid 19.  Here's to Euler and his equation :) 

Jim 

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14 minutes ago, vlaiv said:

What are the odds for example that this imaginary unit that we "invented" fits perfectly within this identity:

image.png.0b16aa8481415b8b83ca6f2675a33c5c.png

I think the word "invented" is way too strong. Number theory quite clearly follows geometric rules. As I said above, everything in Physics can be expressed mathematically but that relationship isn't necessarily a two way street.

The example of imaginary numbers you gave is interesting and certainly beyond my grasp as a maths dunce. But there is a Pi in there and Pi is a geometrical entity if ever there was one. So doesn't that tie the rest of it to geometry?

My first encounter of imaginary numbers was in a now quite well know book by James Gleick: "Chaos", and was my first foray into fractals. Particularly the Mandelbrot Set, which I explored extensively (and very slowly :) ) on my BBC micro using an algorithm I constructed myself based on my reading of that book. Imaginary numbers play a big role in fractals and by my reckoning fractals are geometrical entities. So we come full circle; maths being a language that quantifies the geometrical nature of physics.

Does the Mandelbrot Set reside in geometry or mathematics? (I think I just convinced myself that it's purely mathematical... :) )

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20 minutes ago, Paul M said:

I think the word "invented" is way too strong. Number theory quite clearly follows geometric rules. As I said above, everything in Physics can be expressed mathematically but that relationship isn't necessarily a two way street.

The example of imaginary numbers you gave is interesting and certainly beyond my grasp as a maths dunce. But there is a Pi in there and Pi is a geometrical entity if ever there was one. So doesn't that tie the rest of it to geometry?

My first encounter of imaginary numbers was in a now quite well know book by James Gleick: "Chaos", and was my first foray into fractals. Particularly the Mandelbrot Set, which I explored extensively (and very slowly :) ) on my BBC micro using an algorithm I constructed myself based on my reading of that book. Imaginary numbers play a big role in fractals and by my reckoning fractals are geometrical entities. So we come full circle; maths being a language that quantifies the geometrical nature of physics.

Does the Mandelbrot Set reside in geometry or mathematics? (I think I just convinced myself that it's purely mathematical... :) )

We do quiet regularly "force" or perhaps approximate geometry onto the real world in order to more easily model it mathematically.  What I mean is we often approximate the physical world by reducing it to a more convenient and expressible geometrical device - the classic "consider a spherical cow moment".    Maybe this will make better sense - when describing Irradiance which we define as power per unit area  , we start with the approximation of "a point source".  Light emitted from a point source propagates as a sphere and so the mathematics becomes determinable and produces the inverse square law I = kd^-2  derived from the area of a sphere 4 PI d^2.   So in a way we have artificially forced PI into the relationship and upon nature.   Point sources don't really exist (not in our macro world)  so the real or true relationship between Irradiance and distance is somewhat more complicated (for extended sources) than our simple approximation suggests.  I think I tend to agree with you Paul , from my experience much of our  Physics describing, certainly the classical world, is rooted in geometrical approximations. 

Jim 

Edited by saac
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I suspect asking if maths is discovered or invented is a category error. I.e. the terms are not realy appropriate .

What happens is we propose a set of axioms, say i^2 = -1 and see what happens. So we "invented" and axiom and then "discover" the consequences. 

This is what happened when Riemann discovered new curved geometry by assuming that parallel lines crossed unlike Euclid's axioms where they don't. His new geometry is the backbone of General Relativity.

A brain tease for Sunday morning is what happens if you take as an axiom u^2 = 1 and u not equal to 1

Regards Andrew 

 

 

 

 

OK you get the hyperbolic numbers the backbone of Special Relativity 

Edited by andrew s
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