Posts Tagged ‘maths’
Apologies for this meandering piece of …
from Wikipedia – ‘Lagrange points in the Sun-Earth system (not to scale). This view is from the north, such that Earth’s orbit is anti-clockwise’
Living in Australia is such a relief, for someone who can’t afford to choose where in the world he might live, and happily being far from the madding action. I virtually never look at Aussie politics, being strangely and unhealthily drawn to more dysfunctional (IMHO) regions, such as the USA, Putinland, China, the Middle East etc. From my safe perch I can lecture these dysfunctional and brutal regions without fear of any blowback, and yet I fear, somehow, that nobody’s listening…
Global politics is a bit of a mess these days, to state the bleeding obvious. The USA in particular has gotten what it deserves, in boasting that any citizen can become President. Of course it isn’t true, but it’s just another claim used to make those citizens feel superior, even if they’re living in a prison cell.
Lately I’ve been somewhat pre-occupied with the artificiality of nations – of the nation concept. We make things up, and then come to believe in them as real. There are many examples – back in my university days (I was a late-goer to university, commencing in my thirtieth year and hanging around almost to my fortieth) I had a conversation with a bright young friend who pointed out that human rights were a made-up thing, and so…. Of course this is true, but, as I didn’t point out to him, because I’m rather slow-witted – and that’s the advantage of writing, based on reflections in tranquility – the human world is full of made-up things, like groups of buildings we designate as forming a university, or like formations made of wood and other materials that we call tables and chairs and houses.
So what is real? The land beneath our feet. The planet that gravity pushes us into (so gravity is real, it seems). The sun, the moon and the stars, which we now know are also suns. The mysterious universe that we’ve just begun to explore. The world revealed to us by microscopes and so many other technologies – genes, neurons, hormones, an enormous variety of cells and sub-cellular organelles, and of course myriad bacteria and archaea never known to exist until relatively recently. The near-infinite number of simple and complex organisms we share the planet with, including huge numbers that had their heyday, or relatively brief period on earth, long ago, like the Ediacaran biota (first found here in South Australia – and not so brief, they flourished between 600 and 540 million years ago, while we have so far managed well under half a million years, and we surely won’t manage much longer).
And then there are things we’re not quite sure are real (though some are surely more sure than others), such as photons, which are apparently massless particles, surely a contradiction in terms. Let me see, light comes in wave and also particle forms? For mass and energy are related according to Einstein’s equation, which is godlike in its omnipotence. Waves have energy, doubtless, I’ve felt that energy while floating about in the sea. Energy is, apparently, mass multiplied by the constant c, which is the speed of light, multiplied by itself, though I’ve heard that nothing can move faster than the speed of light, so…we can convert energy into mass only by means of an equation which is…. impossible, sort of? It would be a matter (bad choice of words) of a teeny-tiny mass being converted, who knows how, into a super-duper quantity of energy, and maybe even vice versa, and I’m not sure if knowing this, if we really do know it, helps us to understand – stuff. But everybody who’s very smart says that it helps us very much to understand stuff, like how space and time are intimately related, because… well, maybe because of that equation, which perhaps also explains how particles can get around without having any mass. And I don’t know if all this is very exciting or just mind-numbing or what. And then there are muons, gluons, colliding hadrons, quarks, neutrinos, bosons and maybe even god particles, for those who are still religious, for god knows what it’s all about. But I’m assured that it’s all more or less calculated so I should just shut up.
So, is all this real? Is dark matter really dark? Is dark energy a matter of expansion? Is the multiverse a bad joke? Is quantum tunnelling just a bore? Is theory just stringing us along?
But seriously, we really are clever – or okay, they, the clever ones – because they’ve landed people on the moon and roving machinery on Mars and placed other machinery at Lagrange points…
Now that’s a subject I want to get clear about. There are five Lagrange points created by the gravitational forces of the sun and our planet – the balance of those gravitational interactions. I’ll quote the clever ones from the European Space Agency to be clear:
There are five other locations around a planet’s orbit where the gravitational forces and the orbital motion of the spacecraft, Sun and planet interact to create a stable location from which to make observations.
But not-so-clever me, when he first conceptualised this, assumed there would be only one of those points, much closer to Earth than the Sun of course, due to the Sun’s bigness. How did they manage to locate/calculate five? The ESA’s description and explanation of these five points (L1 to L5) is intriguing, though it doesn’t answer my question. Here’s their description of L1:
The closer an object is to the Sun, the faster it will move. So, any spacecraft going around the Sun in an orbit smaller than Earth’s will soon overtake our planet. However, there is a loophole: if the spacecraft is placed directly between the Sun and Earth, Earth’s gravity pulls it in the opposite direction and cancels some of the Sun’s pull. With a weaker pull towards the Sun, the spacecraft needs less speed to maintain its orbit, so it can slow down.
I do actually get this, I think, but what about the other Lagrange points (named, by the way, after the Italian turned French astronomer, mathematician and physicist Joseph-Louis Lagrange (1736-1813)? And also by the way, Lagrangian mechanics is a whole nother field worth exploring, or not).
So, in my own words (sort of), every two-body gravitational system like that of Earth and Sun has five of these points. L1, L2 and L3 are called unstable points, while L4 and L5 are stable. So, those first three are gravitationally unstable, and they align with the Earth and the Sun. L1, being between Earth and Sun, offers an unimpeded view for solar observations. L2, being behind the Earth vis-a-vis the Sun (by about 1.5 million kilometres), is good for viewing deep space, in a direction away from the Sun-Earth-Moon system. So that’s where the Just Wonderful Space Telescope (JWST) is, and where the Nancy Grace Roman will be. Plenty of space apparently (haha), and other probes are stationed there too.
So, onto L3, L4 and L5, and I’m simplifying everything massively here of course. L3 is behind the Sun, from our perspective, and its orbit is just a bit wider than Earth’s. Any probe in that position can observe the far side of the Sun… which makes me wonder, does the Sun spin? Well, according to AI (never lies) it most certainly does:
Yes, the sun rotates on its own axis.However, because it is a giant ball of plasma and not a solid object like Earth, it spins at different speeds depending on the latitude and depth.
- The Equator: Spins the fastest, taking about 24.5 Earth days to complete one rotation.
- The Poles: Spin the slowest, taking over 34 days to complete one rotation.
- The Core: Rotates as a solid body much faster, turning about once a week.
Also, Lagrange points have much to do with the three body problem, which I may or may not get into. I can guess at least what the three bodies would be, re Lagrange points that is, for our region – the Sun, the Earth, and, say, JWST….
Anyway, I think I’ve meandered long enough. Maybe next time I’ll discombobulate myself with further details.
References
https://www.esa.int/Enabling_Support/Operations/What_are_Lagrange_points
https://www.space.com/does-the-sun-rotate
https://en.wikipedia.org/wiki/Lagrange_point#/media/File:Lagrange_points_simple.svg
algorithms, logarithms and biorhythms
In a recent conversation I was talking about the algorithms used by organisations like Facebook and Google to get people hooked into reading or listening to or otherwise attending to a whole host of stuff related to what they attended to yesterday. My interlocutor continued the conversation, but inadvertantly replaced the word ‘algorithm’ with ‘logarithm’ throughout, which I didn’t correct, perhaps not so much out of politeness but because, though I knew enough about the difference to use the right word, I would have difficulty in defining either term, in case he asked. As for biorhythms, I threw them in just for fun.
Mathematics is probably my weakest subject, and for that reason I’ve often been drawn to it in my dilettantish fashion, as to a kind of secret society I’ve been excluded from due to lack of the requisite. So I’ll start with my extremely limited and probably wrong working understanding of algos and logos (and bios), then provide a more informed account through a bit of research.
An algorithm, I think, is a largely computer-generated aggregate of a person’s data-gathering which provides some predictive power about their future data-gathering, or future interest, so that the data can be presented to them for gathering. Or something like that. A logarithm, I think, has something to do with exponentials. So that if something increases exponentially, it also increases logarithmically. Or logarithmickly? Or maybe a logarithm is some mathematical relation between a number and its exponent. Or something like that. A biorhythm is something to do with your supposed natural cycle of activity/inactivity throughout the day. It was new age stuff of the seventies, my early teen years. It seems to have died the death.
So now for the details and correctives.
An algorithm is indeed most associated with computing and data processing, though it may be used in a broader sense, mathematically or not. In fact it can be used to describe the process to get any result. In that sense a cookbook is a set of algorithms. But even in computing, the term is broader than my working understanding of it implied. I was thinking of particular algorithms, the ones used by certain web presences like Google, Facebook and Netflix to get you hooked into continuing to use them. To get you addicted, in a sense. So that’s all that needs to be said here about algorithms in general. In another post I’ll look at those particular algorithms that have worked so well for their creators (or the hirers of those creators) that they’ve become some of the richest folk in the world. That should be much more interesting.
A logarithm is also something like what I thought it was, but of course my understanding was very vague. It’s described as the inverse function to exponentiation – just as I sort of thought. For example, take the number 64, and think of it as 2 x 2 x 2 x 2 x 2 x 2, or 26 . The 6 here is the exponent (n), while the 2 is the base (b), so the general formulation is bn. The exponent (n) is also called the power, as in two to the sixth power, 26. Now, say that you want to know what power to raise the number two to get 256. That’s to say, 2n = 256, find n. This number that we want to find is the logarithm. So the question or equation can be restated thus:
log2(256) = n , find n
and the answer is n = 8. So the logarithm is the power that we have to raise a specified (base) number to, in order to get to another specified number. That’s all we need to know as a beginning. Obviously it gets more complicated with higher powers – but then, with higher powers, anything’s possible.
A biorhythm chart is a development of ideas concocted by one Wilhelm Fliess, a good mate of the only slightly less eccentric Sigmund Freud, back in the late 19th century. I well remember this fad gaining super-popularity in the seventies, with a paperback on the topic floating around our household. That and ‘speed reading’ were all the go. The basic hypothesis was that our lives – presumably in terms of energy, ‘clarity’ and competence – go in cycles, which can be charted as sinusoidal waves. The standard model argued for a 23-day physical cycle, a 28-day emotional cycle and a 33-day intellectual cycle. I’ve actually met someone who drew up these charts for herself back in the day. You live and learn. Needless to say, there’s no evidence whatever to back these ideas up, but they’re claimed often as ancient learning which has been eclipsed by the monstrous juggernaut of modern science, much like that other font of cyclical predictive wisdom, astrology…..
References
https://computer.howstuffworks.com/what-is-a-computer-algorithm.htm
Khan academy video: intro to logarithms
how to debate William Lane Craig, or not – part 4, on mathematics and gods
Now we come to the argument that God is the best explanation of the applicability of mathematics to the physical world. My intuitive response to this – and of course I’m not a mathematician – is that mathematics appears to me to be be a kind of abstraction from, that’s to say a manipulation of, a play on and further development of, the regularities that exist in the world, and that if no such regularities existed, the world wouldn’t exist. Or at least would not be in any sense describable. For example, the most basic form of regularity required would be a binary contrast, describable in mathematical or logical terms as x and not x. The real world, though , offers far more opportunities for playing on and manipulating regularities than this. So many opportunities have been found in fact, and so many beautiful theorems have been developed from them over the centuries that mathematics has often been given a mystical, miraculous status. One thinks of the Pythagoreans in ancient times, and the mathematically-obsessed philosophers of the seventeenth century, such as Descartes, Spinoza and Leibniz. However, I think it’s fair to say that, historically, when mathematics has been raised to mystical heights, great problems have ensued. So I don’t see anything particularly miraculous in the fact that a tool for understanding the regularities of the world can be developed and manipulated to underpin theories which further deepen or extend that understanding.
Eugene Wigner’s 1960 essay, ‘The unreasonable effectiveness of mathematics in the natural sciences’ is available online, and everyone should be encouraged to read it – though it doesn’t make for easy reading. I think it’s a little unfortunate that Wigner uses the word ‘miracle’ a number of times in the essay, but he certainly doesn’t refer at any time to a god. And while I would hesitate to interpret Wigner from my lay background, I’m not sure I agree with his view in the essay that, while elementary mathematical concepts derive directly from the perceived regularities of the actual world, more complex and abstract mathematical concepts don’t so derive, and yet can be applied with uncanny reliability, or if you like profitability, from our perspective, to that world, as is the case with much modern physics. If that were so, if the mathematical abstractions our minds create were completely removed from the world’s actual regularities, and yet just happened to apply to them to provide us with a richer and more developed view of our universe, then that would indeed be a ‘happy coincidence’. But abstraction doesn’t occur in a vacuum. Just as non-Euclidian geometry derives from the regularities of nature that Euclid strove to axiomise in a set of rules, and just as multi-dimensionality derives from the standard three-dimensional world of our experience, mathematical abstraction is always tied to some underlying actual regularity, however obscured by its overlay. The applicability of maths is not a happy coincidence (which isn’t to say all mathematical abstractions are applicable of course), but that is just because the world has regularity. Thus when we look at Dr Craig’s formal argument:
1. If God did not exist, the applicability of mathematics would be a happy coincidence.
2. The applicability of mathematics is not a happy coincidence.
3. Therefore God exists.
we see once again that the problem lies in the conditional statement – this time statement one. Our world has regularities, without which not. Mathematics is all about the play of regularities, so it isn’t coincidental that some mathematics has applicability. This is not mysterious, and it doesn’t imply anything about supernatural agency. Thus it isn’t reasonable to infer the existence of any god, let alone the human-obsessed, son-begetting god adhered to by Dr Craig.