Algebra - the language of mathematics

I know I’m doing things back-to-front – arse-about - as we say in Oz (and possibly elsewhere) but, considering all the esoteric mathematics I produce on this blog, I thought I should try and explain some basics.

As I mentioned earlier this year in a post on ‘analogy’, mathematics is a cumulative endeavour and you can’t understand calculus, for example, if you don’t know algebra. I’ve come across more than a few highly intelligent people, of both sexes, who struggle with maths (or math as Americans call it) and the sight of an equation stops them in their tracks.

Mathematics is one of those topics where the gap, between what you are expected to know and what you actually learn, can grow as you progress through school, mainly because you were stumped by algebra. You know: the day you were suddenly faced with numbers being replaced by letters; and things like counting, adding, subtracting, dividing, multiplying, fractions and even decimals suddenly seemed irrelevant. In other words, everything you’d learned about mathematics, which was firmly grounded in numbers – something you’d learned almost as soon as you could talk – suddenly seemed useless. Even Carl Jung, according to his autobiography, stopped understanding maths the day he had to deal with ‘x’. In fact, his wife, Emma, had a better understanding of physics than Jung did.

But for those who jump this hurdle, seemingly effortlessly, ‘x’ is a liberator in the same way that the imaginary number i is perceived by those who appreciate its multi-purposefulness. In both cases, we can do a lot more than we could before, and that is why algebra is a stepping-stone to higher mathematics.

Fundamentally, mathematics is not so much about numbers as the relationship between numbers, and algebra allows us to see the relationships without the numbers, and that’s the conceptual hurdle one has to overcome.

I’ll give a very simple example that everyone should know: Pythagoras’s triangle.

I don’t even have to draw it, I only have to state it: a² + b² = c²; and you should know what I’m talking about. But a picture is worth innumerable words.

The point is that we can use actual integers, called Pythagorean triples, that obey this relationship; the smallest being 5² = 4² + 3². Do the math as you Americans like to say.

But the truth is that this relationship applies to all Pythagorean triangles, irrespective of their size, length of sides and units of measurement. The only criteria being that the triangle is ‘flat’, or Euclidean (is not on a curved surface) and contains one right angle (90^o).

By using letters, we have stated a mathematical truth, a universal law that applies right across the universe. Pythagoras’s triangle was discovered well before Pythagoras (circa 500BC) by the Egyptians, Babylonians and the Chinese, and possibly other cultures as well.

Most of the mathematics, that I do, involves the manipulation of algebraic equations, including a lot of the stuff I describe on this blog. If you know how to manipulate equations, you can do a lot of mathematics, but if you don’t, you can’t do any.

A lot of people are taught BIDMAS, which gives the priority of working out an equation: Brackets, Indices, Division, Multiplication, Addition and Subtraction. To be honest, I’ve never come across a mathematician who uses it.

On the other hand, a lot of maths books talk about the commutative law, the associative law and the distributive law as the fundaments of algebra.

There is a commutative law for addition and a commutative law for multiplication, which are both simple and basic.

A + B = B + A  and  A x B = B x A (that’s it)

Obviously there is no commutative law for subtraction or division.

A – B ≠ B – A  and  A/B ≠ B/A (pretty obvious)

There are some areas of mathematics where this rule doesn’t apply, like matrices, but we won’t go there.

The associative law also applies to addition and multiplication.

So A + (B + C) = (A + B) + C  and  A x (B x C) = (A x B) x C

It effectively says that it doesn’t matter what order you perform these operations you’ll get the same result, and, obviously, you can extend this to any length of numbers, because any addition or multiplication creates a new number that can then be added or multiplied to any other number or string of numbers.

But the most important rule to understand is the distributive law because it combines addition and multiplication and can be extended to include subtraction and division (if you know what you're doing). The distributive law lies at the heart of algebra.

A(B + C) = AB + AC  and  A(B + C) ≠ AB + C (where AB = A x B)

And this is where brackets come in under BIDMAS. In other words, if you do what’s in the brackets first you’ll be okay. But you can also eliminate the brackets and get the same answer if you follow the distributive rule.

But we can extend this: 1/A(B - C) = B/A - C/A (where B/A = B ÷ A)

And  -A(B – C) = CA – BA  because (-1)² = 1, so a minus times a minus equals a plus.

If 1/A(B + C) = B/A + C/A then (B + C)/A = B/A + C/A

And  A/C + B/D = (DA + BC)/DC

To appreciate this do the converse:

(DA + BC)/DC = DA/DC + BC/DC = A/C + B/D

But the most important technique one can learn is how to change the subject of an equation. If we go back to Pythagoras’s equation:

a² + b² = c²  what’s b = ?

The very simple rule is that whatever you do to one side of an equation you must do to the other side. So if you take something away from one side you must take it away from the other side and if you multiply or divide one side by something you must do the same on the other side.

So, given the above example, the first thing we want to do is isolate b². Which means we take a² from the LHS and also the RHS (left hand side and right hand side).

So b² = c² – a²

And to get b from b² we take the square root of b², which means we take the square root of the RHS.

So b = √(c² – a²)

Note b ≠ c – a  because √(c² – a²) ≠ √c² - √a²

In the same way that (a + b)² ≠ a² + b²

In fact (a + b)² = (a + b)(a + b)

And applying the distributive law: (a + b)(a + b) = a(a + b) + b(a + b)

Which expands to  a² + ab + ba + b² = a² + 2ab + b²

But (a + b)(a – b) = a² – b²  (work it out for yourself)

An equation by definition (and by name) means that something equals something. To maintain the equality whatever you do on one side must be done on the other side, and that’s basically the most important rule of all. So if you take the square root or a logarithm or whatever of a single quantity on one side you must take the square root or logarithm or whatever of everything on the other side. Which means you put brackets around everything first and apply the distributive law if possible, and, if not, leave it in brackets like I did with the example of Pythagoras’s equation.

Final Example:  A/B = C + D    What’s B = ?

Invert both sides:  B/A = 1/(C + D)

Multiply both sides by A:   B = A/(C + D)   (Easy)

Note: A/(C + D) ≠ A/C + A/D

Time again to talk about time

Last week’s New Scientist’s cover declared SPACE versus TIME; one has to go. But which? (15 June 2013). This served as a rhetorical introduction to physics' most famous conundrum: the irreconcilability of its 2 most successful theories - quantum mechanics and Einstein’s theory of general relativity - both conceived at the dawn of the so-called golden age of physics in the early 20^th Century.

The feature article (pp. 35-7) cites a number of theoretical physicists including Joe Polchinski (University of California, Santa Barbara), Sean Carroll (California Institute of Technology, Pasadena), Nathan Seiberg (Institute for Advanced Study, Princeton), Abhay Ashtekar (Pennsylvania University), Juan Malcadena (no institute cited) and Steve Giddings (also University of California).

Most scientists and science commentators seem to be banking on String Theory to resolve the problem, though both its proponents and critics acknowledge there’s no evidence to separate it from alternative theories like loop quantum gravity (LQG), plus it predicts 10 spatial dimensions and 10⁵⁰⁰ universes. However, physicists are used to theories not gelling with common sense and it’s possible that both the extra dimensions and the multiverse could exist without us knowing about them.

Personally, I was intrigued by Ashtekar’s collaboration with Lee Smolin (a strong proponent of LQG) and Carlo Rovelli where ‘Chunks of space [at the Planck scale] appear first in the theory, while time pops up only later…’ In a much earlier publication of New Scientist on ‘Time’ Rovelli is quoted as claiming that time disappears mathematically: “For me, the solution to the problem is that at the fundamental level of nature, there is no time at all.” Which I discussed in a post on this very subject in Oct. 2011.

In a more recent post (May 2013) I quoted Paul Davies from The Goldilocks Enigma: ‘[The] vanishing of time for the entire universe becomes very explicit in quantum cosmology, where the time variable simply drops out of the quantum description.’ And in the very article I’m discussing now, the author, Anil Ananthaswamy, explains how the wave function of Schrodinger’s equation, whilst it evolves in time, ‘…time is itself not part of the Hilbert space where everything else physical sits, but somehow exists outside of it.’ (Hilbert space is the ‘abstract’ space that Schrodinger’s wave function inhabits.) ‘When we measure the evolution of a quantum state, it is to the beat of an external timepiece of unknown provenance.’

Back in May 2011, I wrote my most popular post ever: an exposition on Schrodinger’s equation, where I deconstructed the famous time dependent equation with a bit of sleight-of-hand. The sleight-of-hand was to introduce the quantum expression for momentum (p_x = -i h d/dx) without explaining where it came from (the truth is I didn’t know at the time). However, I recently found a YouTube video that remedies that, because the anonymous author of the video derives Schrodinger’s equation in 2 stages with the time independent version first (effectively the RHS of the time dependent equation). The fundamental difference is that he derives the expression for p_x = i h d/dx, which I now demonstrate below.

Basically the wave function, which exploits Euler’s famous equation, using complex algebra (imaginary numbers) is expressed thus:  Ψ = Ae ^i(kx−ωt)

If one differentiates this equation wrt x we get ik(Ae ^i(kx−ωt)), which is ikΨ. If we differentiate it again we get d²Ψ/dx² = (ik)²Ψ.

Now k is related to wavelength (λ) by 2π such that k = 2π/λ.

And from Planck’s equation (E = hf) and the fact that (for light) c = f λ we can get a relationship between momentum (p) and λ. If p = mc and E = mc², then p = E/c. Therefore p = hf/f λ which gives p = h/λ effectively the momentum version of Planck’s equation. Note that p is related to wavelength (space) and E is related to frequency (time).

This then is the quantum equation for momentum based on h (Planck’s constant) and λ. And, of course, according to Louis de Broglie, particles as well as light can have wavelengths.

And if we substitute 2π/k for λ we get p = hk/2π which can be reformulated as

k = p/h where h = h/2π.

And substituting this in (ik)² we get –(p/h)²  { i² = -1}

So Ψ d²/dx² = -(p_x/h)²Ψ

Making p the subject of the equation we get p_x² = - h² d²/dx² (Ψ cancels out on both sides) and I used this expression in my previous post on this topic.

And if I take the square root of p_x² I get p_x = i h d/dx, the quantum term for momentum.

So the quantum version of momentum is a consequence of Schrodinger’s equation and not an input as I previously implied. Note that √-1 can be i or –i so p_x can be negative or positive. It makes no difference when it’s used in Schrodinger’s equation because we use p_x².

If you didn’t follow that, don’t worry, I’m just correcting something I wrote a couple of years ago that’s always bothered me. It’s probably easier to follow on the video where I found the solution.

But the relevance to this discussion is that this is probably the way Schrodinger derived it. In other words, he derived the term for momentum first (RHS), then the time dependent factor (LHS), which is the version we always see and is the one inscribed on his grave’s headstone.

This has been a lengthy and esoteric detour but it highlights the complementary roles of space and time (implicit in a wave function) that we find in quantum mechanics.

Going back to the New Scientist article, the author also provides arguments from theorists that support the idea that time is more fundamental than space and others who believe that neither is more fundamental than the other.

But reading the article, I couldn’t help but think that gravity plays a pivotal role regarding time and we already know that time is affected by gravity. The article keeps returning to black holes because that’s where the 2 theories (quantum mechanics and general relativity) collide. From the outside, at the event horizon, time becomes frozen but from the inside time would become infinite (everything would happen at once) (refer Addendum below). Few people seem to consider the possibility that going from quantum mechanics to classical physics is like a phase change in the same way that we have phase changes from ice to water. And in that phase change time itself may be altered.
 

Referring to one of the quotes I cited earlier, it occurs to me that the ‘external timepiece of unknown provenance’ could be a direct consequence of gravity, which determines the rate of time for all objects in free fall.

Addendum: Many accounts of the event horizon, including descriptions in a recent special issue of Scientific American; Extreme Physics (Summer 2013), claim that one can cross an event horizon without even knowing it. However, if time is stopped for 'you' according to observers outside the event horizon, then their time must surely appear infinite to ‘you’, to be consistent. Kiwi, Roy Kerr, who solved Einstein's field equations for a rotating black hole (the most likely scenario), claims that there are 2 event horizons, and after crossing the first one, time becomes space-like and space becomes time-like. This infers, to me, that time becomes static and infinite and space becomes dynamic. Of course, no one really knows, and no one is ever going to cross an event horizon and come back to tell us.

Judi Moylan – a very rare and endangered species of politician

Judi Moylan is that very rare entity: a politician who puts principles before ego and ambition. It’s worth listening to the short audio imbedded in this link.

To me, this is very sad, because Moylan is too empathetic and not ruthless enough to make it to the front bench – she is one of the last of her kind in her party – yet Federal politics needs more people like her and less like our leaders and leaders-in-waiting.

No one in a position of power or influence in Australian politics has the guts to stand up to the paranoid element in our society. In fact, they do the exact opposite, knowing that by pandering to xenophobia and insecurity they can win the next election. Australian electioneering is governed by the politics of fear, when we have the most buoyant economy in the Western world. What does that say about us as a people?

Why there should be more science in politics

This programme aired on ABC's Catalyst last Thursday illustrates this very well. Not only are scientists best equipped to see the future on global terms, they are best equipped to find solutions. I think there is a complacency amongst both politicians and the public-at-large that science will automatically rescue us from the problems inherent in our global species' domination. But it seems to me that our economic policies and our scientific future-seeing are at odds. Infinite economic growth dependent on infinite population growth is not sustainable. As the programme intimates, the 21st Century will be a crunch point, and whilst everyone just assumes that science and technology will see us through, it's only the scientists who actually acknowledge the problem.

Addendum: One of the interesting points that is raised in this programme is the fact that we could feed the world now - it's a case of redistribution and waste management, not production. No clearer example exists where our economic paradigms are in conflict with our global needs. The wealth gap simply forbids it.

Sequel to ELVENE

People who have read ELVENE invariably ask: where’s the next one? Considering Elvene was first published in 2006, it’s been a long time coming. Firstly, I was aware that I couldn’t possibly live up to expectations – sequels rarely do – and I also knew that I would probably never write a book as good as Elvene again.

There are many tensions inherent in storytelling but none are more challenging than the contradictory goals of realising readers’ expectations and providing surprises. Both are necessary for a satisfactory rendition of a story and often have to be achieved simultaneously.

So the sequel to ELVENE both opens and closes with surprises, yet the journey’s end is rarely in doubt. I was often tempted to abandon this exercise and let people imagine their own outcome from the previous novel. That would have been the safe thing to do. But as I progressed, especially in the second half, I was motivated by the opposite desire: to write a sequel so that no one else would write it.

For most writers, the feeling is that the story already exists, like the statue trapped in the marble, and, as the writer, I’m simply the first person to read it. For much of the exercise I wrote it as a serial to myself, not knowing what was going to happen next. This is an approach many writers take – it provides the spontaneity that makes our art come alive – even if I already knew how it was going to end (actually I didn't).

Footnote: I should point out that you don't need to have read ELVENE to read the sequel - it works as a standalone story. All the backstory you need is incorporated into the opening scenes.

Is the universe a computer?

In New Scientist (9 February 2013, pp.30-31) Ken Wharton presented an abridged version of his essay, The universe is not a computer, which won him third prize in the 2012 Foundational Questions Institute essay contest. Wharton is a quantum physicist at San Jose University, California. I found it an interesting and well-written article that not only put this question into an historical perspective, but addressed a fundamental metaphysical issue that’s relevant to the way we do science and view the universe itself. It also made me revisit Paul Davies’ The Goldilocks Enigma, because he addresses the same issue and more.

Firstly, Wharton argues that Newton changed fundamentally the way we do science when he used his newly discovered (invented) differential calculus (which he called fluxions) to describe the orbits of the planets in the solar system, and simultaneously confirmed, via mathematics, that the gravity that keeps our feet on the ground is the very same phenomenon that keeps the Earth in orbit around the sun. This of itself doesn’t mean the universe is a computer, but Wharton argues that Newton’s use of mathematics to uncover a natural law of the universe created a precedent in the way we do physics and subliminally the way we perceive the universe.

Wharton refers to a ‘Newtonian schema’ that tacitly supports the idea that because we predict future natural phenomena via calculation, perhaps the universe itself behaves in a similar manner. To quote: ‘But even though we’ve moved well beyond Newtonian physics, we haven’t moved beyond the new Newtonian schema. The universe, we almost can’t help but imagine, is some cosmic computer that generates the future from the past via some master “software” (the laws of physics) and some initial input (the big bang).’

Wharton is quick to point out that this is not the same thing as believing that the universe is a computer simulation – they are entirely different issues – Paul Davies and David Deutsch make the same point in their respective books (I reviewed Deutsch’s book, The Fabric of Reality, in September 2012, and Davies I discuss below). In fact, Deutsch argues that the universe is a ‘cosmic computer’ and Davies argues that it isn’t, but I’m getting ahead of myself.

Wharton’s point is that this belief is a tacit assumption underlying all of physics: ‘…where our cosmic computer assumption is so deeply ingrained that we don’t even realise we are making it.’

A significant part of Wharton’s article entails an exposition on the “Lagragian”, which has dominated physics in the last century, though it was first formulated by Joseph Louis Lagrange in 1788 and foreseen, in essence, by Pierre de Fermat (in the previous century) when he proposed the ‘least time’ principle for refracted light. A ray of light will always take the path of least time when it goes between mediums – like air and water or air and glass. James Gleick, in his biography of Richard Feynman, GENIUS, gives the example of a lifesaver having to run at an angle along a beach and then swim through surf to reach a swimmer in trouble. The point is that there is a path of ‘least time’ for the lifesaver, amongst an infinite number of paths he could take. The 2 extremes are that he could run perpendicularly into the surf and swim diagonally to the swimmer or he could run diagonally to the surf at the point opposite the swimmer and swim perpendicularly to him or her. Somewhere in between these 2 extremes there is an optimum path that would take least time (Wharton uses the same analogy in his article). In the case of light, travelling obliquely through 2 different mediums at different speeds, the light automatically takes the path of ‘least time’. This was ‘de Fermat’s principle’ even though he couldn’t prove it at the time he formulated it.

Richard Feynman, in particular, used this principle of ‘least action’, as it’s called, to formulate his integral path method of quantum mechanics. In fact, as Brian Cox and Jeff Fershaw point out in The Quantum Universe (reviewed December, 2011) Planck’s constant, h, is expressed in units of ‘least action’, and Feynman famously derived Schrodinger’s equation from a paper that Paul Dirac wrote on ‘The Lagrangian in Quantum Mechanics’. Feynman also described the significance of the principle, as applied to gravity, in Six-Not-So-Easy Pieces - in effect, it dictates the path of a body in a gravitational field. In a nutshell, the ‘least action’ is the difference between the kinetic and potential energy of the body. Nature contrives that it will always be a minimum, hence the description, ‘principle of least action’.

A bit of a detour, but it seems to be a universal principle that appears in every area of physics. It’s relevance to Wharton’s thesis is that ‘…physicists tend to view it as a mathematical trick rather than an alternative framework for how the universe might really work.’

However, Wharton argues that the mathematics of a ‘Lagrangian-friendly formulation of quantum theory [proposed by him] could be taken literally’. So Wharton is not eschewing mathematics or natural laws in mathematical guise (which is what a Lagrangian really is); he’s contending that the Newtonian schema no longer applies to quantum mechanics because of its inherent uncertainty and the need for a ‘…”collapse”, when all the built-up uncertainty suddenly emerges into reality.’

David Deutsch, for those who are familiar with his ideas, overcomes this obstacle by contending that we live in a quantum multiverse, so there is no ‘collapse’, just a number of realities, all consequences of the multiverse behaving like a cosmic quantum computer. I’ve discussed this and my particular contentions with it in another post.

Paul Davies discusses these same issues in the context of the universe’s evolution and all the diverse philosophical views that such a discussion encompasses. Davies devotes many pages of print to this topic and to present it in a few paragraphs is a travesty, but that’s what I’m going to do. In particular, Davies equates mathematical Platonism with Wharton’s Newtonian schema, though he doesn’t specifically reference Newton. He provides a compelling argument that a finite universe can’t possibly do calculus-type calculations requiring infinite elements of information. And that’s the real schema (or paradigm) that modern physics seems to embrace: that everything in the universe from quantum phenomena to thermodynamics to DNA can be understood in terms of information; in ‘bits’, which makes the computer analogy not only relevant but impossible to ignore. Personally, I think the computer analogy is apposite only because we live in the ‘computer age’. It’s not only the universe that is seen as a computer, but also the human brain (and other species, no doubt). The question I always ask is: where is the software? But that’s another topic.

DNA, to all intents and purposes, is a form of natural software where the code is expressed in amino acids and the hardware are proteins that are constructed and manipulated on a daily basis. DNA is a set of instructions to build a functioning biological organism – it’s as teleological as nature gets. A large part of Davies’ discussion entails teleology and its effective expulsion from science after Darwin, but the construction of every living organism on the planet is teleological even though its evolution is not. Another detour, though not an irrelevant one.

Davies argues that he’s not a Platonist, whilst acknowledging that most physicists conduct science in the Platonist tradition, even if they don’t admit it. Specifically, Davies challenges the Platonist precept that the laws of nature exist independently of the universe. Instead, he supports John Wheeler’s philosophy that ‘the laws of the universe emerged… “higgledy-piggledy”… and gradually congealed over time.’ I disagree with Davies, fundamentally on this point, not because the laws of the universe couldn’t have evolved over time, but because there is simply more mathematics than the universe needs to exist.

Davies also discusses at length the anthropic principle, both the weak and strong versions, and calls Deutsch’s version the ‘final anthropic principle’. Davies acknowledges that the strong version is contrary to the scientific precept that the universe is not teleological, yet, like me, points out the nihilistic conclusion (my term, not his) of a universe without consciousness. Davies overcomes this by embracing Wheeler’s philosophical idea that we are part of a cosmological quantum loop – an intriguing but not physically impossible concept. In fact, Davies’ book is as much a homage to Wheeler as it is an expression of his own philosophy.

My own view is much closer to RogerPenrose’s that there are 3 worlds: the mental, the Platonic and the physical; and that they can be understood in a paradoxical cyclic loop. By Platonic, he means mathematical, which exists independently of humanity and the universe, yet we only comprehend as a product of the human mind, which is a product of the physical universe, which arose from a set of mathematical laws – hence the loop. In my view this doesn’t make the universe a computer. I agree with Wharton on this point, but I see quantum mechanics as a substrate of the physical universe that existed before the universe as we know it evolved. This is consistent with the Hartle-Hawking cosmological view that the universe had no beginning in time as well as being consistent with Davies’ exposition that the ‘…vanishing of time for the entire universe becomes very explicit in quantum cosmology, where the time variable simply drops out of the quantum description.’

I’ve discussed this cosmological viewpoint before, but if the quantum substrate exists outside of time, then Wheeler’s and Davies’ version of the anthropic principle suddenly becomes more tenable.

Addendum: I wrote another post on this in 2018, which I feel is a stronger argument, and, in particular, includes the role of chaos.