1. Wrestling with numbers

 

Maths is notorious for inducing mental panic – and yet it is supposed to SIMPLIFY thinking. Suppose that there are n people in this room, to stop fights breaking out, let’s all shake hands. Let t be the time to shake hands, then if everyone shakes hands with everyone else, then each of n people must shake hands with n-1 others, so the total time, big T,  taken by this orgy of peace-making equals  n(n-1)t – or does it? Let’s run a test for a small value of n. Suppose n=2, then the formula yields 2t, but we know only one handshake is required. Aha, our formula has counted every handshake from the point of view of every person- every shake has been double-counted, so the correct formula is n(n-1)/2. Let’s tabulate this for various n, Café Philosophiques do attract a variable attendance -actually, rather more people than I have included below, with our normal attendance of between 20 and 30, we would be a long time shaking.

 

 

You notice I have added some other columns. A new-age three-hug requires three people, and work out the number of different three-hugs we have to divide by 3 times 2, the number of different orders in which the three people can be chosen in order to eliminate double-counting and get the right answer. When counting just the number of peoplem (collumn 2) I have put in a “divide by 1”, as there is no double counting. I have also added a column of constant unity to the left and finally also a row for zero people. These modifications are there to bring out the beautiful simplicity of this table. You can now see that you don’t need to bother with multiplying figures together. As you go to the right, the figure in each cell of the table can be found by simply adding the figure in the cell above to the figure in the cell above and to the left. This means that the figure in each cell is also the sum of the series of ALL the numbers in the column to the left up to and including the cell above and to the left. If you know some algebra, you recognise that the first column tabulates value of a constant, the second of a linear function, the second of a function including a square of n ( a quadratic function), the third a cubic function, and that we could go on adding as many columns to the right as we liked in order to tabulate values of the function  n(n-1)(n-2)……..(n-r+1)/ 1x2x3…xr, which is the number of r-hugs and is a polynomial involving n to the power r, where r can be as large as we like. We call 1x2x3x4…..xr, r factorial, or r! for short, as factorials crop up all over the place in combinatorial mathematics (which is what we have been talking about), probability theory and statistics. For me, this has been a classical and beautiful mathematical exercise, embarking on a potentially endless journey into more and more general results, with wider and wider implications, starting from just one simple idea – that of shaking hands.

 

Does this sort of thing worry you? Well, you are in good company. We can trace back mathematical thinking to, for instance, Babylonian problems still preserved on 4000-year-old inscribed and baked tablets. Already in those days, the master was setting examples that can still terrify us today!

I found a stone but did not weigh it. I weighed out six times its weight and added 2 gin, then added one third of one seventh of this weight multiplied by 24. The total weight was finally 1 man-na. What was the original weight of the stone?

Answer, 4  1/3 man-na .This works out correctly if 1 man-na equals 60 gin.

 

Incidentally, we see just how long ago and far away the basis of our conventional division of hours into 60 minutes and minutes into 60 seconds, both of time and of angle, was established.

 

An Egyptian problem, found on a Papyrus about 3600 years old, seems simpler:

 

If 10 hekat of fat is given out for a year, what is the amount used in a day?   ( c.f. if 104 black bin-bags are given out for a year, how many are used in a week?)

The answer however is not so simple, being expressed as:

1/64 hekat and 3 2/3 1/10 1/2190 ro. (I hekat = 320 ro)

 

Here we see that, with the exception of 2/3, and maybe also ¾, the Egyptian notation and probably the Egyptian mind could not deal with fractions other than “One share of however many”. A fraction like 56/73 (which the above addition of shares equals) was beyond writing down and probably beyond thinking about.

 

Problems with notation have taken much of the following three and a half thousand years to deal with. Modern fractions like 56 73^rds only gradually permeated Europe from Arabia and Italy during the Middle Ages. Present-day algebraic notation, x ‘s and y’s and all that, came into use gradually between the fifteenth and seventeenth centuries; British calculus is said to have been held up for more than a century by Newton’s now largely abandoned notation – patriotism inhibited the adoption of Leibniz’s more adaptable dy/dx expressions.  Maybe there are improvements still to be made. Our own personal struggles with algebra, and mathematics in general, can perhaps be excused when you consider how many centuries even those cleverest-of-the-clever leading mathematicians took to get their notation, and their corresponding thought processes, straightened out.

 

Note that the above-mentioned three and a half thousand years, say 3511 years, to be precise, could have been expressed as: three millennia plus five centuries plus one decade plus one year. It is revealing to quote a Derbyshire sale catalogue dating from 1920 for lands belonging to His Grace the Duke of Rutland: Here is a typical lot description (shortened):

Yeld Wood Farm, Woodlands & Cottage situate close to the Village of  Baslow…, containing an area of about 82 Acres 3 Roods and 22 Perches.

As you will know (?) there are forty perches to a rood and four roods to an acre.

A rood can be well visualised as the typical area of a mediaeval strip (or perhaps half a strip), one furlong long by one pole wide, or 220 yards by 5 ½ yards. A perch is simply a square rod (or pole), 5 ½  yards by 5 ½  yards.

So, you see how the prospective buyer can actually visualise the land area involved, whereas giving it as 82.87265, say 82.88, acres, or 82 and 71/80 acres, requires the farmer to visualise 0.88 of an acre; probably he’d rather have it in roods and perches. An ancient Egyptian might render the area for sale as 82 ½  ¼ 1/8 1/80 acres.

 

All this takes me back to primary school, I used to be able to add, subtract and even multiply in acres, roods and perches, and in miles, furlongs, chains, rods, yards, feet and inches. Measurement systems like this are meant to avoid the need to think in terms of fractions or decimals. Almost any quantity can be expressed as integral (whole number) multiples of units you have a habitual feel for. Same with old money: one pounds, seven shillings and six pence ha’penny. Nowadays, since decimalisation, we still have the various sized coins reflecting our  physical need to hand over change in “roods and perches”, to speak metaphorically, namely the 50p, the20p, the10p, 5p, 2p and 1p coins, but we no longer have common names for them and may get quite confused trying to convert, say, 83p into a convenient palmful of change.

 

Not so the mathematician. He, or I maybe, have surrendered the primitive need to visualise the sizes of the quantities we deal with in exchange for the extreme simplicity of the decimal system. Everything is in multiples or divisors by ten. 82.87265 acres may be hard to imagine, but it can be multiplied by a hundred by a simple double shift of the decimal point. Other multiplications, additions and subtractions take a bit longer than this but are completely straightforward. In contrast, imagine trying to work out how many 4 ounce bags of sweets can be made up from a day’s production of 2 tons, 7 hundredweights, 3 stones, 5 pounds and 12 ounces, which latter is expressed entirely in units I still have a real feel for. 2.372 tonnes = 2372 Kilograms =  23,720 100gramme bags is so much easier!

 

This handy decimal system goes back at least to the India of a couple of thousand years ago. For the next stage of the discussion, we need to simplify it a bit more.

10, 100, 1000, 10000, 100000, 1000000,….

is getting hard to write, let alone distinguish just how many 0’s there are. So generalising on 100 being 10 squared, written 10², we can write the sequence as:

10,10,²10³, … and so on, these being the sort of numbers that appear in successive columns of our handshake and hug table.

Similarly, the sequence of numbers less than 1:

0.1, 0.01, 0.001, 0.0001, 0.00001,…. 

Can be written 10‾1 (that is one over ten), 10ˉ² (one over tensquared), 10ˉ³, and so on.

(For mathematicians to deal with numbers less than 1 in these elegant decimal and power expressions took a lot longer in the adoption than dealing with the larger-than -one numbers. Finally, to join these sequences of powers together we are forced to adopt the convention that the number 1, unity itself,= 10° , even though multiplying a number by itself zero times seems meaningless)

 

 Now, at last we are in a position to start measuring the Universe. 

 

 

 

2. Getting the Measure of the Universe

In reaching out to see the great and the little, we start with a measure of our own size, 1 metre – a stride, an arm’s length, a child’s height. – this is our 10°.

Let’s start going up in scale in powers of ten, or, as practical scientists say “orders of magnitude”

Order 1 10metres; the width of the bookshop

Order2 100 metres; a sprint, how far a shout will carry, or a missile be thrown

Order 3 1 kilometre; a walk to the station, a waving friend visible

Order 4 10 Km; the distance to the nearest town, a long run

Order 5 100Km; a journey to the regional capital, to court, to prison, to the seaside. Now we are beginning to reach the edge of the pre-industrial ordinary person’s experience

Order 6 1000Km; travelling to the national capita city, to another country, or on a pilgrimage

An experience of only a few courtiers, merchants, churchmen, armies

Order 7 10,000Km; The voyage of Christopher Columbus, Marco Polo’s travels

Order 8 100,000Km; A girdle around the Earth.

We are now getting to the edge of the ordinary person’s experience, but not beyond the ingenious measurements of the Greek geometers, who obtained very good estimates for the size of the Earth.

Order 9 1 million Km;  The distance to the Moon; beyond the ordinary mortal, but again not beyond the ingenious Greeks, whose curiosity and method is humbling. They visualised the distance in “stades”, a foot-race distance of about 200 yards.

Order 11 100 million Km; The distance to the sun. Even this the Greeks tried to measure, and their estimate was “correct” to within an order of magnitude; ity was more accurately known by the 17^th. Century. A digression, what is a triumph in science may remain useless in practice!

Order 12 1 billion Km; The distance to Jupiter. Only the invention of the “Gallilean” telescope made this possible to conceive .Galileo first observed Jupiter’s 4 largest moons in 1610. They are easily seen with binoculars and, as early as 1676, small delays in their eclipsing by the planet were used to obtain a good stab at the speed of light, unimaginable at 186, 000 miles (Anglo-Saxon motorway units) per second.

Order 16 1 Light Year, or a third of a parsec. The nearest star is about 4 light-years away. Note the introduction of new units to try to help us visualise the immensities.  The distance to a star (not a planet) was first measured in 1838, using parallax, the slight change in direction from opposite sides of the Earth’s orbit.

Order 21  Diameter of the Milky Way, our galaxy (100,000 light years) Advances in the measurement and understanding of starlight spectrums and in so-called Cepheid variable stars made measurements like this possible by about 1915.

 

A digressional quote from the internet, how to bring the Universe down to size:

 There are an estimated 150 globular clusters that swarm around our galaxy.  Each of them contains 100,000 to 1,000,000 stars in a spherical region ONLY a few hundred light-years in diameter.

Order 22 1 million light years, is the approximate distance to the nearest other galaxy, The Great Nebula in Andromeda, M 31. Controversy about whether “galaxies”, those fuzzy objects, were gas clouds, perhaps forming stars, in the Milky Way, or other collections of stars at a great distance, was finally settled only in 1923 (the approximate birth date of Freeman Dyson)  with the aid of the 100 inch Mount Wilson telescope– individual stars could now be seen in the M31 nebula.

Order 25 In 1929, Edwin Hubble proposed his celebrated “expanding universe” theory. The dimmer the supernovae, the further away the galaxy and the bigger the red shift, explained by its receding from us. A billion light years is about the limit for observations of this type on galaxies.

Order 26 Ten billion light years. The distance to the edge of the observable universe is currently estimated as about 16 billion light years, giving a visible diameter of twice this.

So, the largest number we can come up with, relating the size of the observable Universe to a stretchy human pace-length is some 3 X 10²6

Though the Greeks imagination reached out to their estimate for the Sun’s distance, of order 10, it was not until the 17^th. Century that planetary distances became accepted, order 11 to 12, not until the 19^th. Century that the true remoteness of the fixed stars was revealed, order 16 and not until the lifetime of the parents of many at this meeting that the true scale of the observable Universe, orders 22 to 26, was understood and accepted.

Surely we must question whether any existential philosophy more than 200 years old can have more than inspirational or allegorical significance?

 

 

 

 

3. Smaller and ever-smaller

 Time now to turn from telescopy to microscopy and go down in scale to the smaller and smaller, starting again from our “zeroth” order of I metre.

Order Minus 1 10cms.  A “handy” size, the scale of a handspan, a fist, a stone, a sheet of writing paper, a jug of milk.

Order Minus 2 1 cm.  A finger’s-breadth, a flower, handwriting, an easily snapped  twig, a pebble

Order Minus 3 1 mm. Getting hard to see. Grit, a seed, a pin-head, your nails needing cutting

Order Minus 4 0.1 mm. About as small as can be seen or imagined to be see-able (!) by the naked eye. Small seeds, sand-grains, eye of a needle. From mediaeval times, magnifiable to a more comfortable scale by single-lens “reading glasses” ( as in Name of the Rose)

Order Minus 5 0.01mm. 10 micro-metres. \Silt or “soil” particles; they don’t float but do smear. Pollen grains – may blow about but can and need to settle. The cells of animal and plant tissues are often in this range; first described by Robert Hooke, c. 1670

Order Minus 6 1 micrometre. Dust. As we know, you can’t see it till it settles. In the 17^th. Century, the double-lens microscope allowing X20 to X200 magnification brought this scale into view.

Order Minus 7 0.1 micrometres or  100 nanometres. The wavelength of visible light is in the range 400-700 Nm. and this limits what could be distinguished using the best optical microscopes by late Victorian times.

Order Minus 9   1 nanometre, a billionth of a metre; about the diameter of a sugar molecule. The actual existence of “molecules” became accepted during the 19^th. Century, but the direct investigation of their structure only began c. 1920 with X-ray crystallography, X-rays having a wavelength comparable to molecular sizes.

Order Minus 10 1 Angstrom, a tenth of a nanometre, 100 picometres; typical effective size and separation of atoms.

Order Minus 12 1 picometre , a billionth of a millimetre. The wavelength of the electrons used in microscopy is about 5 picometres. This limits the electron microscope, developed in the 1930’s.

ORDER minus 13  This is where “High Energy Physics” takes over; larger and larger linear and circular accelerators:-, particularly since about 1960, CERN and the infant Large Hadron Collider

Order Minus 15 1 femtometre  roughly the radius (whatever that means) of a proton or electron. The existence of the electron was deduced about 1900, but protons and neutrons not until their tracks could be followed in cloud or bubble chambers from about 1930

Order Minus 18  1 attometre or nano-nanometre. About the feasible limit of High Energy Physics and correspondingly the scale of elementary forces and particles studied.

Order Minus 45  The Planck Length. An entirely hypothetical and especially hard to understand concept. May perhaps be thought of as the ultimate limit of the precision with which a particle’s position could be ascertained in the quantum theory. The energy of the probing particle/wave would be such that a black hole would be formed, so no measurement would result (!?!)

I have gone down to the very, to the 27^th. Power, silly Planck Length just because I want to give the Universe a chance to resist the power of the human mind!

So, the extension of man’s ability to look at very small things has gone from order 7 to order 18 in little more than 100 years. As with the very large, surely our outlook should have changed radically with such an expansion in our ability to observe the sub-sub-microscopic entities of which everyday objects and ourselves are made up. Certainly, we are filled with wonder by television programmes, articles and books, but most of even us “educated classes” experience little outside the everyday scale in our everyday lives. We are mostly scientifically ill-informed and even more inexperienced; we probably do not own or rarely use a microscope or a telescope, let alone an x-ray diffractometer or a linear accelerator! It is very easy, still, for us to live in an unquestioning mental world akain to that of the “ancients” in which only a few visionaries posed fundamental questions. How many of us have ever thought of estimating the moon’s distance by timing the length of a lunar eclipse, as did the Greek natural philosophers.

4. But how BIG is the Universe, actually?

"Space is big. Really big. You just won't believe how vastly hugely mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist, but that's just peanuts to space.”

Douglas Adams: The Hitch-hiker’s Guide to the Universe

 

 That is, how many conceivable points does it contain? EASY!

  4πR³, where R is the radius in Planck Lengths, 1.6πX10 to power71x3, let’s approximate a bit, after all my calculations may not be that precise, 10 to the power 324 is near enough.

5. God’s Numbering System

But the very hairs of your head are all numbered.  Matthew Ch. 10 Verse 30

 

God is supposed to have no problem numbering, that is describing and knowing, every point in the World, now known to be so much bigger than the Evangelist could have thought, and implied in the quotation.

The readers of the Bible are supposed to be awe-struck by this degree of omniscience. Not so Archimedes, who explains, in the 3^rd. century B.C, in a paper addressed to a King Gelon, that has numbering system is perfectly capable of dealing with the number of grains of sand that might be needed to fill the Universe. Our numbering system is also perfectly capable of dealing with the scale of things, as we have already seen.

Another way to look at this is to perform a card trick. Imagine I am holding three  perfectly normal  packs of 52 cards, excluding troublesome jokers, 156 cards which to simplify things can be considered all to be different, each pack having a different design on the back. I assure you that this pack has not be prepared or tampered with in any unfair way, but the cards are of course in one particular order. Now, watch carefully.

Dave fans, shuffles, fumbles and drops the whole pack on the floor.

Oh *******!!  How am I ever to sort them out again? Well, if I work through all the possible orders to find the right one – a 152-letter message, I will need not merely all the time in the world, but much more than that!

There are just two ways of ordering two cards, six of ordering three, twenty-four of ordering four, in short 1x2x3x4x……x156 of ordering them all – its that FACTORIAL again, 156! This number is rather large. I haven’t had time to calculate it, but I do know that 70! Is approximately 10 to the 100^th., so 100 factorial is going to be at least 10 to the 150^th  and somewhere around the three-pack mark the number of ways of ordering the cards is going to exceed 10 to the power 324, the number of “Planck points” in the visible universe. So there on the floor is all that God needs to number all the hairs on the head of all of space.

Now there is a very theoretical minimum conceivable time interval ca;lled the Planck Time, about 10 to the power of minus 43 seconds.( this is about one ten thousandth. of the time it takes light to cross the diameter of an atom) The present age of the Universe is a mere 10 to the 18^th. Seconds, or so, which comes to 10 to the 61 Planck Times. So, there is no way those playing cards could be got into even an “infinitesimal” fraction of all possible orders the right order in the lifetime of the Universe so far.

We can also see that God would not need mnay more playing cards to number not merely each point in the Universe, but each point at each instant in the life of the Universe, each point in space time, with plenty of room left over to describe what is going on at each point, whether empty, or associated with a particle, or with the scale and direction of each possible force field, and finish by giving this point in space-time a fanciful name, perhaps inspired by Peak Rock-climb or Lead-mine nomenclature! Don’t Sneeze Now Arete or Second-Cousin’s Fortune. There are even more names available that orderings of playing cards. Which leads us to:

 

6. Monkeys Typing Shakespeare

Shall I compare thee to a summer’s Day?

Thou art more beautiful and more temperate.

Rough winds do shake the darling buds of May

And summer’s lease hath all too short a date.

 

At this point I had intended to embark on some intricate calculations of the time it would take the proverbial “monkeys” to type even this one supreme sonnet of Shakespeare, but I think you have got my drift by now. Even were the monkeys able to employ, not typewriters, but the still proverbial “Quantum Computer”, they would have no chance of discovering even an early draft by the bard within the lifetime of the Universe. This is, I suppose, a commonplace observation, but it is less commonplace to ask;

 “How then DID the sonnets of Shakespeare ever GET written – starting from the blankish, even if anthropophilic slate of the early Universe”

Particularly as the Universe got off to such a laggardly start in the race to write Shakespeare’s works.. Nearly ten billion years were given over just to forming giant stars, letting them manufacture heavy elements and then blow themselves up, so that the scattered materials could condense into a second generation of stars and dust containing elements useful for forming iron and silicon planets. Another half-billion years at least were required for things to solidify a bit on planet Earth and for Jupiter to vacuum up most of the dangerous impacting meteors. Another mere hundred million years or two sufficed for life to appear, but more than three billion years were used up before it crept out of the sea. Another 300 million years were needed to evolve mammals and 298 million years to evolve the first self-consciously intelligent species. And during all this time, the monkeys can be imagined typing away, by now well in the lead – they’ve written several very beautiful lines of a risqué sonnet. Even the last two million years before the present, about a ten-thousandth of the lifetime of the Universe, have largely been employed in developing language from scratch, honing all our subtle passions, emotions and abstract intelligence and in developing the art of story-telling and aural tradition. Only in the last 50,000 or so years have written alphabets allowed remembered culture to develop and be passed on. Urban living, with all its special crafts, including those of playwright and poet,  seems to extend back no further than 10,000 years before the present, and this period is essentially that which has allowed literature, philosophy and science to be recorded so that the likes of Shakespeare and Newton could rejoice in “standing upon the shoulders of giants” to achieve there own dazzling in- and out-sights.

So we allowed the typing Monkeys a very long start indeed, but we still got there first! This shows the true scale of the wonder of human thought Douglas Adams once again got here first – his super-computer Deep Thought, faced with discovering the QUESTION to which 42 is the ANSWER to the riddle of Life, the Universe and Everything announces that it needs to design:

“A computer which can calculate the Question to the Ultimate Answer, a computer of such infinite and subtle complexity that organic life itself shall form part of its operational matrix…. And it shall be called..the Earth”

7. The Pen is mightier than the Interstellar Sword

 

So much for the scale of the Universe, we can hack it!  Actually, we can have a good stab at bringing the mind-boggling idea (one of our own, it must be allowed ) of INFINITY down to size. Actually, it turns out that there is not one “infinity”, but an infinity of infinities, of which the endless string of numbers, 1,2,3,4,5,6,7,8,9,…….is the littlest sister. These distinctions were clarified by Georg Ferdinand Ludwig Philipp Cantor in the late nineteenth century. Incidentally, it is obvious that his parents knew he was destined for great things!

Well, Masters of the Universe are we? Well, no. We can certainly think, but we cannot do much. The Earth in its present age is our Ivory Tower; we are pub-philosophers, mere impractical theoreticians when faced with the real world. We are trapped in a doomed pub, I mean planet, in an otherwise inhospitable solar system. And Doomsday, for technological civilisation at least, may be nearer than we would like. Don’t for a minute imagine we can terraform and colonise Mars or a Jovian moon – we can’t even stop the spread of the Sahara or the general de-terraforming of our own planet. We need to reach out to other stars in the Milky Way, as depicted in many a Science Fiction novel, or even worked out in multi-disciplinary precision by the British Interplanetary Society in its publication “Project Daedalus” (out of print and hard to find both on the net and somewhere in the roof of this bookshop).

Project Daedalus looked at a “fly-by” (i.e. non-decelerating) trip to Barnard’s Star, taking 50 years at an average speed of 15% of the speed of light. The starship would weigh only 46,000 tons – launchable by only 1000 Saturn Rockets using only a few million tons of fuel. BUT, its propulsion would be by fusion (after decades of experimentation STILL far from achievable in a controlled way) of 26,000 tons of Helium, mined from the atmosphere of Jupiter. And that is the LEAST starry-eyed of the several alternatives they pursued!  I have not checked the calculations (maybe I could), but if the energy needed for such a voyage were not to be excised from Jupiter, who would certainly hardly notice, it would require the siphoning off from domestic use of energy equivalent to more than 50 years-worth of the entire national consumption of the USA. The trouble is that Barnard’s Star and other possibly “hopeful” star systems are about 200 million times as far away as the Moon. If we were content, as we have been so far with Voyager 1 and 2, to let an interplanetary rocket just keep on going, it would take several million years to reach the target star.

Personally, I think this is well worth doing. The von Daniken assertion that Aliens have visited the Earth is just faintly possible if by “aliens” you mean an interstellar probe sent out some time in the last hundred or two million years by another advanced civilisation. Maybe the craft burnt up on entering our atmosphere – maybe you have seen it burn up on a clear night in August, but maybe some blistered remnant got through and is lying buried in the Carboniferous rocks of the Peak District! Or maybe we should send out a rocket containing thousands of lead and insulation-encased spores or viruses, or just some amino acids or RNA. The mother capsule would travel beyond the solar system and then be programmed to shatter explosively, sending its millions of separate seeds to scatter throughout the galaxy. There is all the time in the Universe for them to find a home. Maybe, as Fred Hoyle maintained with his Yorkshire fast-bowler pugnacity, life on earth is itself the progeny of such a seeding from space!

So we can’t get there. It takes too long, or it takes too much energy to get up to speeds comparable with the speed of light. But what about light itself, and the invisible spectrum of electromagnetic radiation of which it is part? That’s travelling at the speed of light already! If we shout loud enough, we can be heard at Barnard’s star in only ten year’s time. An answering “COOEEE” might come back within the lifetime of those of us gathered here tonight. The sending out of messages and the listening for messages that might be aimed at us is, of course known by the acronym SETI, the Search for Extra-terrestrial Intelligence”.

At a conference in 1961, Frank Drake set out his eponymous Drake Equation for the number of interstellar-communicating civilisations at present in our Galaxy.(The fact that the “present” is modified by the long times taken for messages to travel even at the speed of light need not bother us TOO much –though there is a danger that everyone will have gone to sleep before a reply comes through from the other side of the Milky Way). Let N be the number of such civilisations, then:

N = ( Rate of Star Formation )X (fraction with planets) X( number of habitable planets) X( probability of life emerging) X (probability of intelligence evolving) X (probability of electromagnetic technology developing)X(lifetime of electromagnetically-active civilisation)

The value, even the range of values, that all of these factors might take has remained controversial for the last 47 years. There is even major disagreement about the rate of star formation, a “simple” matter of statistics and physics. If there are 200 billion stars in the Milky Way, and the galaxy is 10 billion years old, hey presto, the average rate of formation is 20 per year. Most great minds, using much more sophisticated arguments, have still come out with estimates in the range 4 to 50 per year, but there is a school of thought that thinks the great pioneering days of the Milky Way are over, we are past the galactic menopause! Regarding the fraction with planets, there have been enormous strides since the detection of the first extra-solar planet was announced only in 1995, but discoveries since then have tended to confirm what had long been assumed, that most single (rather than double or multiple) stars will have planets in stable orbits. Habitable planets are, however, rather special beasts, and just how special the Earth is has become more and more evident. It seems we need to be not only the right distance from the sun, but big enough to hold an atmosphere, but not so big that the greenhouse effect spirals. Maybe the right constitution to produce vulcanism and plate-tectonics has been crucial.  Maybe we need the large moon to produce an intertidal zone. Almost certainly we have needed father Jupiter to clean up the solar system of large boulders the impact of which would otherwise produce too frequent and too comprehensive extinctions of multi-cellular lifeforms. Even given this perfect planet, is it inevitable that life will spontaneously appear, or was it an ultimately improbable one-off event? The principle of mediocrity supports the common appearance of life, but can in the absence of further evidence be trumped by the anthropic principle- that the Earth, the Universe and Laws of Nature might be tailor-made to suit our navel-contemplating development , so that our being here says nothing about the likelihood of comparable beings elsewhere – we’re here because we’re here because we’re here, you might say. As for the development of intelligence, speculation goes wild – suppose the dinosaurs had not bee extinguished; suppose mammals had not survived.

The most depressing factor in the Drake Equation however is the lifetime of a technologically advanced civilisation. Our look at how the Universe wrote the works of Shakespeare without employing monkeys makes it seem that we are living in a period of exponentially increasing “progress”, comparable changes occurring in a billion, then a million, then a thousand, then a hundred, then twenty years. Are we approaching an asymptote a very few years hence, when all will revert catastrophically to a much earlier phase – or will the pace of change slow as we arrive on a high plateau of wellbeing? History and current anxieties do not encourage optimism. Imagine that there is a “High Plateau” civilisation somewhere in our galaxy. A million years ago their telescopes identified a promising planet in orbit around Sol; there was even the spectroscopic signature of oxygen in its atmosphere, an almost certain sign of organic life. Straightaway, they trained their radio telescopes on the Earth and initiated both a listening watch and transmission of a beacon signal. For thousands of years, that is for thousands of generations (c.f. Douglas Adams’ priests watching over the computer Deep Thought – he got there first AGAIN), this stable civilisation persevered in watching and calling Earth. If we examine our own culture, you can see how unlikely even this possible scenario is. Finally, they gave up hope and switched their gaze elsewhere – in our year 1933, perhaps, just before we discovered cosmic radio interference. Maybe they will look again in another few thousand year’s time – but will our technological civilisation still be here?

So, even if there can be life out there among the stars, we may have only a thousand billionths of a chance of “catching it in when we knock, especially if it operates a timeshare.

 

 

8.       Some Conclusions – Douglas Adams wuz here.

We have conducted a lightning tour through the historic development of mathematics and how its notations have helped us to tame the immense scale of the Universe, a scale that has only become apparent in the last four centuries or so. In my humble opinion, mankind will never reach a limit to our ability to think about the laws by which this Universe operates, the flexibility of even our so-called “Hunter-gatherer” minds far, far exceeds the demands that physics and cosmology place upon us.

There is a beautiful and dangerous snag, though. I have left out of all this the study of life itself – of ourselves. Here, our very flexibility and complexity means that our studies of ourselves lag ever further and further behind our individual and cultural creations. Our abstract abilities can remain abstract in the ever expanding frontiers of pure mathematics, or abstraction can become realised as Music, as Art, as Literature, as Financial Instruments, as Warfare – as Philosophy. The scope of our ability to outrun our own understanding is another indication of how the physical universe is orders of magnitude simpler than the creations of our own individual and social selves. Is it surprising that, during my lifetime we seem to have turned in on ourselves, becoming obsessed not with exploring the world “out there” but with understanding and trying to control the social and environmental implications of our own species.

Nevertheless, in our big Ivory Tower the earth, and this little ivory tower, The Café Philosophique, we can have so much mental fun. There is no limit to what we might think, and every one of us, and every thought we have is unique and almost infinitely improbable! Douglas Adams always gets there first, we have indeed invented

The Infinite Improbability Drive!