Sunday, 23 September 2012

Induction Exhumed

(Originally published 23 March, 2011, as a Note on my Facebook.)
Will the sun rise tomorrow?  Will electricity still work?  Will the next piece of bread I eat be any good as food?  Are there such things as laws of physics, or is the universe just winging it?  Since 1758 the answer to these philosophical questions has been "We can never know" -- as explained by David Hume, in what has come to be known as the Problem of Induction. 
If a body of like colour and consistence with that bread, which we have formerly eat, be presented to us, we make no scruple of repeating the experiment, and foresee, with certainty, like nourishment and support.  Now this is a process of the mind or thought, of which I would willingly know the foundation...  As to past Experience, it can be allowed to give direct and certain information of those precise objects only, and that precise period of time, which fell under its cognizance: but why this experience should be extended to future times, and to other objects, which for aught we know, may be only in appearance similar; this is the main question on which I would insist.  The bread, which I formerly eat, nourished me; that is, a body of such sensible qualities was, at that time, endued with such secret powers: but does it follow, that other bread must also nourish me at another time, and that like sensible qualities must always be attended with like secret powers?  The consequence seems nowise necessary. 
Formally, we can express inductive reasoning as follows:
  • Bread nourished me on Monday. 
  • Bread nourished me on Tuesday. 
  • Bread nourished me on Wednesday. 
  • Bread nourished me on Thursday. 
  • Bread nourished me on Friday. 
  • Therefore, bread will nourish me tomorrow. 
The conclusion is not a necessary consequence of the premises.  Hence, inductive reasoning is invalid.
Since not only all of science, but every move we make in our daily lives, is based on inductive reasoning, this is a bit of a worry.  Naturally, a number of people have proposed solutions, or at least workarounds.  To date, none have been entirely satisfactory.  The best is a position called "probabilism", which basically replaces the conclusion to the above with
  • Therefore, bread will probably nourish me tomorrow.
...which is not too bad, considering.  We have a very large historical statistical sample of incidences of people eating bread, and it correlates very strongly with nutrition; so that gives us pretty good confidence in there being a connection between the two.
Provided, that is, that we know there is some kind of underlying order which we just have to find.  If there isn't, then the probabilist is in the position of the superstitious gambler who has yet to figure out that "winning streaks" end randomly and without warning; that no matter how many times in a row a coin comes up heads, the chance of its coming up heads next time remains precisely 0.5.
I don't want to make grandiose claims... but I think I have solved this 253-year-old problem.  Not all on my lonesome, obviously; shoulders of giants, and all that.  There are three critical mental tools, all taken from much greater thinkers than myself, without which I couldn't have come near it.
 The first is the Anthropic Principle, which I first met in Stephen Hawking's A Brief History of Time.  This was first developed by astronomers seeking to explain why the Earth provides such suspiciously fine-tuned conditions for the existence of life.  Their answer was: for no reason -- it just so happened that one of the many millions of planets in the universe happened to form in the right conditions.  Necessarily, it was on that one fortunate planet that life then proceeded to evolve.
Hawking expands the principle to explain why our universe just happened to have the right physical constants for life to exist in it somewhere.  There may be thousands of other universes where the conditions weren't right... and those universes will, necessarily, have no-one observing them.  No matter how rare the conditions that allow conscious life may be, it is only in those conditions that conscious life will appear, and therefore conscious beings will always find their conditions just right.

The second tool is the Buddhist principle of sūnyatā, or "emptiness".  Sūnyatā is the recognition that there is no such thing as "essence", no point where the metaphysical buck stops.  Everything is interrelated and interdependent.  The world does not come in discrete, identifiable chunks with fixed natures, no matter how much it might seem so to us.
The opposite view was first formulated by Plato.  Two hats (let's say) are both the same thing, "hats", by virtue of participating in the eidos or "form" of hats.  That is to say, there is an idea or concept of The Hat, and the objective world's seen and felt hats are mere shadows or projections of this concept.  Plato, in other words, believed that reality is object-oriented; that the things we handle and use are mere instances of object classes declared in the World of Forms.
(Ironically, the Mahayana Buddhists later arrived independently at a similar conclusion, interpretingsūnyatā to mean that the essences or fixed natures of things were on a higher plane instead of being found in the world.)
The virtual workspace projected from neural stimuli that we call "consciousness" seems to work in an approximately object-oriented fashion, which I suppose is why philosophies like Plato's feel plausible.  It does reflect, well enough to be useful, how solid objects and living things behave on a human timescale.  But it fumbles even on some everyday things, like clouds; and there's no reason to suppose it's an insight into fundamental reality.
"Essences", "forms", "inner natures" -- whatever you call them, they are sometimes invoked in an attempt to solve the Problem of Induction.  You can be sure the next piece of bread you eat will nourish you, because, if it doesn't, it isn't truly bread but something else.  Alan Musgrave demonstrates the falsity of this reasoning:
We can now be sure that the next piece of bread we eat will nourish us, but that is to be sure of nothing because that is (in part) how we are using the term "bread".  What we now cannot be sure of is that it is a piece of bread we are eating!  It may look like bread, have been bought from the bread shop, have been manufactured in the normal way by the baker, but it will not be bread if it fails to nourish us.  If we define "bread" (in part) by its effects on humans who eat it, then humans cannot be sure that it is bread they are eating until those effects have happened.  We need a new term, say "cread", to refer to what looks like bread, tastes like bread and so forth.  We can be sure from observation (we are assuming) that something is a piece of cread.  But now our original inductive inference must be expressed using this new term:
  • Cread nourished me on Monday (and so turned out to be bread). 
  • Cread nourished me on Tuesday (and so turned out to be bread). 
  • Cread nourished me on Wednesday (and so turned out to be bread). 
  • Cread nourished me on Thursday (and so turned out to be bread). 
  • Cread nourished me on Friday (and so turned out to be bread). 
  • Therefore, cread will nourish me tomorrow (and so turn out to be bread). 
Hume's original conclusion about the invalidity of this argument stands. 
-- Alan Musgrave, Common Sense, Science, and Scepticism

So essences (or "inner natures", or whatever) are of no help in solving the Problem of Induction.  But it gets worse.  Note that cread's bread-nature can only be inferred from whether it nourishes us.  What if it still isn't bread, but fread?  Fread looks, feels, smells and tastes like bread, and has the same nutritional properties as well.  It differs from bread only in having a completely different essence.
Either fread is plausible, or it is not.  If it is, then what essence a thing has cannot be determined in any way whatsoever.  You cannot tell, for instance, that your spouse is human.  They might look and behave in all ways like a human, have human anatomy, physiology, and ancestry... but have the essence of a pod being.  Essence then carries no information; it has no bearing on any question; it is irrelevant to all enquiries.
If fread is not plausible, this must mean that essence can be determined, with complete confidence, by a thorough empirical investigation.  It must then be synonymous with -- indeed, reducible to -- a thing's incidental properties.
If fread is plausible, essences can never be found because no conceivable event can test for them.  If it is not, essences do not exist over and above incidental properties.  In either case, essences as distinct from incidental properties are, for all practical and theoretical purposes, non-existent.  Things are empty -- sūnya -- of any fixed nature or essence.
Doesn't feel like we've got very far, does it?  If anything, sūnyatā seems almost to be a reiteration of the Problem of Induction itself.  Don't worry; we'll get there.

The third tool is Daniel C. Dennett's idea, from Freedom Evolves, dubbed the Library of Democritus; which he himself builds from Jorge Luis Borges' Library of Babel, a concept Dennett expands upon in Darwin's Dangerous Idea.  Let's start with the Library of Babel.
Imagine a library containing millions upon billions of books.  Each is 100 pages long, with 40 lines of text on each page and 80 characters to a line.  Actually, "millions upon billions" is an understatement: this library holds every possible 320,000-character combination that can be typed with a keyboard.  (I'm adapting slightly, as did Dennett; I gather Borges limited himself to the letters commonly used in Spanish.)
How many is that?  Twenty-six letters, each in upper and lower case: 52.  Ten digits, plus the ten characters you get by holding Shift down: 72 so far.  Eleven additional character keys (` - = [ ] \ ; ' , . /) and their shift-alternatives (~ _ +  { } | : " < > ?): 94.  And the space-bar: 95.  Not the Return or Tab keys -- those can be simulated by pressing the space-bar the appropriate number of times.  95 it is.  Actually, let's add in ‘ ’ “ ” –, automatically generated by Microsoft Word, just to round it up to 100.
So the total number of books in the Library is 100 to the power of 320,000, that is to say 1 followed by 640,000 zeroes.  For comparison, there are fewer than 1-with-80-zeroes atoms in the known universe.  Dennett terms this kind of scale "Vast", capital V, meaning "Very much more than Astronomical".  The overwhelming majority will look like this:
xeQK!l/- rC=@{_w*!.R!nl=eQ;87zZx]+z1\YDOYK^b2B_}1nGjmWF^A}”e’L^fm3|ax'JBh_1”2@'f
dq`H4@4<X*P,D(M\(+AyB&uC?gu=#;<7CKykza 4]/ W<IH8EjcCkN=<WBKb7u)[rYf%&e{['–.A>ks4
But every so often -- in a tiny fraction of the Library, which still amounts to a Vast number of books -- there will be a passage like this:
or this:
Jw~)}D>dG?4uGiO‘g!`nd7”h_[w is the totality of the world-as-it-is for them at an
y given time. So when wetB.$’pEO~8I/pA}fF<Yr%7laXwzb$]9di@g*]:hnE|ymly
or this:
passage is evidently earlier thais simply the observable correspms to be doing fi
ne without an or took a census of all the pondsven if we accept Aristotle’s cri
or, yes, this:
and mambas aren’t poisonous – they’re venomous. It’s an important distinction:
poison harms you when you swallow it, venom when it’s injected into you. So,    
While no volume in the Library is more than 100 pages long, every longer book can be sliced into 100-page excerpts, each and every one of which will be found in the Library.  (The last one will have lots of white space after the text ends, but that's allowed.)  Let your imagination wander here for a while longer; an appreciation of the scale we're dealing with is vital to the argument.
Somewhere in the Library is the story of your life, told with complete accuracy; all your secrets laid bare.
Somewhere else is the story of your life, except that all the things you don't want people to know are replaced by whatever you'd rather they believe.
Somewhere else again is the same story, except that, at 4:59pm on Saturday 26 November 2005, you turn into a small blue china teapot for a total of 57.38 seconds.
The Bible is in there, of course.  There is also a Bible with all the commandments reversed.  And a Bible that replaces all the Hebrew names with Klingon ones.  And a Bible where Jesus turns out to be a Time Lord.  The Lord of the Rings is in the Library, and so is Wuthering Heights, and so is a sort of mashup of the two where Heathcliff is secretly an incarnation of Sauron.
That's assuming we put all the volumes in the right order, of course.  Well, somewhere in the Library, there will be a book with that order printed in it.  (And books with every other possible order printed in them.)
Now for the Library of Democritus.  Where the Library of Babel is a library of books, the Library of Democritus is a library of universes.  Dennett conceived them as very simple universes: two-dimensional grids of squares, each square in one of two states: empty or full (a space, or a particle).  Where the Library of Babel has every possible 100-page book, the Library of Democritus has every possible universe-grid of a given size.
Let's say that each universe is a 2-dimensional 1500x1500 grid.  That makes 2^2,250,000 universes: my computer's version of Microsoft Excel stops at 2^1023, but we can estimate it at slightly over 1 followed by 675,000 zeroes.
Just as we could construct longer books in the Library of Babel by concatenating the short ones, we can create larger universes by sticking the grids together at the edges.  Or stacking them like plates, if we want more than two spatial dimensions.  Or flicking through them like video frames, if we want a time dimension.  Or any combination or permutation of the above, as big and as complicated as we like.
As with the Library of Babel, the Library of Democritus will be mostly random noise -- nonsense.  As with the Library of Babel, there will also be every possible non-random combination of events.  Some of them will have laws of physics: rules describing the behaviour of the particles in them, which hold good in every single instance.  These will be a vanishing minority, just as the books in sensible English were a vanishing minority in the Library of Babel; but that minority will still be a Vast number.
You may think the universes are still limited to one kind of particle.  You'd be wrong.  What distinguishes "kinds" of particles?  Their behaviour as they interact.  You can have different rule-sets for different particles and still call your universe "deterministic", provided it's one where each particle sticks with one rule-set.  We can reify patterns of behaviour as "properties" of the particle, such as "mass", "charge", "spin", or whatever.  (If you're thinking "But they wouldn't be its real properties", remember sūnyatā.)
The Library of Democritus contains our universe.  Be warned, however: it also contains the universe which was exactly like our universe until 2:47pm and 48.29 seconds yesterday (Greenwich Mean Time), at which point the entire Solar System suddenly became a uniform disc of purple goo.  And the universe where that is going to happen, precisely five hours, three minutes, and forty seconds from now.  Check your watch.

This -- I imagine you're thinking -- is worse than before!  How can we possibly be sure that we're not in a universe where we'll end up as purple goo in a couple of minutes' time, for no reason at all?
Well, we'll see in a moment.  But first I want to open up your mind just a crack further.  Just as we are uncertain about the future (as Hume says), so we must also be uncertain about the past.  After all, we don't have it to look at directly.  What we have is present memories, present video recordings and sound and books and drawings and footprints and so on, from which we reconstruct the past.
"I remember yesterday," said Wen thoughtfully.  "But the memory is in my head now.  Was yesterday real?  Or is it only the memory that is real?  Truly, yesterday I was not born." 
-- Terry Pratchett, Thief of Time

For all we know, we might have been purple goo until two minutes ago, then suddenly popped into existence exactly as we are.  Indeed, there are billions of universes in the Library where precisely this has happened!  The number of universes in which our reconstructions accurately reflect the past is, by comparison, vanishingly tiny.  Going by probability, we'd have to say that we almost certainly popped out of nowhere moments ago, and will almost certainly disappear again moments from now!
Until, that is, we bring in sūnyatā.  Effectively, we've been asking which of the universes is the "real" one, as if we knew for a fact that there was one "real" one and all the rest were mere hypotheses.  But what does "real" mean here?  Does this universe have the "essence" or "nature" of being real, which all the other ones lack?  Thanks to the Buddhists, we know what to think of essentialistic hair-splitting.  We are in this universe; that, and that alone, distinguishes it from the rest of the Library.
What was the real past?  By what criterion does any particular succession of past moments count as "real"?  By being reconstructible on the basis of consistent rules -- low-level laws such as "My memory is accurate", with complications added in such as "except between my going to bed and my waking", or "unless it contradicts video evidence" (since our memories contain many more instances of our memory having been wrong than of videos changing behind our backs).
You see how the anthropic principle applies here.  As creatures living moment by moment at local points in the universe, our only way of extending our knowledge beyond our immediate instantaneous sensations is by applying rules.  Therefore, the universe we thereby discover will of necessity be rule-bound; that is to say, deterministic.  And other fragments of the Library of Democritus count as belonging to our universe, or not, by virtue of whether they fit into the rules we apply.

To round it all up: how do we know that we will still be living in the same rule-bound universe tomorrow?  Let's rephrase the question.  What, within the Library of Democritus, counts as "us tomorrow"?
The answer is simplicity itself.  A "future-me" counts as "future-me" by virtue of the fact that his rule-bound reconstruction of the past includes present-me.  Therefore, any sequence of events leading from present-me to future-me must be rule-bound.  Therefore, contrary to Hume, the universe we live in is certainly deterministic; therefore, the probabilists' calculations do apply; and therefore, I can be very nearly sure that the next piece of bread I eat will nourish me.


  1. Deleted my comment eh? That's the question isn't it: no comments, or a comment pointing out what a pathetic existence you lead. Tough one. I do pity you though...and judging from what i saw skimming a little more of your blog, i dont think you quite "get" Kant. Ah well, and so it goes...

    1. I haven't deleted any comments that I can recall, apart from the ones in my spambox. I am quite bemused that this is the first one I've had on this post. (We had quite a decent discussion going on the Facebook note that this is copied from, which reminds me, I still haven't got round to the question of whether we deduce the laws of "our" universe from observation or know them a priori.) If you are the same troll who left a contentless sneer on my last Narnia post, it's still there.