Known Unknowns

This month’s post may make a valid point.  Or it may not.  Or it may be impossible to tell, the concept of which itself may or may not make sense by the end of the piece!

How do we handle things we don’t know?  More precisely, how do we cope with things we know we don’t know?  All right then: how do we handle things we know we can’t know?

As is the nature of this blog, the examples we’re going to discuss are (at first, at least) taken from the fields of computer science and mathematics; but there are plenty of analogies in the other sciences.  This certainly isn’t a purely theoretical discussion.

On the whole, we like things (statements or propositions) in mathematics (say) to be right or wrong: true or false.  Some simple examples are:

• The statement “2 > 3” is false
• The statement “There is a value of x such that x < 4” is true
• The proposition “There are integer values of x, y and z satisfying the equation x³ + y³ = z³” is false

OK, that’s pretty straightforward but how about this one?

• “Every even number (greater than 2) is the sum of two prime numbers”

Now, we can start off by trying this out easily enough.  Working up: 4 = 2 + 2; 6 = 3 + 3; 8 = 3 + 5.  Jumping in at random, 100 = 11 + 89 (and a number of other ways) so it looks promising.

In fact, this Goldbach’s Conjecture (GC) has been shown by calculation (computation) to be true for a huge range of even numbers, and no-one’s ever found an example where it’s not.

However, that’s not quite the same thing as proving it’s always true.  We’d need a cleverer logical argument for that (because we can’t try out an infinite number of numbers individually) and no-one’s managed that yet.

So, at the moment, we don’t actually know if GC is true or false.  If someone can supply the general proof, it will be true; if anyone finds an exception (an even number, which isn’t the sum of two primes) then it’s false.

Or something else might happen …

Let’s take another example, which we’ve looked at recently in a more festive context …

Without labouring on detail (because we’ve already done that), there are an infinite number of whole numbers (integers), positive or negative.  We can think of the set of these numbers as having a certain (infinite) size.

Somewhat counter-intuitively, if we then extend the integers to include the fractions (1/3, 11/4, etc.), then, although we might argue that there are more of them, in a strictly mathematical sense, the set of fractions has the same (infinite) size as the set of integers because we can set up a direct one-to-one mapping from the integers to the fractions.

Now, we might attempt to regain some sanity here by suggesting this is so because there’s only really one infinity.  However, this line of reasoning falls apart when we look at the set of all numbers (the reals), now including all the things that can’t be expressed as fractions (√2, π, etc.).

Unfortunately, it turns out we can’t set up a one-to-one mapping from the integers to the reals so we have to conclude that we’re dealing with a different infinity.  If the set of integers is of a certain infinite size, then the set of reals is a different infinite size.  And, once we’ve managed this, it emerges that it doesn’t stop there.  In fact, there’s an endless sequence of infinite set sizes, getting larger and larger without end.

Now, we don’t need to worry here about where that sequence goes but, now that we know there is such a sequence, it’s interesting to return to the set of integers and the set of reals and consider their places in the sequence.  In particular, are they consecutive in this sequence?  In other words, are they next to each other in terms of infinite set sizes or is there something that produces a set between the two?  Is there an infinity between the integer infinity and the real infinity.  This (proposing that there isn’t) is the Continuum Hypothesis (CH).

Well, no-one’s found an infinity between the integers and the reals.  On the other hand, no-one’s proved there isn’t one.  So, on the surface, we seem to be in the same position as with the GC.  However, for the CH, mathematicians have gone one stage further.  The question has been shown to be undecidable.  What?

So, what does this mean?  Well, it sort of depends on how you look at it … In terms of the previous paragraph, no-one’s ever going to find an infinity between the two but no-one’s ever going to prove there isn’t one either.  In terms of pure logic, this means both of the following:

• Starting from everything we know of numbers and sets, there’s no logical sequence of steps that will show that the CH is true … or false.  It’s logically ‘distant’.
• The CH  can be either true or false, arbitrarily, and it doesn’t interfere with the rest of what we know of numbers or sets.  It’s logically ‘independent’.

And, really, we can paraphrase these (equally valid) viewpoints as:

1. ‘It’s impossible to tell’, or
2. ‘It doesn’t matter’

Now here’s the thing … This is actually what most of us do subconsciously when we think about things that we don’t know.  Even with the day-to-day stuff we could know, individually we often make an intuitive distinction between what we can be bothered to find out what we can’t.  But, when we enter the realm of the truly unknown, it takes on a whole new significance.

How do we deal with being told (possibly proved) that something is beyond all human knowledge?  Very often, we find ourselves adopting a position of either (1) or (2) above and, which it may be can be as much a social or philosophical stance as a scientific one.  In fact, we might even rewrite them again as sort of:

1. ‘Wow, really?’, or
2. ‘I don’t care!’

Take Heisenberg’s Uncertainty Principle, for example, which, in simple terms, says there’s a limit to what we can measure at the quantum level: if we look too closely, we change what we’re looking at.  Here it’s pretty easy to see a likely difference between a scientific amazement (possibly frustration) that we can’t get any closer and a wider public lack of interest.

Similarly, Turing‘s Algorithmic Halting Problem and Gödel‘s Algebraic Incompleteness may have the same effect.

But, to finish with, let’s consider a really big question, one that’s often considered unknowable.  What happened before the Big Bang?  Or perhaps, What was there before the Big Bang?  On one level or another, this seems to be out of our grasp.

We have various tools to try to help … or hinder our attempt.  We can search for subatomic particles capable of appearing from nowhere and catastrophic singularities in space.  We can tie together notions of space and time into a single concept so we’re not even sure what ‘before’ means anymore.  (“Don’t ask what was before the BB: there was no ‘before’ the BB”.)  But, at the end of the day, we’re left not knowing.  It’s not that our science has failed, as such, but our ability to apply it from where we are in the universe is limited.  We don’t have the necessary ‘vision’.  Perhaps we can’t ‘see’ far enough (in space and/or time).  Perhaps we don’t live in enough dimensions.  If there was a ‘before’, then we can’t really comprehend it; if there wasn’t, then we don’t really know what that means.  And, again, how we cope with this uncertainty (and many other similar scientific unknowns) reflects our philosophy more than our science.

Because we can either say:

1. “That’s interesting.  Perhaps we’re part of a bigger something that we can’t see by scientific means, from where we are”, or
2. “Don’t ask stupid questions!”

And that may say more about us than the science!