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Science fiction trades on disproving stuff. The wonder gadgets and gizmos of your average SF story require shoving a lot of pesky scientific theories onto a rubbish pile to make way for them. The warp drives and transporters of Star Trek require overthrowing a lot of awkward physics theories like Heisenberg’s Uncertainty Principle and Special and General Relativity. In order for the psychics in your average Philip K Dick story to become a reality, you would have to disprove all the neuroscience and physics that says you can’t have telepathy and precognition. Weirdly enough there is nothing unscientific about this, we got the sun centred system by overthrowing the Earth centred one and we got evolution by overthrowing creationism so it’s only natural for the SF author to extrapolate forward. Perhaps one day Einstein will be overthrown by FTL. Maybe conventional neuroscience will be overturned by someone who can predict the future or survive death. Given that it’s possible to disprove theories in physics, chemistry or biology, it’s natural to wonder (in typical SF run-with-it logic!) whether it’s possible to disprove the language of these sciences, mathematics. Are there aliens who design spaceships without using numbers or reason with contradictory logic? Could we one day discover that 1+1=3?
The conventional wisdom says that this is impossible. It says that there’s something special about math, something that makes it undisprovable. Like an indestructible fortress, mathematics is an edifice that, when it’s built, nothing known to science can shift. The conventional wisdom has a pretty strong pedigree, consisting of the whole of Western science and philosophical speculation. You would be turning your guns on Plato, Descartes, Leibniz, Bertrand Russell, hell even real scientists like Einstein… Okay okay! But there’s an obvious question: what’s so special about math that you can’t disprove it? And the obvious answer is nothing! You should be able to disprove math like you can anything else.
There are two main arguments conventionally put forward to justify the undisprovability of mathematics. First, mathematicians and philosophers argue that you can’t imagine it being disproved. How would it work if 2+2=5, or squares had five sides instead of four? If you can’t even imagine two plus two not equalling four, how do you imagine actually disproving it? Anyone who has any imagination can see that this argument doesn’t hold. The fact that you can’t imagine something to save your life does not prove that it doesn’t exist. You can prove this with an example from – wait for it – mathematics. Imagine a fourth spatial dimension. You can’t, it’s impossible, you can’t even visualise it… This doesn’t stop the fourth dimension from being mathematically and physically possible. Mathematicians use higher dimensional spaces all the time, even though they can’t imagine them. Given this, what’s to stop a five sided square being an analogous case, something that’s possible even though all the mathematicians in the world can’t imagine it to save their lives? Further, the whole not being able to imagine it being disproved shtick only works for intuitive and elementary stuff. It doesn’t work for anything complicated. What about the graph for y=x4+2x3+3x – 9; could that be false?
Second, mathematicians and philosophers argue that over 2-3,000 years of recorded history, no mathematical statement or theory has been disproved. We still believe in 2+2=4 or Pythagoras’ theorem as much now as when they were first discovered. How can these statements have withstood the test of such time if they weren’t true? There are two major problems with this argument. One, the fact that scientists have believed something for over a thousand years does not make it any more likely to be true: Earth centred cosmology was believed for 1,500 years, and it was even mathematically accurate! Why couldn’t the same be true of Pythagoras’ theorem? Moreover, the persistence of a belief system does not imply that it is logically or empirically true, in fact it very probably implies the opposite. Consider the belief in god, maintained by religious authority, or the belief in star signs.
Two, if you study the history of mathematics carefully you find that it too has its analogues of revolutions. Before the 17th century, mathematics did not consist of the equations and formulae that every schoolchild knows today. Instead it was done by drawing, using a straightedge and compass to construct geometrical shapes and work out their interrelationships. The rise of algebraic methods swept the geometrical approach away. Things that were possible algebraically were not possible with a straightedge and compass. Conversely, though mathematicians are loath to admit it, things you could do with a straightedge and compass cannot be done by algebra. Another, perhaps more powerful example is complex numbers. Before Cardano discovered them, it was impossible to square root -1, afterwards it was possible, completely changing mathematician’s concept of number. Mathematical revolutions may not be as visible as scientific revolutions but they have the same effect, sending shockwaves through existing mathematics that irrevocably change its nature.
The ‘disprovability’ of mathematics would alter the way it is used in science. Currently it is regarded as a fixed frame of reference with which to ‘calibrate’ the rest of science. Experiments may be faulty and theories false but the equations used to describe them are never in error. A statistician can always rely on a random variable being bell shaped and a physicist doing orbital calculations can rely on the theorems of calculus. In the new way of thinking this can no longer be maintained. The theorems of calculus may be revised or the concept of random variable may change. (Both these senarios actually have happened to some extent.) If mathematics is the lens through which we view the physical world then that lens might be less like a telescope, with carefully calibrated optics producing a steady image, and more like a kaleidoscope, producing different images each time it turns, none of which is ‘true.’
If mathematics was disproved, where might we go next? One philosopher who has seriously considered the possibility that mathematics is radically false is Henry Flynt. Flynt argues that mathematics is a social construct whose ‘acolytes’ – mathematicians – act to preserve and maintain its core values and hypotheses. Remember how conventional wisdom said it didn’t change? Now we know why; math doesn’t change for exactly the same reason the church doesn’t! Because of this, Flynt argues that further intellectual and ‘scientific’ progress requires overthrowing, or negating, mathematics. To continue the analogy of the previous paragraph, centuries of ossification have made mathematics a duff lens that obscures more than it reveals with layers and layers of aberrations. Flynt believes we need to throw down the lens and look with new eyes. What will we see when we “look with new eyes?” It depends what ‘eyes’ we look with! Flynt suggests the creation of “meta-technology,” conceptual tools that would allow humans to ‘parse’ reality in ways that transcend classical logic. Replacing mathematics with meta-technology would be a bit like replacing the James Webb telescope with a viable form of astral projection. Flynt’s ideas are of course speculative, in fact, they are pure science fiction. However like all good science fiction they are based on scientific evidence and shouldn’t be dismissed because of their unconventional nature.
As well as having consequences for mathematics and science per se, the idea of a wholesale mathematical revolution forces us to reevaluate the way we think about mathematics and the type of questions we ask about it. The title question of this article was so simple it was almost childish, but poke it a little and the simplicity falls away to reveal a morass of thorny complications. Questions which most people never see fit to think about burst into consciousness. What do you mean by mathematics? What do you mean by disprove? What do you mean by possible?
One pressing objection to my argument is that the question of disproving mathematics is so broad it cannot possibly make sense. A bomb that is so powerful it can destroy reality itself seems to be the ultimate fantasy but it in fact lapses into incoherence and self contradiction. How could you “destroy reality” and if you did wouldn’t you destroy the “reality” of the destructive event itself therebye…? In a similar vein the idea of disproving mathematics is like ‘disproving science’, suggestive, but when you think about it for two seconds it’s nonsense.
There are several answers one might give to this. One is to pull back and say we’re only considering disproving sub theories of mathematics - arithmetic, Euclidean geometry, classical logic – mathematics itself will still carry on, albeit with different content. This would be analogous to how we see empirical science where an overarching field such as physics experiences periodic revolutions (Newton Einstein, quantum mechanics.) Here though, an important distinction comes to light between science and (conventional) mathematics which allows me to put my foot through the door I originally wanted to put it through. Because empirical science undergoes periodic revolutions in its content, it’s possible to draw a distinction between science and its content. General Relativity may be replaced by Warp Field Theory but the Star Trek universe still obeys the laws of physics (just not quite our laws.) Mathematics (conventionally) is different. Because it doesn’t experience periodic revolutions on its theories there is no way to draw a distinction between ‘mathematics’ and the sum total of mathematical theories. Furthermore, all existing ideological justifications of mathematics are premised on the non occurrence of such revolutions. A large part of mathematics’ self identity as a discipline consists in it being ‘undisprovable.’ In this way, if you could overturn arithmetic, complex numbers, Euclidean and Non-Euclidean geometry etc. then you would have overturned mathematics as a discipline. You would have literally disproved mathematics, like my schoolboy self dreamed.
Another issue is one that I alluded to above. Mathematics is traditionally understood to be so basic, so fundamental that its frameworks determine thought, experience and reality themselves. In disproving mathematics, wouldn’t we be in some sense undercutting, thought, experience and reality themselves? Let’s think about this a minute. On the one hand there are some mathematical notions that seem so basic and elementary that they are part of our basic thought patterns. One plus one equals two. A square has four equal sides. Space has three dimensions. It seems like if we tried to nix these mathematical concepts we wouldn’t be able to think straight or even see straight (what would the world look like if we could see four dimensionally?) On the other hand many mathematical theories are highly technical calculational schemas and algorithms that exist more as a superstructure than a basis for thought. When you multiply out a quadratic equation, you manipulate symbols almost mechanically according to algebraic rules. It seems that we could imagine different rules for multiplying out a quadratic equation without having a major reality dysfunction. In this sense then, mathematical theories might be disproved wholesale and we would still have ten fingers!
Perhaps the idea of ‘disproving mathematics’ is one of those ideas that doesn’t quite ‘make sense.’ In considering it we are poking at the basic framework of concepts that undergirds the way we think about knowledge and science. But not quite making sense is different from nonsense. Science is full of concepts that ‘don’t quite make sense’, a hole in space, an unobservable wavefuction, quantum entanglement. These are all concepts that stretch the boundaries of our existing linguistic frameworks. But they are valid concepts and they advance knowledge.
Science fiction is exactly what we need to explore the consequences of a mathematical revolution. However – and perhaps this shows the pernicious influence of Plato and Bertrand Russell – I can think of no science fiction novels based on the idea of ‘impossible’ mathematics. I can only think of three stories: Greg Egan’s Dark Integers is about a parallel universe that is ‘so far up the number line’ that it obeys different mathematical laws. Ted Chiang’s Division By Zero is about the consequences of discovering a proof that 1=2. To my mind the most intriguing is an online story called Self Destruct Mode by Stephen Watkins which has aliens coding signals in ‘anti-mathematics” described as “a field of mathematics that begins with the assumption that everything we know about mathematics is false.” Apart from these, the disprovability of mathematics is an unexplored blank. If any QM readers have any other suggestions please comment.
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