Zach Weber on Paraconsistent Logics

 

Here is a dilemma you may find familiar. On the one hand, a life well lived requires security, safety and regularity. That might mean a family, a partner, a steady job. On the other hand, a life well lived requires new experiences, risk and authentic independence, in ways incompatible with a family or partner or job. Day to day, it can seem not just challenging to balance these demands, but outright impossible. That’s because, we sense, the demands of a good life are not merely difficult; sometimes, the demands of a good life actually contradict. ‘Human experience,’ wrote the novelist George Eliot in 1876, ‘is usually paradoxical.’
 
One aim of philosophy is to help us make sense of our lives, and one way philosophy has tried to help in this regard is through logic. Formal logic is a perhaps overly literal approach, where ‘making sense’ is cashed out in austere mathematical symbolism. But sometimes our lives don’t make sense, not even when we think very hard and carefully about them. Where is logic then? What if, sometimes, the world truly is senseless? What if there are problems that simply cannot be resolved consistently?
 
Formal logic as we know it today grew out of a project during the 17th-century Enlightenment: the rationalist plan to make sense of the world in mathematical terms. The foundational assumption of this plan is that the world does make sense, and can be made sense of: there are intelligible reasons for things, and our capacity to reason will reveal these to us. In his book La Géométrie (1637), René Descartes assumed that the world could be covered by a fine-mesh grid so precise as to reduce geometry to analysis; in his Ethics (1677), Baruch Spinoza proposed a view of Nature and our place in it so precise as to be rendered in proofs; and in a series of essays written around 1679, G W Leibniz envisioned a formal language capable of expressing every possible thought in structure-preserving, crystalline symbols – a characteristica universalis – that obeys precise algebraic rules, allowing us to use it to find answers – a calculus ratiocinator.
 
Rationalism dreams big. But dreams are cheap. The startling thing about this episode is that, by the turn of the 20th century, Leibniz’s aspirations seemed close to coming true due to galvanic advances across the sciences, so much so that the influential mathematician David Hilbert was proposing something plausible when in 1930 he made the rationalist assumption a credo: ‘We must know, we will know.’
 
Hilbert’s credo was based in part on the spectacular successes of logicians in the late 19th century carving down to the bones of pure mathematics (geometry, set theory, arithmetic, real analysis) to find the absolute certainty of deductive validity. If logic itself can be understood in exacting terms, then the project of devising a complete and consistent theory of the world (or at least, the mathematical basis thereof) appeared to be in reach – a way to answer every question, as Hilbert put it, ‘for the honour of human understanding itself’.
 
But even as Hilbert was issuing his credo and elaborating his plans for solving the Entscheidungsproblem – of building what we would now call a computer that can mechanically decide the truth or falsity of any sentence – all was not well. Indeed, all had not been well for some time.
 
Already in 1902, on the verge of completing his life’s work, the logician Gottlob Frege received an ominous letter from Bertrand Russell. Frege had been working to provide a foundation for mathematics of pure logic – to reduce complex questions about arithmetic and real analysis to the basic question of formal, logical validity. If this programme, known as logicism, were successful then the apparent certainty of logical deduction, the inescapable truth of the conclusions of sound derivations, would percolate up, so to speak, into all mathematics (and any other area reducible to mathematics). In 1889, Frege had devised an original ‘concept notation’ for quantified logic exactly for this goal, and had used it for his Basic Laws of Arithmetic (two volumes of imposing symbolism, published in 1893 and 1903). Russell shared this logicist goal, and in his letter to Frege, Russell said, in essence, that he had liked Frege’s recent book very much, but had just noticed one little oddity: that one of the basic axioms upon which Frege had based all his efforts seemed to entail a contradiction.
 
Frege had assumed what he called ‘Basic Law V’ which says, in effect: Sets are collections of things that share a property. For example, the set of all triangles is comprised of all and only the triangles. This seemed obvious enough for Frege to assume as a self-evident logical truth. But from Basic Law V, Russell showed that Frege’s system could prove a statement of the form P and not-P as a theorem. It is called Russell’s Paradox:
 
  “”Let R be the collection of all things with the property of ‘not being a self-member’. (For example, the set of triangles is not itself a triangle, so it is an R.) What about R itself? If R is in R, then it is not, by definition of R; if R is not in R, then it is, again by definition. It must be one or the other – so it is both: R is in R and R is not in R, self-membered and not, a contradiction.”
 
 
The whole system was in fact inconsistent, and thus – in Frege and Russell’s view – absurd. Nonsense. In a few short lines, Frege’s life work had been shown to be a failure.
 
He would continue to work for another two decades, but his grand project was destroyed. Russell would also spend the next decades trying to come to terms with own his simple discovery, first writing the monumental but flawed Principia Mathematica (three volumes, 1910-13) with Alfred North Whitehead, then eventually pivoting away from logic without ever really solving the problem. Years would pass, with some of the best minds in the world trying mightily to overcome the contradiction Russell had found, without finding a fully satisfactory solution.
 
By 1931, a young logician named Kurt Gödel had leveraged a similar paradox out of Russell’s own system. Gödel found a statement that, if provable true or false – that is, decidable – would be inconsistent. Gödel’s incompleteness theorems show that there cannot be a complete, consistent and computable theory of the world – or even just of numbers! Any complete and computable theory will be inconsistent. And so, the Enlightenment rationalist project, from Leibniz to Hilbert’s programme, has been shown impossible.
 
Or so goes the standard story. But the lesson that we must give up on a full understanding of the world in which we live is an enormous pill to swallow. It has been almost a century or more since these events, filled with new and novel advances in logic, and some philosophers and logicians think it is time for a reappraisal.
 
If the world were a perfect place, we would not need logic. Logic tells us what follows from things we already believe, things we are already committed to. Logic helps us work around our fallible and finite limitations. In a perfect world, the infinite consequences of our beliefs would lie transparently before us. ‘God has no need of any arguments, even good ones,’ said the logician Robert Meyer in 1976: all the truths are apparent before God, and He does not need to deduce one from another. But we are not gods and our world is not perfect. We need logic because we can go wrong, because things do go wrong, and we need guidance. Logic is most important for making sense of the world when the world appears to be senseless.
 
The story just told ends in failure in part because the logic that Frege, Russell and Hilbert were using was classical logic. Frege assumed something obvious and got a contradiction, but classical logic makes no allowance for contradiction. Because of the classical rule of ex contradictione quodlibet (‘from a contradiction everything follows’), any single contradiction renders the entire system useless. But logic is a theory of validity: an attempt to account for what conclusions really do follow from given premises. As contemporary ‘anti-exceptionalists about logic’ have noted, theories of logic are like everything else in science and philosophy. They are developed and debated by people, and all along there have been disagreements about what the correct theory of logic is. Through that ongoing debate, many have suggested that a single contradiction leading to arbitrary nonsense seems incorrect. Perhaps, then, the rule of ex contradictione itself is wrong, and should not be part of our theory of logic. If so, then perhaps Frege didn’t fail after all.
 
Over the past decades, logicians have developed mathematically rigorous systems that can handle inconsistency not by eradicating or ‘solving’ it, but by accepting it. Paraconsistent logics create a new opportunity for theories that, on the one hand, seem almost inalienably true (like Frege’s Basic Law V) but, on the other, are known to contain some inconsistencies, such as blunt statements of the form P and not-P. In classical logic, there is a hard choice: give up any inconsistent theory as irrational, or else devolve into apparent mysticism. With these new advances in formal logic, there may be a middle way, whereby sometimes an inconsistency can be retained, not as some mysterious riddle, but rather as a stone-cold rational view of our contradictory world.
 
Paraconsistent logics have been most famously promoted by Newton da Costa since the 1960s, and Graham Priest since the 1970s. Though viewed initially (and still) with some scepticism, ‘paraconsistent logics’ now have an official mathematics classification code (03B53, according to the American Mathematical Society) and there have been five World Congress of Paraconsistency meetings since 1997. These logics are now studied by researchers across the globe, and hold out the prospect of accomplishing the impossible: recasting the very laws of logic itself to make sense of our sometimes seemingly senseless situation. If it works, it could ground a new sort of Enlightenment project, a rationalism that rationally accommodates some apparent irrationality. On this sort of approach, truth is beholden to rationality; but rationality is also ultimately beholden to truth.
 
That might sound a little perplexing, so let’s start with a very ordinary example. Suppose you are waiting for a friend. They said they would meet you around 5pm. Now it is 5:07. Your friend is late. But then again, it is still only a few minutes after 5pm, so really, your friend is not late yet. Should you call them? It is a little too soon, but maybe it isn’t too soon … because your friend is both late and not late. (What they’re not is neither late nor not late, because you are clearly standing there and they clearly haven’t arrived.) Whatever you think of this, paraconsistent logic simply counsels that, at this point, you should not conclude, however provisionally, that the Moon is made of green cheese, or 2+2=5, or that maybe aliens did build the pyramids after all. That would be just bad reasoning.
 
Now, such situations are so commonplace that perhaps it seems implausible that some fancy system of non-classical logic is needed to explain what is going on. But maybe we are so enmeshed in contradictions in our day-to-day lives, so constantly pulled in multiple conflicting directions at once, that we don’t even notice, except when the inconsistency becomes so insistent that it can’t be ignored.
 
Paraconsistent logics help us find structure in the noise. Most strikingly, a subfield within paraconsistent logic has emerged that focuses on its applications to mathematics. One idea here would be to go back to Frege’s grand system in his Grundgesetze and recast it in a paraconsistent logic. Classical approaches mandate finding a way so that Russell’s paradox is no longer derivable (as has been attempted by many). A paraconsistent approach, on the other hand, will allow the paradox to go through, just in a way that does not do (too much) damage. Richard Sylvan, an early visionary in inconsistent mathematics, proposed in the late 1970s an axiomatic set theory that ‘meets the paradoxes head on’. In recent years, there have been several good, though inconclusive, steps in this direction. The idea is that, in this way, the foundations of mathematics can be set back on the path to finding unshakable (if paradoxical) certainty at its bottom.
 
An immediate concern about a paraconsistent approach is that it looks like a kind of cheating. It seems to sidestep the hard work of philosophical theorising or scientific theory-building. The worry, articulated recently by the philosopher of science Alan Musgrave in ‘Against Paraconsistentism’ (2020), is that:
 
  “It can plausibly be maintained that the growth of human knowledge has been and is driven by contradictions. More precisely, that it has been and is driven by the desire to remove contradictions in various systems of belief.”
 
If paraconsistent logics allow us to rest easy with an inconsistent theory, then there would be no impetus to improve. Another way to put the objection is that paraconsistency seems to offer an easy way out of difficult problems, a way to shrug off any objection or counter-evidence, to maintain flawed or failed theories long after they’ve been discredited. Does archaeological evidence contradict ancient-alien theory? No worries! This is just a contradiction, no threat to the theory. Rational debate seems stymied, if not destroyed.
 
This methodological objection points back to some of the assumptions that operate below the surface in our scientific and philosophical theories, from the Enlightenment and earlier. Often, there are two or more competing theories to explain some given data. How do we decide which to adopt? A standard account from Thomas Kuhn, in 1977, is that we weigh up various theoretical virtues: consistency, yes, but also explanatory depth, accord with evidence, elegance, simplicity, and so forth. Ideally, we might have all of these, but criteria such as simplicity will be set aside if it is outweighed by, say, predictive power. And so too for consistency, say paraconsistent logicians such as Priest and Sylvan.
 
Any of the theoretical virtues are virtuous only to the extent that they match the world. For example, all else being equal, a simpler theory is better than a more complicated one. But ‘all else’ is rarely equal, and as people from Aristotle to David Hume point out, the simpler theory is only better to the extent that the world itself is simple. If not, then not. So too with consistency. The virtue of any given theory then will be a matter of its match with the world. But if the world itself is inconsistent, then consistency is no virtue at all. If the world is inconsistent – if there is a contradiction at the bottom of logic, or at the bottom of a bowl of cereal – a consistent theory is guaranteed to leave something out.
 
What does prompt progress, then, in cases where we might decide that an inconsistent theory is allowable? There are many ways one theory may be better than another. In many cases, consistency will still win out (eg, your friend is not arriving at both 5:12 and 5:20), and your best waiting-for-a-friend-theory should not say that they are. But that theory choice is more to do with facts about people and time than logical consistency. Using inconsistency as a catch-all, as classical logic does, looks – to a paraconsistent logician – like a too-blunt avoidance of the real hard work: of thinking things through on an individual basis. ‘How then are we to determine whether a given contradiction in a given context is rationally acceptable?’ Priest and Sylvan asked in 1983. There is a simple solution: ‘A preliminary answer is that, at this stage, we need to consider each sort of case on its merits.’
 
A more on-the-ground answer to Musgrave’s methodological worries – and a warning to anyone tempted by paraconsistency as some kind of free pass – is that working within paraconsistent logics makes things more difficult, not less. The paraconsistent idea is that classical logic makes too many arguments valid, too many proofs go through when they shouldn’t, and so these validities and proofs are removed from the logical machinery. That makes drawing conclusions in a paraconsistent framework much harder because there are fewer inferential paths available. Someone who attempts a paraconsistent ancient-aliens theory may find that constructing valid arguments in their new ‘more permissive’ system is too challenging to be worth the effort.
 
And that points to some serious practical problems for paraconsistency that have emerged since it was proposed. Perhaps Frege need no longer worry that Russell’s contradiction will lead to 0=1 in his theory. But Frege also wants his theory to prove that 1+1=2 and support other elementary arithmetic. A paraconsistent Frege may start to worry, with good reason, that these true results are no longer derivable either. Depending on your views about the role of logic in mathematics, this issue, sometimes called ‘classical recapture’, is a serious problem. Sylvan put forward the idea of ‘rehabilitating’ mathematics using paraconsistency – trying to regrow established truths rather in the way one might rehabilitate a damaged ecosystem. As of today, much of Sylvan’s project remains undone. Work on this problem has been one of the most active and challenging areas of paraconsistent research.
 
What would a rehabilitated rationalist project look like? Gödel proved there cannot be a theory that answers every question in a consistent and computable way. The prevailing wisdom is that we will need to make do with theories that are either incomplete, or uncomputable (which comes to much the same thing as incompleteness, since, even if there is an answer, we have no effective way to get at it). A paraconsistent way forward would be to look for a system with precise and effective rules, that does answer every question after all – a complete description of the world (or at least the mathematical bit) – but where the system sometimes ‘over’-answers the question, saying both YES and NO. Because sometimes, maybe, the answer is both YES and NO.
 
The deep unease about paraconsistency, beyond methodological issues about scientific progress or practical problems about devising proofs, is what it would philosophically mean to accept a worldview that includes some falsity (where ‘false’ means having a true negation). How can a false theory be acceptable? And if it is or could be, then once consistency is no longer inviolable, is there any hard ground any more? If falsity is possible, then maybe everything is possible, and not in a good way. Perhaps this is what Musgrave is gesturing at when he says ‘an inconsistent theory provides no good explanation of anything’. If this is correct, any restored paraconsistent Enlightenment project will be a Pyrrhic victory, or worse.
 
The explanations an inconsistent theory provides, it must be admitted, may not look like what traditional philosophers have been expecting. But the expectations of traditional philosophers have not come to pass; indeed, Gödel gave us a mathematical proof that they will never come to pass. In the meantime, there are other kinds of valuable explanation right in front of us. As Schrödinger put it: ‘The task is not so much to see what no one has yet seen, but to think what no one has yet thought, about that which everybody sees.’
 
In 1921, a young Ludwig Wittgenstein’s Tractatus Logico-Philosophicus was published, after Russell’s qualified achievement with Principia but in advance of Gödel’s limiting theorems. Wittgenstein announces in the book’s forward that he has solved all the problems of philosophy. He marches to this conclusion through a relentless sequence of numbered propositions, appearing to lay out the nature of logic, what it can accomplish and, most crucially, what it cannot. By the end, his march arrives at the coastline, as it were, where we can look out at the vastness of the ocean, though logic will take us no further. According to Wittgenstein, the limits of logic mean that what is truly important in life can be shown but not said. He writes: ‘The solution of the riddle of life in space and time lies outside space and time.’ But then, he adds: ‘The riddle does not exist.’
 
And so, Wittgenstein is forced to conclude that all of his talk about showing and saying and riddles has itself been illegitimate. Yes, ‘there are, indeed, things that cannot be put into words … They are what is mystical’ – but the mystical is itself logically impossible, a senseless contradiction. Wittgenstein must admit that his whole beautiful book has been, by his own lights, nonsense, and the problems of philosophy are not so much solved as passed over in silence.
 
Wittgenstein was looking, like many, for an Archimedean point, a place ‘outside’ the world. This would be the view sub specie aeternitatis (borrowing a phrase from Spinoza). Only from there, he thought, could the world be explained. In seeking even to articulate that, to draw the limits of what we can understand, Wittgenstein contradicts himself, and inevitably so. He finds a contradiction, just as Frege and Russell and Gödel did when they attempted a complete theory. Wittgenstein took this as a kind of failure. But what if he had found what he was looking for and just didn’t recognise it? Perhaps Wittgenstein, like many others, felt pushed to make a false choice between a mysticism that provides some all-encompassing but inarticulate sense of the world, and a rational theory that is rigorous and precise but must be forever incomplete, inadequate.
 
This is a false choice if there can be a theory of the world that does both. Paraconsistency today does not have such a theory ready, but it holds out the (im)possibility of one, someday. It has recently been applied to religious worldviews (to Buddhism by Priest, to Christianity by Jc Beall). Maybe the lesson to take is that the philosophical Archimedean point at the end of the Enlightenment project will need to be both in, and not in, the world. ‘The proposition that contradicts itself,’ wrote the later Wittgenstein, ‘would stand like a monument (with a Janus head) over the propositions of logic.’
 
If we are living in an inconsistent world, in a world with contradictions from the foundations of mathematics to the triviality of dinner appointments, then its logic will leave room for falsity, doubts and disagreements. That is something many philosophers outside analytic and logical traditions have been urging for some time. As Simone de Beauvoir wrote: ‘Let us try to assume our fundamental ambiguity … One does not offer an ethics to a god.’ God would have no need of a logic, not even a paraconsistent one. But maybe we do.
 
 
This paradoxical life. By Zach Weber. Aeon, January 11, 2022.

Creativity is, and Always Will Be, a Human Endeavor

On March 31, 1913, in the Great Hall of the Musikverein concert house in Vienna, a riot broke out in the middle of a performance of an orchestral song by Alban Berg. Chaos descended. Furniture was broken. Police arrested the concert’s organizer for punching Oscar Straus, a little-remembered composer of operettas. Later, at the trial, Straus quipped about the audience’s frustration. The punch, he insisted, was the most harmonious sound of the entire evening. History has rendered a different verdict: the concert’s conductor, Arnold Schoenberg, has gone down as perhaps the most creative and influential composer of the 20th century.

You may not enjoy Schoenberg’s dissonant music, which rejects conventional tonality to arrange the 12 notes of the scale according to rules that don’t let any predominate. But he changed what humans understand music to be. This is what makes him a genuinely creative and innovative artist. Schoenberg’s techniques are now integrated seamlessly into everything from film scores and Broadway musicals to the jazz solos of Miles Davis and Ornette Coleman.

Creativity is among the most mysterious and impressive achievements of human existence. But what is it? Creativity is not just novelty. A toddler at the piano may hit a novel sequence of notes, but they’re not, in any meaningful sense, creative. Also, creativity is bounded by history: what counts as creative inspiration in one period or place might be disregarded as ridiculous, stupid, or crazy in another. A community has to accept ideas as good for them to count as creative.

As in Schoenberg’s case, or that of any number of other modern artists, that acceptance need not be universal. It might, indeed, not come for years—sometimes creativity is mistakenly dismissed for generations. But unless an innovation is eventually accepted by some community of practice, it makes little sense to speak of it as creative.

Advances in artificial intelligence have led many to speculate that human beings will soon be replaced by machines in every domain, including that of creativity. Ray Kurzweil, a futurist, predicts that by 2029 we will have produced an AI that can pass for an average educated human being. Nick Bostrom, an Oxford philosopher, is more circumspect. He does not give a date but suggests that philosophers and mathematicians defer work on fundamental questions to “superintelligent” successors, which he defines as having “intellect that greatly exceeds the cognitive performance of humans in virtually all domains of interest.”

Both believe that once human-level intelligence is produced in machines, there will be a burst of progress—what Kurzweil calls the “singularity” and Bostrom an “intelligence explosion”—in which machines will very quickly supersede us by massive measures in every domain. This will occur, they argue, because superhuman achievement is the same as ordinary human achievement except that all the relevant computations are performed much more quickly, in what Bostrom dubs “speed superintelligence.”

So what about the highest level of human achievement—creative innovation? Are our most creative artists and thinkers about to be massively surpassed by machines?

No.

Human creative achievement, because of the way it is socially embedded, will not succumb to advances in artificial intelligence. To say otherwise is to misunderstand both what human beings are and what our creativity amounts to.

This claim is not absolute: it depends on the norms that we allow to govern our culture and our expectations of technology. Human beings have, in the past, attributed great power and genius even to lifeless totems. It is entirely possible that we will come to treat artificially intelligent machines as so vastly superior to us that we will naturally attribute creativity to them. Should that happen, it will not be because machines have outstripped us. It will be because we will have denigrated ourselves.

Also, I am primarily talking about machine advances of the sort seen recently with the current deep-­learning paradigm, as well as its computational successors. Other paradigms have governed AI research in the past. These have already failed to realize their promise. Still other paradigms may come in the future, but if we speculate that some notional future AI whose features we cannot meaningfully describe will accomplish wondrous things, that is mythmaking, not reasoned argument about the possibilities of technology.

Creative achievement operates differently in different domains. I cannot offer a complete taxonomy of the different kinds of creativity here, so to make the point I will sketch an argument involving three quite different examples: music, games, and mathematics.

Music to my ears

Can we imagine a machine of such superhuman creative ability that it brings about changes in what we understand music to be, as Schoenberg did?

That’s what I claim a machine cannot do. Let’s see why.

Computer music composition systems have existed for quite some time. In 1965, at the age of 17, Kurzweil himself, using a precursor of the pattern recognition systems that characterize deep-learning algorithms today, programmed a computer to compose recognizable music. Variants of this technique are used today. Deep-learning algorithms have been able to take as input a bunch of Bach chorales, for instance, and compose music so characteristic of Bach’s style that it fools even experts into thinking it is original. This is mimicry. It is what an artist does as an apprentice: copy and perfect the style of others instead of working in an authentic, original voice. It is not the kind of musical creativity that we associate with Bach, never mind with Schoenberg’s radical innovation.

So what do we say? Could there be a machine that, like Schoenberg, invents a whole new way of making music? Of course we can imagine, and even make, such a machine. Given an algorithm that modifies its own compositional rules, we could easily produce a machine that makes music as different from what we now consider good music as Schoenberg did then.

But this is where it gets complicated.

We count Schoenberg as a creative innovator not just because he managed to create a new way of composing music but because people could see in it a vision of what the world should be. Schoenberg’s vision involved the spare, clean, efficient minimalism of modernity. His innovation was not just to find a new algorithm for composing music; it was to find a way of thinking about what music is that allows it to speak to what is needed now.

Some might argue that I have raised the bar too high. Am I arguing, they will ask, that a machine needs some mystic, unmeasurable sense of what is socially necessary in order to count as creative? I am not—for two reasons.

First, remember that in proposing a new, mathematical technique for musical composition, Schoenberg changed our understanding of what music is. It is only creativity of this tradition-defying sort that requires some kind of social sensitivity. Had listeners not experienced his technique as capturing the anti-­traditionalism at the heart of the radical modernity emerging in early-­20th-century Vienna, they might not have heard it as something of aesthetic worth. The point here is that radical creativity is not an “accelerated” version of quotidian creativity. Schoenberg’s achievement is not a faster or better version of the type of creativity demonstrated by Oscar Straus or some other average composer: it’s fundamentally different in kind.

Second, my argument is not that the creator’s responsiveness to social necessity must be conscious for the work to meet the standards of genius. I am arguing instead that we must be able to interpret the work as responding that way. It would be a mistake to interpret a machine’s composition as part of such a vision of the world. The argument for this is simple.

Claims like Kurzweil’s that machines can reach human-level intelligence assume that to have a human mind is just to have a human brain that follows some set of computational algorithms—a view called computationalism. But though algorithms can have moral implications, they are not themselves moral agents. We can’t count the monkey at a typewriter who accidentally types out Othello as a great creative playwright. If there is greatness in the product, it is only an accident. We may be able to see a machine’s product as great, but if we know that the output is merely the result of some arbitrary act or algorithmic formalism, we cannot accept it as the expression of a vision for human good.

For this reason, it seems to me, nothing but another human being can properly be understood as a genuinely creative artist. Perhaps AI will someday proceed beyond its computationalist formalism, but that would require a leap that is unimaginable at the moment. We wouldn’t just be looking for new algorithms or procedures that simulate human activity; we would be looking for new materials that are the basis of being human.

A molecule-for-­molecule duplicate of a human being would be human in the relevant way. But we already have a way of producing such a being: it takes about nine months. At the moment, a machine can only do something much less interesting than what a person can do. It can create music in the style of Bach, for instance—perhaps even music that some experts think is better than Bach’s own. But that is only because its music can be judged against a preexisting standard. What a machine cannot do is bring about changes in our standards for judging the quality of music or of understanding what music is or is not.

This is not to deny that creative artists use whatever tools they have at their disposal, and that those tools shape the sort of art they make. The trumpet helped Davis and Coleman realize their creativity. But the trumpet is not, itself, creative. Artificial-intelligence algorithms are more like musical instruments than they are like people. Taryn Southern, a former American Idol contestant, recently released an album where the percussion, melodies, and chords were algorithmically generated, though she wrote the lyrics and repeatedly tweaked the instrumentation algorithm until it delivered the results she wanted. In the early 1990s, David Bowie did it the other way around: he wrote the music and used a Mac app called Verbalizer to pseudo­randomly recombine sentences into lyrics. Just like previous tools of the music industry—from recording devices to synthesizers to samplers and loopers—new AI tools work by stimulating and channeling the creative abilities of the human artist (and reflect the limitations of those abilities).

Games without frontiers

Much has been written about the achievements of deep-learning systems that are now the best Go players in the world. AlphaGo and its variants have strong claims to having created a whole new way of playing the game. They have taught human experts that opening moves long thought to be ill-conceived can lead to victory. The program plays in a style that experts describe as strange and alien. “They’re how I imagine games from far in the future,” Shi Yue, a top Go player, said of AlphaGo’s play. The algorithm seems to be genuinely creative.

In some important sense it is. Game-playing, though, is different from composing music or writing a novel: in games there is an objective measure of success. We know we have something to learn from AlphaGo because we see it win.

But that is also what makes Go a “toy domain,” a simplified case that says only limited things about the world.

The most fundamental sort of human creativity changes our understanding of ourselves because it changes our understanding of what we count as good. For the game of Go, by contrast, the nature of goodness is simply not up for grabs: a Go strategy is good if and only if it wins. Human life does not generally have this feature: there is no objective measure of success in the highest realms of achievement. Certainly not in art, literature, music, philosophy, or politics. Nor, for that matter, in the development of new technologies.

In various toy domains, machines may be able to teach us about a certain very constrained form of creativity. But the domain’s rules are pre-formed; the system can succeed only because it learns to play well within these constraints. Human culture and human existence are much more interesting than this. There are norms for how human beings act, of course. But creativity in the genuine sense is the ability to change those norms in some important human domain. Success in toy domains is no indication that creativity of this more fundamental sort is achievable. 

It’s a knockout

A skeptic might contend that the argument works only because I’m contrasting games with artistic genius. There are other paradigms of creativity in the scientific and mathematical realm. In these realms, the question isn’t about a vision of the world. It is about the way things actually are.

Might a machine come up with mathematical proofs so far beyond us that we simply have to defer to its creative genius?

No.

Computer­s have already assisted with notable mathematical achievements. But their contributions haven’t been particularly creative. Take the first major theorem proved using a computer: the four-color theorem, which states that any flat map can be colored with at most four colors in such a way that no two adjacent “countries” end up with the same one (it also applies to countries on the surface of a globe).

Nearly a half-century ago, in 1976, Kenneth Appel and Wolfgang Haken at the University of Illinois published a computer-­assisted proof of this theorem. The computer performed billions of calculations, checking thousands of different types of maps—so many that it was (and remains) logistically unfeasible for humans to verify that each possibility accorded with the computer’s view. Since then, computers have assisted in a wide range of new proofs.

But the supercomputer is not doing anything creative by checking a huge number of cases. Instead, it is doing something boring a huge number of times. This seems like almost the opposite of creativity. Furthermore, it is so far from the kind of understanding we normally think a mathematical proof should offer that some experts don’t consider these computer­-assisted strategies mathematical proofs at all. As Thomas Tymoczko, a philosopher of mathematics, has argued, if we can’t even verify whether the proof is correct, then all we are really doing is trusting in a potentially error-prone computational process.

Even supposing we do trust the results, however, computer-assisted proofs are something like the analogue of computer-assisted composition. If they give us a worthwhile product, it is mostly because of the contribution of the human being. But some experts have argued that artificial intelligence will be able to achieve more than this. Let us suppose, then, that we have the ultimate: a self-reliant machine that proves new theorems all on its own.

Could a machine like this massively surpass us in mathematical creativity, as Kurzweil and Bostrom argue? Suppose, for instance, that an AI comes up with a resolution to some extremely important and difficult open problem in mathematics.

There are two possibilities. The first is that the proof is extremely clever, and when experts in the field go over it they discover that it is correct. In this case, the AI that discovered the proof would be applauded. The machine itself might even be considered to be a creative mathematician. But such a machine would not be evidence of the singularity; it would not so outstrip us in creativity that we couldn’t even understand what it was doing. Even if it had this kind of human-level creativity, it wouldn’t lead inevitably to the realm of the superhuman.

Some mathematicians are like musical virtuosos: they are distinguished by their fluency in an existing idiom. But geniuses like Srinivasa Ramanujan, Emmy Noether, and Alexander Grothendieck arguably reshaped mathematics just as Schoenberg reshaped music. Their achievements were not simply proofs of long-standing hypotheses but new and unexpected forms of reasoning, which took hold not only on the strength of their logic but also on their ability to convince other mathematicians of the significance of their innovations. A notional AI that comes up with a clever proof to a problem that has long befuddled human mathematicians is akin to AlphaGo and its variants: impressive, but nothing like Schoenberg.

That brings us to the other option. Suppose the best and brightest deep-learning algorithm is set loose and after some time says, “I’ve found a proof of a fundamentally new theorem, but it’s too complicated for even your best mathematicians to understand.”

This isn’t actually possible. A proof that not even the best mathematicians can understand doesn’t really count as a proof. Proving something implies that you are proving it to someone. Just as a musician has to persuade her audience to accept her aesthetic concept of what is good music, a mathematician has to persuade other mathematicians that there are good reasons to believe her vision of the truth. To count as a valid proof in mathematics, a claim must be understandable and endorsable by some independent set of experts who are in a good position to understand it. If the experts who should be able to understand the proof can’t, then the community refuses to endorse it as a proof.

For this reason, mathematics is more like music than one might have thought. A machine could not surpass us massively in creativity because either its achievement would be understandable, in which case it would not massively surpass us, or it would not be understandable, in which case we could not count it as making any creative advance at all.

The eye of the beholder

Engineering and applied science are, in a way, somewhere between these examples. There is something like an objective, external measure of success. You can’t “win” at bridge building or medicine the way you can at chess, but one can see whether the bridge falls down or the virus is eliminated. These objective criteria come into play only once the domain is fairly well specified: coming up with strong, lightweight materials, say, or drugs that combat particular diseases. An AI might help in drug discovery by, in effect, doing the same thing as the AI that composed what sounded like a well-executed Bach cantata or came up with a brilliant Go strategy. Like a microscope, telescope, or calculator, such an AI is properly understood as a tool that enables human discovery—not as an autonomous creative agent.

It’s worth thinking about the theory of special relativity here. Albert Einstein is remembered as the “discoverer” of relativity—but not because he was the first to come up with equations that better describe the structure of space and time. George Fitzgerald, Hendrik Lorentz, and Henri Poincaré, among others, had written down those equations before Einstein. He is acclaimed as the theory’s discoverer because he had an original, remarkable, and true understanding of what the equations meant and could convey that understanding to others.

For a machine to do physics that is in any sense comparable to Einstein’s in creativity, it must be able to persuade other physicists of the worth of its ideas at least as well as he did. Which is to say, we would have to be able to accept its proposals as aiming to communicate their own validity to us. Should such a machine ever come into being, as in the parable of Pinocchio, we would have to treat it as we would a human being. That means, among other things, we would have to attribute to it not only intelligence but whatever dignity and moral worth is appropriate to human beings as well. We are a long way off from this scenario, it seems to me, and there is no reason to think the current computationalist paradigm of artificial intelligence—in its deep-learning form or any other—will ever move us closer to it.

Creativity is one of the defining features of human beings. The capacity for genuine creativity, the kind of creativity that updates our understanding of the nature of being, that changes the way we understand what it is to be beautiful or good or true—that capacity is at the ground of what it is to be human. But this kind of creativity depends upon our valuing it, and caring for it, as such. As the writer Brian Christian has pointed out, human beings are starting to act less like beings who value creativity as one of our highest possibilities, and more like machines themselves.

How many people today have jobs that require them to follow a predetermined script for their conversations? How little of what we know as real, authentic, creative, and open-ended human conversation is left in this eviscerated charade? How much is it like, instead, the kind of rule-following that a machine can do? And how many of us, insofar as we allow ourselves to be drawn into these kinds of scripted performances, are eviscerated as well? How much of our day do we allow to be filled with effectively machine-like activities—filling out computerized forms and questionnaires, responding to click-bait that works on our basest, most animal-like impulses, playing games that are designed to optimize our addictive response?

We are in danger of this confusion in some of the deepest domains of human achievement as well. If we allow ourselves to say that machine proofs we cannot understand are genuine “proofs,” for example, ceding social authority to machines, we will be treating the achievements of mathematics as if they required no human understanding at all. We will be taking one of our highest forms of creativity and intelligence and reducing it to a single bit of information: yes or no.

Even if we had that information, it would be of little value to us without some understanding of the reasons underlying it. We must not lose sight of the essential character of reasoning, which is at the foundation of what mathematics is.

So too with art and music and philosophy and literature. If we allow ourselves to slip in this way, to treat machine “creativity” as a substitute for our own, then machines will indeed come to seem incomprehensibly superior to us. But that is because we will have lost track of the fundamental role that creativity plays in being human.

A philosopher argues that an AI can’t be an artist. Creativity is, and always will be, a human endeavor. By Sean Dorrance Kelly.  Technology Review,   February 21, 2019