Wittgenstein once argued that “if a contradiction were actually found in arithmetic, then that would only prove that an arithmetic with such a contradiction in it could render very good service”. He also claimed that “the laws of logic are arbitrary”.
The Austrian-British philosopher Ludwig Wittgenstein (1889–1951) laid his cards on the table when he warned against “the superstitious dread and veneration of contradiction”. Indeed, at one point, Wittgenstein even advocated “a new logic of contradictions”.
In Lectures, Cambridge 1930–1933, Wittgenstein can also be found stating the following:
“[T]he laws of logic, for example, excluded middle and contradiction, are arbitrary. This statement is a bit repulsive, but nevertheless true.”
To some readers it may seem like a jump to apply a mode of reasoning about mathematical systems (see later section) to the laws of logic themselves. Alternatively, to other readers the application of this line of reasoning (i.e., about mathematical systems) may seem like a (well) logical consequence of such a position.
Thus, if only on this reading of Wittgenstein, it’s literally all about systems or human creations, then why not create a new logic of contradictions?
Take the cases of Nikolai Lobachevsky (1792–1856) and Bernhard Riemann (1826–1866) and their alternative non-Euclidean geometries.
If Riemann and Lobachevsky felt no qualms about breaking free of the system of Euclidean geometry, then perhaps Wittgenstein might have felt at ease about endorsing the adoption — or creation — of alternative logics. Indeed, at this juncture at least, it doesn’t really matter if such alternative logics are self-contradictory or if they tacitly assume — and even tacitly employ — the aforementioned laws of logic.
That said, what if a contradiction could (in Wittgenstein’s own words) “render very good service”?
Wittgenstein’s following words can be found in the Remarks on the Foundations of Mathematics:
“[I]f a contradiction were now actually found in arithmetic — that would only prove that an arithmetic with such a contradiction in it could render very good service.”
Wittgenstein also seemed to be a dialetheist when he argued that “an arithmetic with such a contradiction in it could render very good service”. That is, philosophers, mathematicians and even laypersons can indeed work with contradictions.
It’s always possible, of course, that a mathematician (or, more likely, a philosopher) may say that if there were any genuine contradictions, then — by definition — they simply couldn’t “render very good service”. So, in that case, we’d be wrong to class them as workable contradictions at all.
So how, exactly, could we make use of contradictions?
Well, dialethic logicians and philosophers do. What’s more, they cite various concrete examples. For example, dialethic logic can make sense of inconsistent theories in science, moral and political beliefs which clash with each other, the paradoxes of set theory, etc. (See note at the end of this essay.)
However, I don’t really believe that Wittgenstein had the same kinds of thing in mind as the dialetheists. That’s primarily because his central positions on contradictions and the “arbitrary” nature of the laws of logic were purely abstract and metaphilosophical.
Wittgenstein also advanced an (as it were) mathematics-is-use (or pragmatic) position when he argued the following:
“The point is that we all make the SAME use of it. To know its meaning is to use it in the same way as other people do. ‘In the right way’ means nothing.”
When Wittgenstein used the phrase “in the right way” above, it can be taken to imply that some mathematicians and philosophers believed that there’s a right way that’s completely separate from mathematicians and their systems (or completely separate from conventions, symbols and their meanings, etc.). That is, they believed that there could be a right way without literally anyone actually knowing anything about it. (This is the position of mathematical realism.)
Yet Wittgenstein might well have asked what purpose this hidden right way serves.
In other words, if no one actually knows that right way, and no one can even say anything about it, then what role does it have? Indeed can it even be known to exist (or have being) at all?
If we return to Wittgenstein’s position on mathematical systems.
Systems, Systems and More Systems
As found in the book Wittgenstein: Lectures, Cambridge 1930–1933, Wittgenstein wrote (or actually spoke) the following words:
“Reason only applies within a system of rules [] It is nonsense to ask for reasons for the whole system of thought. You cannot give justification for the rules.”
Wittgenstein believed that (to use his own words) the “game” of contradiction-spotting was “pointless” because any such paradoxes or contradictions which do occur (or which even exist) only do so because they’re the consequences of the “system of rules” which the logicians and mathematicians created in the first place. That is, these contradictions and/or paradoxes don’t exist separately from such systems.
That shouldn’t be a surprise.
Take Kurt Gödel’s incompleteness theorems.
These theorems only apply to mathematical systems. What’s more, the so-called “Gödel truths” only occur within the mathematical systems of which they are parts.
So systems (whether logical or mathematical) are in the driving seat here.
Moreover, if such systems are indeed creations — human creations, then one can see where Wittgenstein was coming from.
Wittgenstein’s position seems to have been that those in search of paradoxes and contradictions believed that such things existed separately from the systems in which they were embedded — indeed separately from all human minds.
So even if it can be accepted that such contradictions and paradoxes are hidden, then they’re still only hidden within the systems created by logicians and mathematicians.
The upshot here is that all this is loosely equivalent to someone creating an antidote which, unknown to the creator, would also bring about brain damage when administered. Thus the creator doesn’t know that it can bring about brain damage. However, the antidote is still his own creation. And without the administration of his antidote, such specific cases of brain damage would never occur.
Wittgenstein became even more concrete and radical about systems when he wrote the following words:
“[T]here is nothing there for a higher intelligence to know — except what future generations will do. We know as much as God does in mathematics.”
In other words, there are no mathematical equations or timeless abstract numbers in the Platonic ether waiting for mathematicians to discover.
That said, it’s of course possibly — or actually — the case that mathematical structures or equations contain content that no mathematician alive today is actually aware of (i.e., just like the aforementioned antidote).
For example, in The Emperor’s New Mind, Roger Penrose wrote:
“The Mandelbrot Set is not an invention of the human mind: it was a discovery. Like Mount Everest, the Mandelbrot Set is just there!”
However, even in this case it will still require mathematicians (or computer programmes) to bring out that now-hidden content.
Of course a Platonist (or mathematical realist) may now ask this question:
How can you accept that there’s anything hidden (or out there) at all before such so-called “creations” of mathematicians?
In other words, according to Wittgenstein’s (as it were) anthropological position (see here), there can’t be anything hidden. That’s because that something would need to be independent of all mathematicians and all mathematical systems.
Wittgenstein himself uses the words “higher intelligence” as a virtual synonym for Platonic realm. So just as there’s no Platonic realm which contains all mathematics and all mathematical truths (or equations), so there’s no higher intelligence that (as it were) has all of mathematics in His head.
So it can be argued that, at least as far as contradictions and paradoxes are concerned, Wittgenstein was a systemist (rather than, say, a structuralist). That is, he believed that mathematicians and logicians (as well as philosophers of mathematics) can never step outside of the systems they construct.
[See my ‘Kurt Gödel, Vacuous Paradoxes and Self-Reference’ and ‘Why Empty Logic Leads to the Liar Paradox’ in which it’s argued that the discussed paradoxes are a result of the systems in which they are embedded.]
Wittgenstein’s Anti-Platonism
Platonism has already been mentioned a couple of times in this piece. So it’s important to note here that not only mathematical Platonists have had problems with Wittgenstein’s philosophy of mathematics.
In any case, the term “Platonism” is — at least partly — used for convenience’s sake in that the words mathematical realist could also have been used.
That said, Wittgenstein does indeed explicitly express his anti-Platonism in the following passage (which was aimed at Alan Turing):
“If you say, ‘The mere fact that a proof could be found is a fact about the mathematical world,’ you’re comparing the mathematician to a man who has found out something about a realm of entities, the physics of mathematical entities.”
Wittgenstein was arguing that the Platonist (or Alan Turing himself) believes that there’s a “mathematical world” which contains “entities”. And this is equivalent to a someone discovering a physical world (or a new country) which is full of physical entities (e.g., animals, trees, mountains, rivers, etc.).
Yet this is a misleading analogy.
In fact the problem is that Platonists don’t even see it as being an analogy at all!
Most people would accept that the physical world (or country) in this analogy would exist regardless of its discovery. However, nothing is known about it. Despite that, it would still exist (or be real).
So can we glide so easily over from that position on an undiscovered physical place (or country) to the existence of an abstract entity in an abstract mathematical world which has so far been undiscovered?
Wittgenstein believed that such a mathematical entity is actually created, constructed or even invented — not discovered. That is, it is brought into existence by the very act of construction (or invention).
This seems to make Wittgenstein an intuitionist or constructivist. (See mathematical constructivism and mathematical intuitionism.) However, as with all “Wittgenstein scholarship”, there are some scholars who accept this conclusion and just as many scholars who don’t. (This isn’t the place to run through what the legions of “Wittgenstein experts” have said on this subject.)
All that said, we still have at least one clear example of Wittgenstein expressing his literal (for want of a better word) constructivism.
Constructivism and Wittgenstein’s Roads
Take Wittgenstein’s image of an unbuilt road. Wittgenstein said:
“Professor Hardy says, ‘Goldbach’s theorem is either true or false.’ — We simply say the road hasn’t been built yet.”
Put simply, Goldbach’s theorem (or “conjecture”) is neither true nor false until a road has been built which leads to it being known to be either true or false. That is, the truth or falsity of Goldbach’s theorem must be constructed! After all, how could its truth or falsity be established without such a construction?
Yet there’s a rather obvious response to that last paragraph.
That response is to argue that although human beings need a road that leads to the destination that is the truth or falsity of any theorem, that truth or falsity still exists even without such a road. So, sure, we don’t know if it’s true or false without such a road (or construction), but it’s still (ontologically) either true or false without it.
Indeed Wittgenstein himself conceded that
“[a]t present you have the right to say either; you have a right to postulate that it’s true or that it’s false”.
Of course that’s not much of a concession.
That is, until you built the road, you simply don’t know if a theorem is true or false. So the claim that it’s either true or false regardless is (almost) without substance. Again, according to Wittgenstein’s own logic, it can’t be either true or false until the (or a) road to it is built…
Yet how can a road lead to somewhere that doesn’t, as yet, exist?
In this analogy, then, the building of the road seems to be (at least part of) the actual creation of the place it leads to!
In any case, Wittgenstein elaborated in this way:
“For which road you build is not determined by the physics of mathematical entities but by totally different considerations [].”
In other words, these analogical roads (as just hinted at) literally construct the “mathematical entities”: they don’t “discover them”.
To continue with Wittgenstein’s analogy.
We are free to construct any road we like. Yet this “isn’t because the mathematics says that the road goes there”. It’s “because the road isn’t built until mathematics [or a mathematician] says it goes there”.
It’s of course the case that the place itself doesn’t (as it were) tell the roadbuilders to build the road to it. To use Wittgenstein’s own words: the place doesn’t “say [] that the road goes there”. Instead, a decision has to be made to build the (literal) road. And it’s the roadbuilders who build the road, not the place it’s going to.
Surely (as already hinted at) this analogy is too strained.
That’s because no road is built unless whatever it leads to already exists. In the case of literal roads, no road is built unless it leads to somewhere which already exists (even if not a town or city, then just a given place or area). The road, either during or after being built, doesn’t literally bring into being the place it leads to. The road is built precisely because such a place already exists…
In the end, then, perhaps this is a (scare-quoted) “misreading” of Wittgenstein’s words. (This is often the case when it comes to interpreting the Austrian philosopher’s very-odd prose.)
Finally, if the grammar (or analogy) of a road constructing something is a little odd, then settle for the simple idea that mathematical entities are constructed, not discovered.
All the above contains echos of Alan Turing’s well-known dispute with Wittgenstein.
Alan Turing’s Bridge
In 1939, Alan Turing (1912–1954) attended Wittgenstein’s ‘Lectures on the Foundations of Mathematics’ at Cambridge University. At these lectures, Wittgenstein discussed mathematical contradictions.
More relevantly, Turing strongly disagreed with Wittgenstein’s argument that mathematicians should happily allow contradictions to exist within mathematical systems.
When it came to Wittgenstein himself, he stressed two things:
1) The strong distinction which needs to be made between accepting contradictions within mathematics and accepting contradictions outside of mathematics.
2) The supposed applications and consequences of these mathematical contradictions and paradoxes outside mathematics.
As for 1) above, Wittgenstein said (as quoted by Andrew Hodges):
“Why are people afraid of contradictions? It is easy to understand why they should be afraid of contradictions in orders, descriptions, etc. outside mathematics. The question is: Why should they be afraid of contradictions inside mathematics?”
Wittgenstein can be read as not actually questioning the logical validity or status of these paradoxes and metatheorems. He was making a purely philosophical point about their supposed — as well as numerous — applications and consequences outside of mathematics. (These consequences — if not always applications — usually include the subjects of consciousness, human uniqueness, God, human intuition, meaning, purpose, the universe, religion, arguments against artificial intelligence, etc.)
In terms specifically of contradictions and paradoxes.
All the above means that if mathematics is a human invention, then any contradictions and paradoxes there are (i.e., within mathematics) must be down to… mathematicians and their systems. And if they’re down to mathematical systems alone, then they aren’t telling us anything about the physical world (which includes Alan Turing’s bridge — see later) or even about a Platonic world of numbers.
Turing stated (to put it simply) that if there are any contradictions embodied in the mathematics used to construct a bridge, then that bridge would end up falling down! Yet, according to Wittgenstein, the bridge wouldn’t fall down even if the maths used in its construction did contain a contradiction. (This last statement may actually be an overextension of Wittgenstein’s position.)
Turing himself stated the following words:
“The real harm will not come in unless there is an application, in which a bridge may fall down or something of that sort [] You cannot be confident about applying your calculus until you know that there are no hidden contradictions in it.”
It does seem bizarre that Turing should have even suspected that “hidden contradictions” (his debate with Wittgenstein was primarily about the Liar Paradox) could lead to a bridge falling down. That is, Turing believed — if somewhat tangentially — that a bridge may (or even would) fall down if some of the mathematics used in its design somehow instantiated a paradox or a contradiction.
But surely it’s hard to imagine the precise route from any paradox (especially the Liar Paradox) to the practical (or concrete) applications of mathematics of any kind — let alone to the building of a bridge and then that bridge falling down.
Yet perhaps Turing’s argument is that there could never be such a concrete manifestation. And that’s precisely because the builders wouldn’t even get as far as finishing the bridge because the contradiction would have already shown itself.
So what about Wittgenstein’s response to this line of reasoning by Turing?
Wittgenstein responded by saying that “[b]ut nothing has ever gone wrong that way yet”. That is, no bridge has ever fallen down due to a paradox or contradiction in mathematics.
More generally, of course Wittgenstein was well aware of many of the arguments against embracing contradictions. And he played all of them down.
For example, when discussing the possibility of anything and everything being deduced from a contradiction (see the principle of explosion), Wittgenstein stated the following:
“[W]ell then, don’t draw any conclusions from a contradiction; make that a rule.”
Wittgenstein wasn’t actually arguing that everything and anything couldn’t be drawn from a contradiction. He was simply arguing that we shouldn’t draw such conclusions. That is, such ridiculous and/or irrelevant conclusions (see irrelevant conclusion) may indeed be drawn — so simply don’t draw them!
Interestingly enough, there may be a (as it were) Platonist assumption hidden in Wittgenstein’s riposte above. That’s because these bizarre conclusions must — or simply may — still exist in a (as it were) Platonic space. However, we still have a choice as to whether or not to draw (or deduce) them.
It’s worth noting here, then, the many comments written and spoken about “everyday mathematicians” not caring too much about “the foundations of mathematics”, “hidden contradictions” or, indeed, Kurt Gödel’s incompleteness theorems. That’s because — as many mathematicians and others have argued — such things simply don’t impinge on everyday mathematics. (I’m willing to be told otherwise by mathematicians, rather than by philosophers.)
All this is summed up by the mathematician and physicist Alan Sokal when he focussed on Gödel’s metatheorems.
Firstly, Sokal stressed the difference between “metatheorems” and “conventional mathematical theorems” in the following way:
“[] Metatheorems in mathematical logic, such as Gödel’s theorem or independence theorems in set theory, have a logical status that is slightly different from that of conventional mathematical theorems.”
And it’s largely because of that difference that Sokal continues with these words:
“It should, however, be emphasized that these rarefied branches of mathematics have very little impact on the bulk of mathematical research and almost no impact on the natural sciences.”
So if such metatheorems (if not the logical sport of contradiction-spotting) have almost zero “impact on the natural sciences”, then surely they have less than zero impact on, say, Turing’s bridge.
*******************************
Note: Wittgenstein’s Dialetheism?
Dialetheists “embrace contradictions”. (See ‘Setting Contradictions Together’.)
Take the dialetheist Graham Priest (1948-).
Is Graham Priest’s own dialetheism about the world (i.e., an ontological position) or is it about what we can — or even should — say about the world? Clearly, if it’s about the world, then it’s far more radical than merely being about our epistemic, scientific or logical takes on the world.
These questions are asked because there’s virtually nothing in Priest’s work where he explicitly states that contradictions exist in the world. In basic terms, it’s all about how “contradictions can be embraced” in logic, epistemology and the philosophy of science — as well as in science itself.
Yet Priest does argue that dialetheism isn’t (purely) a formal logic. He states that it’s “a thesis about truth”.
One way to put all this is in terms of set-theoretic paradoxes, as discussed by Bryson Brown.
Brown says that “the dialetheists take paradoxes such as the liar and the paradoxes of naïve set theory at face value”. That is, it may be the case that dialetheists choose — for logical and/or philosophical reasons — to accept paradoxes and/or contradictions even though they also believe that, ultimately, they aren’t true of the actual world. (What of the abstract world?)
As dialethic logicians have stated, we can accept inconsistent scientific theories if they still prove to be useful and have predictive consequences. In fact scientists (especially physicists) have been fine with this situation for a long time. Priest himself puts it this way:
“Inconsistent theories may have physical importance too. An inconsistent theory, if the inconsistencies are quarantined, may yet have accurate empirical consequences in some domain. That is, its predictions in some observable realm may be highly accurate. If one is an instrumentalist, one needs no other justification for using the theory.”
Bryson Brown also says that our
“best theory of the structure of space-time, Einstein’s general theory of relativity, is inconsistent with our quantum-mechanical view of microphysics”.
Brown also mentions the Danish quantum physicist Niels Bohr within this context.
So did Bohr believe that there were inconsistencies — let alone contradictions — actually in the world? Philosophers have disagreed on this. However, it can be argued that Bohr (as an epistemic anti-realist) might have found it difficult to grasp what an inconsistent world (or any aspect of the world) would — or could — be like. This is at least the case when it comes to worldly contradictions. That is, it is hard to see how A & not-A can be the case at one and the same time (despite the wave/particle duality, etc.).
This means that we need to decide if embracing theoretical inconsistencies (in science, logic, etc.) also means embracing worldly contradictions.
Bryson Brown also acknowledges the fact that, in general, we “accept non-trivial but inconsistent obligations and/or beliefs”. Here again this is really an epistemic and moral matter. More explicitly, it isn’t being argued that, for example, we can believe that “Cats have tails” and also believe that “Cats don’t have tails” at one and the same time. No one believes that an individual cat both has a tail and doesn’t have a tail. So, in that case, A works may work in one context; and ¬A may work in another context. In other words, “Cats have tails” (which, grammatically, doesn’t entail “All cats have tails”) is about cats in the plural; whereas any statement about an individual cat won’t be about all cats. Yet an individual cat both having and not having a tail means that the statements “Cats have tails” and “Cats don’t have tails” are both true. (This, admittedly, is also about cats in the plural.)
In Priest’s own terms, this truth-value indeterminacy (though p’s being indeterminate is itself be seen as a truth value) results in a scientific theory (or logical system) being inconsistent. And that inconsistency is a result of the logician or scientist being unable (at least at a given point in time) to erase certain inconsistencies from his system or theory. (This is something that’s often been said about Bohr’s well-known model of the atom.)
What we say about the world (whether in science, philosophy, mathematics, logic or everyday life) may well be consistent or inconsistent. However, the world itself can neither be consistent nor inconsistent.
Yet if dialethism is about the world, then Priest (to use his own example — see here) could indeed be both in New York and not in New York at one and the same time! If it’s about what we can say, on the other hand, then it’s simply a logic that can helpfully capture and formulate such things as the inconsistencies in scientific theories. Alternatively, dialetheic logic can advance the pragmatic option of seeing two contradictory positions as both being true (or simply usable) — at least for the time being!
Thus nearly all Priest’s examples of dialetheic contradictions are really about human perceptions of — or attitudes towards — the world, not the world itself. They also concern inconsistencies in scientific and (another area mentioned by Priest himself) legal theories about the world. This makes dialetheism either a position in epistemology and/or one in the philosophy of science. So, if all that is the case, then surely dialetheism isn’t a “robust ontology” which happily embraces contradictions-in-the-world at all.
Of course Priest’s next move may be to question this possibly bogus distinction between the world and our statements about — and knowledge of — the world. Actually, he does question this distinction. He even uses the term “social constructionism” about his own position.
All in all, then, it can be said that Priest uses positions in epistemology and the philosophy of science as means to back up (or defend) his (seemingly) very-radical logical dialetheism.
But what about Wittgenstein himself?
Wittgenstein wasn’t a dialetheist primarily because he wasn’t talking about contradictions-in-the-world, but contradictions-as-creations-of-systems. Indeed it can be suspected that Wittgenstein would have had a problem with the very notion of contradictions-in-the-world.
However, it’s still not clear (to me at least) if dialetheists actually do believe that there are contradictions-in-the-world.
Take the words of Bryson Brown again, in his paper ‘On Paraconsistency’. He writes:
“[Dialetheists] hold that the world is inconsistent, and aim at a general logic that goes beyond all the consistency constraints of classical logic.”
Deriving the notion of an inconsistent world from our psychologistic and/or epistemological limitations (as well as from accepted notions in the philosophy of science) is problematic (see psychologism). In other words, the epistemological acceptance of inconsistencies can’t also be applied to the world itself.
So it may be the case that dialetheists choose — for logical and/or epistemological reasons — to accept paradoxes and contradictions (just like Wittgenstein) even though they also believe that, ultimately, they aren’t true of the world (or of bridges). Then again, Brown continues by saying that dialetheists “view these paradoxes as proofs that certain inconsistencies are true” …
But true of what?
True only of the logical, linguistic and systemic expressions of the paradoxes or true of the world itself?
Nonetheless, Graham Priest does mention “reality” when he talks of consistency and inconsistency. For example, when discussing the virtue of simplicity, Priest asks the following question:
“If there is some reason for supposing that reality is, quite generally, very consistent — say some sort of transcendental argument — then inconsistency is clearly a negative criterion. If not, then perhaps not.”
As it is, how can the world be either inconsistent or consistent? Indeed a position can be taken on this which is similar — or parallel — to Baruch Spinoza’s philosophical — and perhaps moral — point that the world can only… well, be.
All this may be betraying my implicit (naïve) realism. Yet it must still be stated that what we say about the world (whether via science, philosophy, mathematics or logic) may well be consistent or inconsistent. (We may also say, with Spinoza, that the world is “beautiful” or “ugly”.) However, the world itself — along with its bridges — can be neither consistent nor inconsistent.
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