The Ladder of Abstraction (with added rungs)

The following  logically belongs in the “Abstraction” section of our essay Consilience in Mathematics.  But as that effort is growing overlong,  we begin to cultivate here a particular idea  building upon that of abstraction simpliciter :  namely, the tendency, in modern mathematics -- and indeed this may serve virtually as the defining characteristic of modern (even: modernist) mathematics -- to abstract from any given abstraction, layer upon layer, rise upon rise, to a virtual (topless/cloud-topped) Babel, reaching to the Beyond.

(Oh, and here again we have a term from the arts, Modernism, which, as it includes “abstract art”, metaphorically applies to mathematics.  Compare our earlier essay on Minimalism in Mathematics.)

In normal practice, mathematicians mostly talk to one another -- and indeed, mostly just to those within their own hyperspecialized neck of the woods.  But occasionally, one writes an undergraduate textbook, and thus must descend to earth, if only for the nonce, and address the laity.  Thus:

This “intrinsic” formulation of Calculus, due to its greater “abstraction”, and in particular  to the fact that, again and again, one has to leave the initial spaces, and to climb  higher and higher  to new “function spaces” (especially when dealing with the theory of higher derivatives), certainly requires some mental effort, contrasting with the comfortable routine of the classical formulas.  But we believe that the result is well worth the labor, as it will prepare the student to the still more general idea of Calculus on a differentiable manifold.

-- Jean Dieudonné, Foundations of Modern Analysis (1960), p. 141

We dub this the “ladder of abstraction”, taking the phrase from our teacher of yore. Referring likewise to ascent into functions-of-functions, and function spaces, and functions from one function space to another, and to the duals of all that:

Detached from any context, this construction is a pointless formality.  But as we move up the ladder of abstraction, we find that constructions such as this  become commonplace …

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 43

The metaphor of ascent is well attested.  Gödel speaks of

the infinite series of ever stronger axioms of infinity, each of which expresses a new idea or insight.

-- quoted in Hao Wang, From Mathematics to Philosophy  (1974), p. 325

A mathematician writes of

the inferential staircase  leading from the laws of physics  to the world that lies about us …

-- David Berlinski, “The End of Materialist Science”, collected in:  The Deniable Darwin (2009), p. 160

~

Saunders MacLane,  in his book Mathematics:  Form and Function (1986), p. 36ff, has a section called “Mathematical Activities”, structured somewhat like our own in the Consilience essay.  Some of the topics are the same (analogy, abstraction, generalization), while others, not relating to consilience especially, differ (conundrums, axiomatization, proof).  One, intrinsic structure, seems to relate to consilience, but is only briefly developed; and the last, completion, we have treated under the more Quinean label of rounding out.

Now, abstraction and generalization are related notions, but neither entails the other.  MacLane acutely adduces the example of group theory.  Originally, this grew out of the concrete examples known as groups of transformations.  Later, algebraists abstracted into abstract groups.  Whether a generalization has thereby been achieved, is (as Chomsky likes to put it) “an empirical question”;  and in this case, it turns out, it has not. “No new groups turn up in this process, in view of the famous theorem of Cayley, which asserts that every (abstract) group is isomorphic to a group of transformations.”  Thus, in this case, the ladder of abstraction has only one rung.  (By contrast, abstract rings do turn out to generalize upon their original model, rings of integers.)

~

Seeking analogues of the Ladder of Abstraction  outside of mathematics proper, I happened upon this:

Quine suggests that levels of abstractness, modeled on Russell’s Theory of Types, might be established.  “In the beginning  there are only concrete objects.”  These constitute type zero and are the values of bound individual variables.  “To be is to be a value of a variable.”  Next comes first-order classes and relations:  they constitute entities of type 1 and are the values of bound predicate variables.  Classes of classes, and relations, constitute entities of type 2;  and so on.

-- Harold Lee, “Discourse and Event”, in: Hahn & Schilpp, eds., The Philosophy of W. V. Quine (1986), p. 297

The resemblance to Russell’s Theory of Types had not escaped me, but I rejected mention of it, since, rather than leading -- as the Ladder does -- to ever greater depth (the metaphor is here in distress -- maybe think of it as a ladder down a mineshaft), it seems to lead mostly to More of the Same.  In other words, forming those strata, as described above, is less like the dizzying and ethereal Abstract Ascent of mathematics, than simply forming new sets via the Power Set operation (a new and larger set consisting of all the subsets of the original set).  Now this, if we start with a finite set, leads absolutely nowhere.  It’s just like counting.  If you start with the whole of the Natural Numbers, now the Power Set operation does become more powerful, leading to new and incomparable levels of infinity.    Whether this leads to true new depth, or is rather a mere formal exercise, I do not know, since I lack all intuition of any infinities beyond the countable, let alone the Power of the Continuum or Measurable Cardinals.  Perhaps it does;  espresso-sodden Berkeley conversations about Quality emerging out of Quantity, return to mind.

Still, I am inclined to doubt it.  The very fact that the fellow can say “and so on”  virtually proves as much.  For there is no “and so on” to true mathematical abstraction.   There is nothing mechanical about such ascent -- it is more like a miracle.  You can proceed only one step -- nay rather, one leap at a time;  and the interval between leaps may take decades or even centuries.   Above the calculus lies Function Theory;  above that, Topology.  Above that, Algebraic Geometry. Far, far above us, hovers Category Theory, unreachably aloft.  And far, far above and beyond that, soars Topos Theory.  What comes next  is known only to angels.

To vary Nestroy’s celebrated epigram -- “Bis die Topologie gehts noch, aber von da bis sheaf theory  zieht sich der Weg.”

Additionally, Quine introduced the term semantic ascent.  There is some similarity to Gleason’s ladder of abstraction, but the ascent doesn’t go very high, and Quine himself -- perhaps surprisingly for a logician -- is wary of the upper reaches, preferring basic-level entities  behaviourally grounded.

Here the Russian author A. D. Aleksandrov, instead of envisaging a ladder,  uses the metaphor of layers  or (appropriately enough) of nesting, like Russian dolls, in the procession to affine or projective geometry and on to topology:

The properties of space are stratified … with respect to their depth and stability.  The ordinary Euclidean geometry was created by disregarding all properties of real bodies other than the geometrical;  here we perform yet another abstraction within geometry.

-- Aleksandrov et al, eds, Mathematics: Its Content, Methods, and Meaning (publication in the original Russian: 1956;  Eng. tr. publ. 1963), vol. III, p. 133

~

The more I think about it, the more this Ladder of Abstraction idea seems possibly fruitful.  Not so much as in the Theory of Types, but as in the scala naturae, which encompasses angelology.  (Compare also graded algebras.)

By contrast, mere ungraded “abstractness” in itself is of little interest. Thus, to take MacLane’s Group Theory example:  the so-called “abstract” groups (MacLane himself uses the sneer-quotes here) mean to lift aloft from Groups of Transformations, in that they retain the laws (associativity, inverses, and all that) while becoming agnostic as to the nature of the elements of the group.  But, first of all, groups of transformations are, compared with, say, pickles, already quite Abstract;  so the word adds, really, nothing.  Indeed, as soon as you say that two apples plus two apples are four apples, and that in the same sense  two penguins plus two penguins make four penguins (well, and a few more, after a while, if the sex mix is right), you are already indulging in such abstraction.

~

I tried looking up “abstraction” in the index of the various math textbooks and philosophy treatises on my shelves, and basically came up with  bupkes.  Thus, in Dummett’s omnibus volume, Truth and Other Enigmas (1978), we find no reference to abstraction per se, let alone to the Ladder of Abstraction, but only to “abstract objects” -- i.e., pickles versus the Meaning of ‘Pickle”,  the Idea of a Pickle, the set-containing-a-pickle, the… sandwich containing a pickle, the -- but enough.  Mathematics is so far beyond this, no comment is required.

~

The ethic -- even, the aesthetic -- of abstraction for its own sake, sociologically chronicaled here (“On Vulgar Numbers”), eventually evoked a backlash.

The Bourbaki group sought to present the entire abstract structure of all mathematical concepts in one set of volumes, the Eléments de Mathématique. In that treatise, the real numbers, which most of us regard as a starting point, only appeared midway into the series, as a special “locally compact topological group”.

An opposing idea, promoted especially in the Russian school, is that a few well-chosen examples can illuminate an entire field.

-- David Mumford, Forward to Mircea Pitici, ed., The Best Writing on Mathematics 2012, p. xv

~

For the latest in fine reading, check this out:

http://www.linguasacrapublishing.com/justice.html

For more about abstraction, here:
         http://worldofdrjustice.blogspot.com/search/label/abstraction

The “Idea” Idea (with an excursus on ideation and subvocalisation)

Much of the most important and vital work done in the last half-century  depends [not upon experiment or brute calculation, but] upon new ideas;  and new ideas are notoriously exceedingly difficult to grasp.

-- Louis J. Mordell, Reflections of a Mathematician (1959), p. 11

We previously stated that mathematics is best characterized as the science, not of number, but of structure (or of pattern -- at this level of generality, either term will do).   As MacLane phrases it:

This chapter introduces the idea of the formal  in terms of certain basic structures:  Set, transformation, group, order, and topology.  With Bourbaki, we hold that Mathematics deals with such “mother structures”.  Against the historical order, we hold that they arise directly from the basic stuff of Mathematics.

Saunders MacLane,  Mathematics:  Form and Function (1986), p. 7

That last bit, you will note, is unabashedly Platonist, counterposing contingent human praxis  to transcendent time-independent Truth.  (We discuss this contraposition here.)

Voilà  le hic

But beyond that, or rather as an animating force within it,  and distinguishing mathematics from such structure- or pattern-centered enterprises as architecture or the plastic arts, is the central role of ideas. 

MacLane puts the matter well.  Re the derivation of Hamilton’s equations from Lagrange’s:

What appears as a trick is in fact an idea -- an idea which must have been clear to Hamilton when he did it.  But we claim that in general  most of the formal tricks appearing in Mathematics  are really ideas in disguise -- ideas presented as manipulations  because the manipulations can be made explicit, while the ideas are a bit nebulous.

-- Saunders MacLane,  Mathematics:  Form and Function (1986), p. 284

In a previous series of essays, we put forward certain particular “mother ideas”.  Here we reserve a meditation-space  for musing about “Ideas -- the very idea”.

~

Hadamard comments on Rodin’s testimony that, throughout the process of sculpting, he must keep the “global idea” in mind, even while working on the smallest details;  and that “this cannot be done without a very severe strain of thought.”

I do not feel that I have understood [a mathematical argument] as long as I do not succeed in grasping it in one global idea; and, unhappily, as with Rodin, this often requires a more or less painful exertion of thought.

-- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 65

Hadamard scoffs at the account given by Souriau in his Théorie de l’Invention:  “Does the algebraist know what becomes of his ideas when he introduces them, in the form of signs, into his formulae?  Undoubtedly not,”  but just turns the crank of mechanical calculation.  Apparently Souriau never consulted an actual mathematician, says Hadamard:  the mathematician trusts his idea, his insight, his intuition, more than he does his calculations, which after all are not infrequently in error  (Hadamard confesses that he, like Poincaré, was but an indifferent numerical calculator):  If these clash, you first redo the calculations, before tossing overboard the Idea that motivated the whole thing.

~

Ideation and subvocalisation

Hadamard then makes an excursus  rather off the path our our principle inquiry;  yet we shall follow him a little ways.  He confronts the question of whether language be the key to thought;   and waxes indignant at those who, like Max Müller, dogmatically assert that, without language, thought itself must needs collapse:

I had a first hint of this when I read in Le Temps (1911):  “The idea cannot be conceived otherwise than through the word, and only exists by the word.”  My feeling was that the ideas of the man who wrote that  were of a poor quality.

-- -- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 66

Likewise, the behaviorist J.B. Watson says somewhere that “thinking is nothing but our talking to ourselves”.

The devotees of this position  point to the dual meaning of the early Greek word logos -- ‘word, language’ and ‘reason, thought’;  and would by implication deny that our diminutive and prickly friend, the humble hedgehog, could really know One Big Thing or even a little weentsy one.

Hadamard, by contrast, is virtually a militant in the opposite camp:  “I fully agree with Schopenhauer when he writes, ‘Thoughts die  the moment they are embodied in words.”  This even applies to algebraic symbolism:  too cumbersome to actually think with;  you mostly only use them when checking your work.

The Dutch Intuitionist mathematician L.E.J. Brouwer is of similar mind:

De woorden van uw wiskundig  betoog zijn slechts de begeleiding van een woordloos wiskundig bouwen …

 

(Caption quotation from Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), p. 38.)

The Neothomist philosopher Etienne Gilson  seconds the opinion of his countryman:

Si un linguiste me dit que c’est notre langue qui modèle d’abord  le monde que nous pensons,  je sais qu’il ne me parle pas en linguiste, mais en philosophe, qui se dispenserait d’ailleurs de me donner aucune justification philosophique de son opinion.  Non seulement je ne sais pas si elle est vraie, mais je ne sais même pas pourquoi elle lui semble vraie.

-- Etienne Gilson, Linguistique et philosophie (1969), p. 51

A noted Freudian psychiatrist agrees:

Every single thought, before formulation, has gone through a prior wordless state.

-- Otto Fenichel,  The Psychoanalytic Theory of Neurosis (1945), p. 46

A contemporary philosopher goes even further:  some ideas may be not only pre-linguistic, but even pre-conscious:

We may not be aware of our ideas.  An idea  in this sense  is a tendency to accept routes of thought .. that we may not recognize in ourselves, or even be able to articulate.

-- Simon Blackburn, Being Good (2001), p. 3.

The epigram "We may not be aware of our ideas" is deliberately paradoxical.  Blackburn means "idea", not in the sense of the completely conscious  "I have an idea, let's...", but of something like the often tacit metaphysical underpinnings of mentation and investigation, which we treated of earlier.  -- Blackburn extends this notion (in a way reminiscent of, but antedating, Freud):  "A permanent strand in Christian thought  is that we have no insight, or even lie to ourselves, about our heart's desires." (id., p. 30)

We close this excursus with an epigram of William Hamilton  which Hadamard quotes:

Speech is thus not the mother,

but the godmother of knowledge.

~

The reason such musings lie off our main track, is that we are largely uninterested in psychology, or thought-processes, or any of the hunches & hiccups that fallen Man is heir to  as he struggles to comprehend all that His hand hath made.  With Hadamard, we conceive that there are cognitive activities for which vocalization is neither required nor especially helpful:  say, playing Go, or basketball.  

There is an epigram, variously ascribed, that has always fascinated me:

“How can I know what I think

 until I see what I say ?”

On the face of it, this would appear to be anecdotal evidence for the thought-needs-language thesis.  But upon nearer inspection, it might argue rather the opposite:  That thought rose from some wordless region of the self, and only became an object to critical consciousness after having been concretized by transformation into words.

For us, the key question is to what extent an Idea -- one worthy of the majuscule -- can even be adequately expressed in our language.   Certainly the higher mathematics cannot be expressed in ordinary human language.  It has invented for itself a more or less arcane system of signs, obeying no human syntax;  you may, if you like, par abus de langage, call that too a “language”, but it is no natural human language, but rather an aide-mémoire cobbled together to express ideas that observe their own semantics, call that language or not.   Hadamard himself attests that human language does not serve him especially well, when he must express mathematical ideas.  Whenever he must hold forth on a mathematical topic, even one of his own devising and thus, to him, abstractly clear as a bell, he must write out the text of his lecture beforehand, lest he be left gasping and groping for words.

There is another old adage, current among linguistic philosophers:

“Whatever can be meant

can be expressed.”

At this point we hear the shade of that crusty critic of Le Temps, growling:  All that you mean, maybe. 

~

Let us put the point even more starkly.  Ask Not  (we channel Kennedy here) whether our (necessarily human versions of) ideas  could be adequately communicated to some other rational species.  Ask whether the Idea, as pre-existent in Platonic paradise, has been adequately incarnated in us.

(There now swims within my vision  the image of a category-theoretic Universal Object, with arrows slanting downwards  this way and that, as in Blake’s great painting.)

~

This is becoming interesting.  Hoping that your appetite has been whetted as well, we link to a couple of math-related installments of the “Any Ideas?” series:

Induction

The free invention of structures

Actual Infinity

~

We have tried to outline a capitalized or pregnant sense of the everyday word idea, which in most contexts certainly does not bear such freight.  (“I’ve got an idea, let’s go get pizza.”)  There is, however, another sense, which is still scientific/intellectual, yet which bears no Platonic or foundational flavor:  what is sometimes called a “bright idea”.   A bright idea is what causes a light-bulb to appear over the cartoon character’s head.  And it does represent some genuine cleverness, though its success is by no means guaranteed (and in the case of Donald Duck, will almost certainly come to grief.)

This more powerful form of inductive construction  can be deduced rather simply from the older form.  The trick is to construct, not the sequence of values, but the sequence of partial functions…

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 145

A “trick” is to an idea  as tactics is to strategy. 

Similarly:

We could prove the inequality by a limit argument from the known inequality for finite sums, but the following reasoning involves a very interesting technical device.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 195

~

We have noted before  that, once you set out to focus on Ideas per se, you keep winding up back in mathematics -- if only because there are so many of them there.  Yet more:  In our own lifetime, math itself has spawned a subfield  whose task, it would seem, is precisely the study and development of Ideas -- for their own sake, almost, and beyond such practicalities as computing the area of the field of Farmer Brown (or rather, Farmer Enkidu, since this concern goes back to Babylonia and beyond) or even its offspring, geometry, or the handmaiden of that, the calculus, or …  This field is called Category Theory, which (as faithful readers of this tragic blog will already know)  I do not personally understand:  but do note, that a recent introduction to same (subtitled “A first introduction to categories” -- the style of the title is that of children’s books;  and God willing, someday toddlers will study this stuff), by Lawvere & Schanuel, is titled:

Conceptual Mathematics

C’est un titre astutieux.  For again (this is a phenomenon which we have treated, in these essays, under the label “faux-naïf”), on the surface this might seem to be one of those liberal-feelgood substitutions for the actual hard work of thought, meant to bolster the self-esteem of slow-learners;  whereas in actual fact, it points at concepts -- what underlies such relatively superficial activities as real analysis, point-set topology, algebraic geometry (you with me, kids?), and all the rest.

[Excelsior]   There is a vast philosophical literature (and a smaller, but still substantial, linguistic literature) concerning the relations between language and thought.   To rehearse this would be pointless;  to attempt to enrich it, quixotic.   Still we may feel our way forwards, and conceivably (eventually) contribute some minim of value, by taking as our paradigm area of Thought -- mathematics, rather than cats being on mats, and that sort of thing.   And Language as comprising, not only natural human languages, but any attempt at symbolic and communicable representation of Thought. 

(For this quest, I request:  God’s guidance and Grace.  Since, sine qua, non.)

An initial linguistic bridge is provided by our remarks above about the notion idea in the sense of ‘bright idea’.   A bright idea is no mere clothing of a perception;  it is closer to an invention.   And the key term it brings us up next to is:  insight.

[TBC?  Solâ gratiâ … ]

The Ontology of Logic (updated)

Since logic and set theory themselves -- at least as I imbibed them at the bosom of Quine -- are all about ontology, this may seem a queer thing to single out.   One reason to do so  is that the question of the ontological status of various abstract entities  is, in mathematics, seen through a glass darkly (typically going undiscussed unless challenged), whereas here --  face to face.

This will, like its sister essay The Ontology of Physics, be simply a corkboard for posting stray thoughts and choice quotes.  For a more general take on all this, click here.  As for the ontology of mathematics proper, we need no special post, since that is an ongoing theme in the "Theologia Mathematica" series. For that, click here.

Here we go.

Frege took all classes as rock-bottom objects  on a par with individuals.

-- W.V.O. Quine, “On Frege’s Way Out” (1954)

This is the bold, the manly path, of Cantorian Realism.   We do not merely accept sets (classes, collections) as, well, okay, “real” in some pale, some Meinongian sense, though in no wise privileged to belly up to the bar as equals with such unquestionably real individuals as Piglet:  No, they are rock-bottom objects, pardner, fit to drink with any man.

*

Commercial Break

A private detective  confronts the uncanny;

an ecclesiastical mystery:

Murphy Calls In a Specialist

[Kindle] [Nook]

*

~

If  to be is to be the value of a quantified variable, then it matters what sort of quantification we are talking about.  Quine frequently raises this knotty issue.

Substitutional quantification makes good sense … no matter what substitution class we take -- even that whose sole member is the left-hand parenthesis.  To conclude that entities are being assumed that trivially, and that far out, is simply to drop ontological questions.

-- W.V.O. Quine, “Existence and Quantification”

Quine footnotes this:  “Lesniewski’s example, from a conversation of 1933 in Warsaw.”

The heart stops cold.  1933 in in central Europe -- not a good place and time.  Yet to have been Quine, even Quine, young and thrusting, in Warsaw with Lesniewski -- very heaven!

Quine goes on: 

Lesniewski did not himself relate his kind of quantification [i.e., substitutional as opposed to objectual] to ontological commitments.

And indeed was right to do so, since

Where substitutional quantification serves, ontology lacks point.

~

Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 1, offers a notably practical definition of set from the standpoint of a working mathematician:

A set is any collection of mathematical objects which is sufficiently well defined to be the subject of logical analysis.

Since the word collection itself is often used by mathematicians as a synonym of set, you might deem this definition circular.  But here collection is being used entirely informally, referring to our pre-theoretic intuitions;  he could just as well said bunch or (even better) passel.

Indeed, from this perspective, Gleason is offering what amounts to an ostensive definition.  It is comparable to such a classic ostensive scenario as this:  “See that creature over there?  Assuming that it is not a mirage or an animatronic gimmick, that is what we refer to as a capybara.”

~

In Set Theory, you start with any object, which can be anything.  Indeed, even one will do: it can be zero, x, or even Piglet, for soon you contrast the-set-containing-Piglet, and off you go.  Indeed, in the most abstract set-theory, you don’t even need objects, just start from the empty set.   Even so, the enterprise smacks of ontology, since you have … let’s call them ‘thingies’, so as not to be dragged into any philosophical presuppositions about objects (thingies being more like the Cheshire-Cat-smile memories of objects), but even so, we are forming sets which consist of such thingies, and ask which contain which, and which are equal to which, and how many of them are there -- tangible things like that.

In Category Theory, you put childish things aside, and say Goodby to All That.  As one practitioner puts it:

Since a category consists of arrows, our subject could also be described as learning how to live without elements, using arrows instead.

-- Saunders MacLane, Categories for the Working Mathematician (1971; 2^nd ed. 1998), p. vii

[Update -- Lent 2013 -- Let us try, for a time, to live without elements...]

~

Of all the sciences -- nay, of all human cognitive activities -- mathematics is ontologically the most venturesome.   For among its key tools is the ancient method of reductio ad absurdum or modus tollens.  Here we work -- calmly and logically -- with Impossible Posits;  and when the smoke has cleared, all is once again as it should be, and we know something new.

Thus, take the question of how many prime numbers exist.  Lots, no doubt;  but it is not initially obvious that they go on forever.    The bigger a number gets, the harder it is for it to pass through the inflexible Sieve of Eratosthenes:  the mesh gets finer and finer as the number of primes -- of your possible submultiples -- grows and grows.

A direct way of proving the infinitude of the primes would be to come up with a formula that could serve as a primal generator:  plug in a number, out pops a prime.   But no such formula is known, and probably none exists.  

So the standard move, known already to the ancient Greeks (men like gods, like very gods) is to turn on your heel and spin about and say:  Fine!  Be like that!  Let’s assume that there are not infinitely many primes.  (Your finitist opponent -- a fat and greasy Nominalist -- emits an oily smile;  but he soon shall taste the wrath of Modus Tollens.)  So there are only finitely many (smile, nod);  so there’s a biggest one (nod, but a fading smile, as our finitist realizes that something is about to go very, very wrong);  let us call this largest one M.  (Suspecting a trap, the finitist objects;  you don’t make an issue of it; let’s call it N instead.)

Our mathematician now has what proves to be a powerful weapon: N, the Biggest Prime in the Universe.   That no such number exists (as we soon shall find out) does not lessen its devastating effect, while we hold it in mind and operate with it -- working in an anti-universe, as it were, on the far side of Alice’s looking-glass, and yet where otherwise all the usual laws continue to hold.

So we form the factorial of N and add a unit:  N! + 1.  It is larger than N;  and yet it must itself be prime, since, by elementary arithmetic, no number (let alone a prime number) no larger than N, can divide it.   Our quixotic posit of a largest prime  has managed to unhorse itself.

Modus Tollens, skewering an Impossible Posit

Note that we really have been operating according to God’s own laws of logic, yet operating -- temporarily, like Jack Bauer saving the day by working in a radioactive chamber -- in a universe not made by His hand:  namely, one in which a largest prime exists.   Like the Devil’s inventions, this universe self-destructs -- but in a fashion that is constructive for ourselves.

   "Your bait of falsehood takes this carp of truth."
   -- Hamlet

This process of assuming a falsity to come up with a positive result  is really quite extraordinary.  It is generally stated that, if a logical system allows you to derive a certain statement as well as its negation, then the whole system falls apart, since then you can derive anything at all,  (p & ~p) => q   being analytic.  But obviously the structure of mathematics is more resilient than that.

In vain might one seek such fruitful use of counterfactuals outside of math.  It is as though, in order to solve some particularly perplexing crime, one were to falsely accuse someone and put him on trial; then, in the course of the proceedings, the forensics and cross-examinations, the real truth would out.  (Come to think of it, that is exactly the plot of many a courtroom drama.)

[Footnote]  G. H. Hardy, in A Mathematician’s Apology, gave the classic expression of the audacity of this move:

The proof is by reductio ad absurdum,  and reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons.  It is a far finer gambit  than any chess gambit:  a chess player may offer the sacrifice of a pawn or even a piece,  but a mathematician offers the game.

 
A serious-humorous example of positing an impossible object -- yet here, not with modus tollens in mind, but as a kind of calculational convenience:

Assume that the bubbles in a foam  are regular polyhedrons, whose faces are regular polygons with n sides, and that the angles between these sides are all 109° 28’.  Since no such object exists, let us call it the ‘follyhedron’, and pretend that it does anyway.

-- Ian Stewart,  How to Cut a Cake (2006), p. 126

[Meta-footnote]

Logic … operate[s]  in terms of quite unidentified  but none the less definite ‘objects’, which then  do or do not fall into classes … I do not know whether it can deal with jelly-like or cloud-like entities, with mushy viscous messes  that do not break up into manifest units.  I suspect that nothing is beyond the technical ingenuity of men …

-- Ernest Gellner, Language and Solitude (posthum. 1998), p. 53

An expert review of Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), by Sam Adams on Amazon, summarizes: 

Brouwer's position arises from the assertion that the ontology of mathematics is determined solely by epistemology, and thus a statement which has not been determined to be true or to be false cannot be determined to refer to a state of affairs … Since mathematical meaning derives from emergent, epistemologically generated ontology, it follows that logic itself is not a formative and central structure of cognition, but rather is itself an emergent, structural overlay.