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Creativity in Mathematics
Noah Litvin
St. John's College, Annapolis MD
February 4th, 2012
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I. Introduction
Mathematicians are creative. We find ourselves surprised by proofs, baffled by
conjectures, and amazed by algorithms. But this is not the only perspective from which we
consider mathematicians. Since the Pythagoreans, philosophers have held up mathematics as a
paradigm of certainty. Mathematicians appear to be discovering a set of laws that govern the
universe. We are certain of these laws despite the varying accuracy of our application of them to
the physical world through the sciences. From this latter perspective, the former requires further
scrutiny. In what sense can mathematicians be creative?
This question becomes more pressing when considering mathematicians move to create
a fully formal mathematics in the twentieth century. The possibility of generating all
mathematical truths through an algorithm did not invalidate the idea of a creative mathematician
but put it in a tenuous position. The mathematician could be creative in choosing which chain of
valid logical deductions to follow but is not creating anything new.
In hisRemarks on the Foundations of Mathematics, Wittgenstein offers an alternative
perspective. For him, the mathematician is an inventor, not a discoverer (168).1 Their
inventions can be understood in two non-exclusive categories: some offer a new ability and some
provide a new psychological aspect. Additionally, Wittgensteins conception of mathematics
dissolves the metaphysical and ontological questions that surround mathematics.
II. Mathematical Formalism
In the twentieth century, many mathematicians attempted to uncover the foundations of
mathematics. Since EuclidsElements, it was implicit in the presentation of mathematical works
1Ludwig Wittgenstein,Remarks on the Foundations of Mathematics, trans. G. E. M. Anscombe, ed. G. H. von
Wright, R. Rhees, G. E. M. Anscombe, (Cambridge, MA: MIT Press, 1983). Further citations will refer to this work
unless otherwise noted.
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that mathematicians should always begin with basic principles and, from them, deduce other
valid propositions. Yet geometers in particular found themselves unsatisfied with the basic
principles they were working from. Euclids definitions, common notions, and postulates lacked
rigor. That is, Euclid and many of the mathematicians that followed him were making
assumptions that were not made explicit. Hilbert sought to fix these oversights by choosing a
simple and complete set ofindependentaxioms (Foundations of Geometry, Introduction).2
To most mathematicians, this task could seem entirely superfluous. Mathematicians may
very well have been assuming as an axiom, for instance, that if two planes , have a point A in
common, then they have at least a second point B in common (Foundations of Geometry, I, 6).
Making this particular axiom explicit is inconsequential to the development of mathematics. But
ifevery axiom of mathematics were listed and the task of fully formalizing mathematics were
achieved, an interesting possibility would arise. It would seem that if one were to establish a
complete set of axioms and rules of inference, an algorithm could be developed which could
generate all mathematical truths. This appeared to be a feasible task until the publication of
GdelsIncompleteness Theorem which demonstrated that it could not be realized; some
undecidable propositions will exist in a consistent system capable of expressing just the basic
propositions of arithmetic. But even without the development of Gdels theorem, the task of
generating a complete set of axioms and laws of inference could still seem questionable. The
criterion by which one is to decide which propositions deserve the status ofaxiom is unclear.
Hilberts three criteria supply only a general idea. If one were to come into dispute over the
simplicity of a given axiom (or its worthiness for inclusion in general), it is not clear how this
matter could be settled.
2David Hilbert, Foundations of Geometry, trans. E. J. Townsend, (Whitefish, MT: Kessinger Publising, LLC, 2010).
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One can temporarily put these difficulties associated with the choosing of axioms aside.
Even without complete certainty regarding the validity of ones axioms, one can still be
confident that the laws of inference will generate truepropositions if ones axioms are true.
Wittgenstein suggests that this is indicative of a mindset of one for whom reality is something
very abstract, very general, and very rigid. Logic is a kind of ultra-physics, the description of the
logical structure of the world, which we perceive through a kind ultra-experience (with the
understanding e.g.) (8).
From this perspective, mathematics would allow for creativity, but in a limited sense. The
mathematician would be akin to an explorer, traversing this logical structure of the world. He
could be creative insofar as he could choose which path he would like to follow, but he could not
forge his own path (as this would be to defy the laws of inference). Turing summarizes this well:
When working with a formal logic ingenuity will then determine which steps are the most
profitable for the purpose of proving a particular proposition (Systems of Logic Based on
Ordinals, p. 208).
III. Wittgensteins Conception of Logic
Wittgenstein offers an alternative perspective on mathematicsone in which
mathematicians are genuinely creative. For him, the mathematician is an inventor, not a
discoverer (168), a creator of essences (32). His view is contrary to that of the mathematical
formalists of the twentieth century. His disagreement rests in his conception of mathematical
axioms as well as that of logical inference.
Rather than seeing logic as an ultra-physics which consists of the laws that govern reality,
Wittgenstein understands logic as a description of what we call thinking (that is, what we call
thinkingproperly).
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The propositions of logic are laws of thought, because they bring out the
essence of human thinking to put it more correctly: because they bring out, or shew,
the essence, the technique, of thinking. They shew what thinking is and also shew kinds
of thinking.
Logic, it may be said, shews us what we understand by proposition and bylanguage (133-134).
Here, Wittgenstein is applying his understanding of words meanings as use, introduced
in the Philosophical Investigations, to logic. Rather than thinking of words as symbols that refer
to things (be they physical objects, concepts, etc.), words are akin to tools which have different
uses in different contexts. This view encourages one to keep in mind how children learn the
meanings of words as well as the situations in which one uses words.
For example, one may assert the proposition, Socrates is mortal. That man may assert
another proposition, IfSocrates is mortal then Socrates will die. Now consider what would
occur if the man were to state Socrates will not die with utter confidence. One would likely say
that this man could not possibly think these three propositions at once, despite the fact that he
can clearly say aloud all three of these propositions. But, for Wittgenstein, this would merely be
indicative of the way in which one uses the word thinking. This is to say, it is for us an
essential part of thinking that in talking, writing, etc.he makes thissort of transition
(116). So if someone were to say that he could actually think all three of these propositions,
then a disagreement would arise regarding in which contexts it is proper to use the word
thinking.
One may insist that this is not a linguistic issue, but a psychological one. Despite the
words varying uses among people and contexts, one could study thinking, the mental
phenomenon, in some sort of laboratory setting to determine scientifically the possibility of
contradictory thought. But, for Wittgenstein, this is only to push the issue a step back. In
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determining what sort of empirical result would serve as ample evidence, one is making clearer
what sorts of things one calls thinking. And disagreement could just as well be met at this
stage.
One may still take recourse in pointing out that anyone could very well assert that
Socrates will not die, but it simply would not be true. Use and meaning, one might insist, are
related but not interchangeable. To better understand Wittgensteins stance, the inquiry should
now focus on the use of the word true. One need not maintain that truth is some sort of ethereal
property which is associated with some propositions and not others. Instead, one should consider
how the word is taught and used in ordinary language. For example, from this perspective, it
should be clear that the proposition: It is true that this follows from that means simply: this
follows from that (5). There is, perhaps, an interesting and subtle nuance in meaning between
the two propositions. And this would come out in thinking about the situations in which one
would explicitly assert a propositions truth. In this way, one can come to understand the rich
variety of meanings that the word truth carries.
So, although logical inference can be thought of as a method which generates new true
propositions from other true propositions, it may be more appropriate to understand logical
inference as a transformation of our expression (9). That is, the distinguishing characteristic
of a proposition which is a logical inference from one that is not is that the former can be
generated from previously asserted propositions through the use of certain rules. From this
perspective, the rules themselves should be understood as propositions regarding grammar and
language. Take, for example, Modus Ponens: (P(PQ))Q. This would appear to be a rule by
which reality abides. Alternatively, one could see Modus Ponens as part of a definition for the
word therefore or implies.
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One might object that this latter perspective does not take into account the fact that one is
compelled by logical deductions. Certainly logical deductions compelbut in what sense? In
ordinary language, it comes out that a proof has compelled someone in the fact that once [one
has] got it [one goes] ahead in such-and-such a way, and [refuses] any other path (34). If an
individual who is compelled by a proof is met with disagreement regarding the proofs
conclusion or legitimacy, the individual would likely insist Dont you see?! without providing
further argument. Additionally, the sense in which one is compelled by a proof is similar to the
way in which a man is compelled by another pointing him down a single path (117). That is, if
the man were to point down two different paths at once (providing two different options), one
would say the man being directed is not compelled to go down the particular path he eventually
walks. Similarly, a law of inference compels in the sense that it insists one particular conclusion;
no choice is offered whether or not one accepts Modus Ponens, for instance, at any particular
point in the proof.
Still one may insist that this explanation is insufficient. This explanation may make it
seem as if one is not truly compelled. One mustgo a certain way according to the laws of
inference. No one can break these laws. But this isnt actually the case. As mentioned previously,
someone could very well assert Socrates will not die after claimingSocrates is mortal and
If Socrates is mortal then Socrates will die. One might call this individual a madman or insist
that he cant really think that. Here, again, this latter statement would more accurately be
interpreted as a remark about what the critic calls thinking. Perhaps one is trying e.g. to say:
he cant fill it with personal content; he cant really go along with itpersonally, with his
intelligence (116). And this certainly makes sense. But the limiting factor here is not stemming
from a law of inference. It is a practical limitation.
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From these considerations, it would appear that the laws of inference compel, but in the
same sense, that is to say, as other laws in human society (116). This becomes clearer when
one considers a situation in which someone actually draws an inference. In 17, Wittgenstein
offers an example: A regulation may state that all men taller than five foot, six are to join a
particular section. A clerk is told that a particular man is five foot, nine. He now attempts to draw
an inference. All men taller than five foot, six are to join this section; This man is taller than five
foot, six; Therefore, this man should notjoin this section. Here, the clerk has broken a law of
inference. For this, the clerk will perhaps suffer certain practical consequencesno more, no
less.
One still may insist that though the clerk, in this case, could break a law of inference, the
law itself is inexorable. Keep in mind that, of course, a law does not apply itself. Rather, in
calling a law inexorable, Wittgenstein suggests that one is employing an image, a picture of a
single inexorable judge, and many lax judges (118). This inexorable judge makes no
exceptions when enforcing the law. Similarly, the laws of inference never allow for
discrimination (i.e. provide an option) to the person applying them. As with the man pointing
another down one path rather than many, this is the sense in which a law of inference is
inexorable.
So the conception that logic allows one to traverse the facts that constitute reality and, in
doing so, make discoveries may not be entirely justified. Rather, through a reconsideration of
meaning in language, it would seem that logical propositions work to describe what we call
thinking, proposition, and language. The sense in which the laws of inference compel us are no
different than the sense in which the other laws of our society do. The structure which logic
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provides should not be thought of as foundational. Rather, logic seems to be a system of analysis
which can be applied to ordinary language and activities.
IV. Mathematics as Activity
Regardless of how one conceives of logic, one may still hold on to the basic propositions
of arithmetic and geometry as undeniable truths. One may think of these propositions as
fundamental to the constitution of reality, separate from human experience. Wittgenstein, again,
offers an alternative perspective. For him, the basic propositions of mathematics are best
understood as describing activities.
One should begin by considering how to teach the basic propositions of arithmetic. One
would likely begin by teaching a student the series of cardinal numbers. A teacher would use a
group of objects to demonstrate different quantities. Once the student had mastered the technique
of counting, the student could move on to addition. Two apples could be set on a table and
correctly counted by the student. Two more apples could subsequently be placed on the table.
Then, the student would count four total apples. With a brief explanation of arithmetic
vocabulary, it could be said that the student now understands what the proposition Two plus
two equals four means. Now imagine a strange situation in which this same process is
conducted, but six apples are counted rather than four. Assuming the student had not made an
oversight, one must say that apples are no good for teaching sums. It is essential to summation
that two and two yield four. If all solid bodies were to express the same irregularity the erratic
apples had, that would be the end of all sums (37).
So, in the same sense one is compelled to make logical inferences, one is compelled to
count (and perform all sorts of mathematical operations) the way one does. Counting (and that
means: counting like this) is a technique that is employed daily in the most various operations of
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our lives (4). One could certainly count however one pleases, but the particular technique of
counting society employs is not arbitrary. One would very quickly run into practical difficulties
if one were to count otherwise. Further, mathematical operations are as inexorable as logical
inference insofar as both leave no ambiguity regarding proper execution. In the series of cardinal
numbers, for example, three follows two every time. Counting does not admit of a choice on the
part of the one counting.
From this perspective, the privileged epistemological status of mathematics begins to
fade. It cant be said of the series of natural numbers any more than of our languagethat it is
true, but: that it is usable, and above all, it is used (4). It is undeniable that there is a great
quantity of men across various cultures who assent to the basic propositions of mathematics. But
it remains that basic mathematical propositions are not qualitatively different than others in
regards to their truth. They are just as much rooted in experiment as the sciences . In the usual
presentation of mathematics, this is obscured.
One could insist that the practical applications of mathematics and the consensus of the
mathematical community are only consequences of apropositions validity. This validity stands
apart from human activity and relies solely on the propositions proof. But, for Wittgenstein, all
symbols (including those used in mathematical proofs) carry meaning only insofar as they are
used. For instance, even this simple arrow, , could be understood as pointing left with the stem
springing out of the head in the direction one is supposed to understand it (if one were taught to
interpret it in this way). The arrow points only in the application that a living creature makes of
it (Philosophical Investigations, 454).3 This applies to proofs and even very simple diagrams
which may serve as proof.
3Ludwig Wittgenstein, Philosophical Investigations, trans. G. E. M. Anscombe, P. M. S. Hacker, Joachim Schulte,
ed. P. M. S. Hacker, Joachim Schulte, (Chichester, West Sussex, U.K., 2009).
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Wittgenstein most succinctly demonstrates this in 38:
Figure A Figure B
One might suggest that Figure A proves that 2 + 2 = 4. But one could just as well use
Figure B to prove that 2 + 2 + 2 = 4. This example draws out how, even in this seemingly simple
example, there is an implicit procedure and set of rules which lead to certain outcomes. In this
case, one is to draw four Xs. One must then circle them in groups of two. It is crucial during this
step that the circles do not overlap. One must also trust that no single X will be left outside of a
circle. Add 2 for each circle. This should yield the same result as the total number of Xs one
counts. This description should make clear how this example could be better understood as a
picture of an experiment (36) rather than a demonstration of an undeniable intuition of space.
One can also consider this from the perspective of a mathematician devising a proof. A
mathematician may conjecture that all hexagons consist of six equilateral triangles (c.f. Euclids
Elements, IV.15). He may become convinced of this through the use of triangular shaped pieces
or perhaps through the drawing of various examples with a compass and ruler. He could then
translate what he is accepting as proof into mathematical language. That is, he would not write,
Take a hexagon and overlay six triangles with sides equal to a side of the hexagon. Rather, the
proof would likely begin, Let there be a Hexagon ABCDEF. The latter form of expression
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hides the temporal, active nature of what the mathematician is discussing. Because of this shift in
language, one is inclined to say that the latter expresses a mathematical truth which stands apart
from human activity. Yet, it would seem that the former account is more honest to what the
mathematician has really done.
This former mode of expression would seem entirely unacceptable for valid mathematical
proof. But it is not far from Euclids use of superposition and coincidence in his proof of
Proposition I.4 in The Elements. This proof has caused discomfort among mathematicians
because it points out the temporal and experimental origins from which many basic mathematical
proofs arise. Hilbert was able to avoid using coincidence as a method in the proof of this
proposition in his Foundations of Geometry. Yet, that this was possible changes nothing.
Euclidean geometry would have progressed all the same if it were not possible (or without
Hilberts publication) as it had for millennia.
Calculus employs motion in its fundamental proofs as well. In Lemma 6 of Newtons
Principia, the two points A and B approach each other. Although now there has been a shift
away from Newtons vocabulary in favor ofone of limits, it has been just that: a shift in
vocabulary. It is indicative of mathematicians distaste for their use of physical experience and
intuitions. After all, both a Newtonian proof and a more modern proof should be equally
persuasive if their basic principles are accepted and their steps are fully understood.
Wittgenstein uses a simple example to draw out the implicit activity in geometric proof
and the subsequent adjustment of language in 25:
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Figure C Figure D
The pattern of lines depicted in Figure C is ascertained to be like-numbered with a
particular pattern of angles, Figure D, through a process of correlation. An image of this
correlation is offered in Figure E:
Figure E
If Figure E were supplied without this context dictating how one is to interpret this
diagram, it could just as well be a star with threadlike appendages (25). But the reader is
given a context. The lines which connect our initial lines to the angles were drawn in an effort to
discover whether or not they were like-numbered. The outcome was unknown. It may be
tempting to say that this is not the case, but this is only for the fact that the patterns each consist
of so few elements. If there were a large number of lines in Figure C and a very irregular figure
in place of Figure D, as in 27, the outcome of a one-to-one correspondence is more obviously
unknown at first glance. Once one has established that there is a like number of angles and lines
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in 25, one can further extrapolate that, if one were to have men standing in the pattern depicted
by Figure D and wands arranged on a wall as in Figure C, each man could have one wand. In this
way, the diagram is a schematicpicture of[ones] giving the five men a wand (26).
There is an alternative conception of this diagram:
I can however conceive figure [E] as a mathematical proof. Let us give names to
the shapes of the patterns [C] and [D]: let [C]be called a hand, H, and [D] a pentacle,
P. I have proved that H has as many strokes as P has angles. And this is once more non-
temporal (27).
One would be tempted to take this latter conception as primary. That is, one could say that our
mathematical proof has revealed an essence of hands and pentacles (namely, that their lines and
angles are same in number). And it is for this reason that the one-to-one correlation process is
successful between these two figures. From this perspective, one must insist that it is impossible
for a hands lines and a pentacles angles to be different in number.
But, as was the case with logical inference, one could imagine a man coming up with this
impossible result. There are two ways in which one could address a man who asserted that the
lines of a hand and the angles of a pentacle are not same in number. One would first assume that
this man had made an error in the process of drawing correlative lines. Perhaps, one would
assume that he omitted a line or he miscounted the correctly drawn lines. Or one might say that
this man must not have understood what it means to correlate these figures. But if there were no
problem in this mans act of correlation, then one would conclude that the subjects of his
correlation were not, in fact, a hand and a pentacle. In other words, it is necessary for a figure to
be called a hand or a pentacle that it has the same number of lines or angles as the other.
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The proof doesnt explore the essence of the two figures, but it does express
what I am going to count as belonging to the essence of the figures from now on.Ideposit what belongs to the essences among the paradigms of language.
The mathematician creates essences (32).
One might insist that this is still a discovery, as one is now entirely certain that this like-
numbered property always will be and always has been the case. Undoubtedly in an enormous
majority of cases [one] will always get same result, and, if [one] did not get it, [we] would think
that something had put [one] out (31). But what is the motivation behind asserting that this
result has uncovered something that was, in some sense, already there? One may imagine that if
men were standing in a pentacle and wands were arranged in a hand, each man would be able to
receive one and only one wand. Alternatively, one could imagine drawing Figure C and Figure D
on a sheet of paper and then connecting each line and angle with one and only one line. Neither
of these examples is contingent on a particular time period. There is no reason to think either of
these activities, when executed correctly, would yield a different result today, tomorrow, or
yesterday. This should make clear that the atemporal sense of this proposition is predicated on
the expectation of certain results of various human activities.
V. Surprise in Mathematics
Wittgensteins perspective may draw into question how surprise could be possible in
mathematics. If mathematicians are not making discoveries (but, rather, inventing), it is not
immediately apparent what could cause surprise. It would seem that someone should not be
surprised by ones own creation. Wittgenstein identifies two different roles surprise plays in
mathematics.
Many would say that the results of mathematical proof are surprising because the results
show the depths to which mathematical investigation penetrates; - as we might measure the
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value of telescope by its shewing us things that wed have had no inkling of without this
instrument (Appendix II, 1). This role, which is dominant at the present day (Appendix II,
1), is illegitimate. The results of any chain of inferences are not like those of an experiment.
That is, if a proof is fully understood, no surprise should be had at any particular step, including
the final one.
The other role of surprise in mathematics that Wittgenstein describes is often experienced
when one comes to a proof of a proposition or finds the solution to a mathematical puzzle. This
is a pleasant surprise; and it is of psychological interest, for it shows a phenomenon of failure to
command a clear view and of the change of aspect of a seen complex (Appendix II, 2). In the
case of a mathematical puzzle, this change of aspect is a shift from a failure to see the solution to
success. The result itself does not surprise, but that one did not think to try the solution surprises.
Similarly, in mathematical proof, this surprise is not at the result itself, for once the result is
obtained, the mathematician is no longer surprised at the result of his proof, say, upon reading it
later on.
One could object that each time someone reads a particular proof, this individual
continues to be surprised by the result. Wittgenstein suggests that this surprise is often brought
on by [thinking oneself] into the situation of seeing the result after having expected something
different (60). For instance, one may be able to imagine that a circle and a right triangle with
legs equal to the circles circumference and radius do not have the same area, even after reading
and understanding a proof which demonstrates that their areas are equal. Since this individual
could imagine the conclusion being otherwise, it is possible for him to be surprised each time he
reads the proof. If someone were to say that one could not imagine the areas to be different, the
likelihood of this persons being surprised in reading the proof would be greatly diminished.
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Here it could be said: What the proof made me realize thats what can surprise me (69). An
individual may be surprised by the fact that these two areas are similar after thinking this was not
the case. He is not surprised by the result itself.
One could also experience surprise when re-reading a mathematical proof as one does
when hearing the turn of a theme one is already familiar with. Here, one should consider what
meaning it would carry for someone to object that it should not be surprising (Appendix II, 3).
Certainly, one might experience some feeling of surprise each time. And this feeling can
accompany a discovery. But it doesnt always. This is to draw out that there are two related, but
different, senses of the word surprise being mixed here: one, a feeling, the other, an experience
of the unexpected.
So, rather than thinking of proofs themselves (that is, chains of inferences) as surprising,
one should rather understand proofs as bringing to light something that surprises us: - because it
is of great interest, of great importance, to see how such and such a kind of representation of it
makes a situation surprising, or astonishing, even paradoxical (Appendix II, 1).
VI. Creativity in Mathematics
From these considerations, an interesting framework for understanding creativity in
mathematics emerges. It would seem that creative mathematicians provide a new aspect, a new
ability, or both. In other words, one can say This mathematical investigation is of great
psychological interest or of great physical interest (Appendix II, 2).
The psychological interest that mathematical innovations can provide stems from a
change in aspect. Wittgenstein addresses aspect change in Part II of the Philosophical
Investigations. Jastrows duck-rabbit is offered as an example of a picture which, for most
people, prompts a change of aspect (PIPt. II, 118).
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Jastrows Duck-Rabbit
One could imagine someone who could only see this picture as a picture of a duck.
Subsequently, for whatever reason, this person could see this picture as a picture of a rabbit.
What was formerly the ducks bill is now perceived as a pair of ears and the eye now appears to
be looking right, rather than left. The image on the page does not change, but this person
describes the change just as if the object had changed before [ones] eyes (PIPt. II, 129).
The expression of a change of aspect is an expression of a new perception and, at the same time,
an expression of an unchanged perception (PIPt. II, 130).
So it is with the solution to a mathematical puzzle. Wittgenstein introduces a Chinese
tangram as an example of mathematical puzzle in 42.
Figure F Figure G
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In a Chinese tangram, one is given a set of puzzle pieces (Figure F) which must be
arranged to create a given shape (Figure G). After extensive trial and error, one could become
convinced that the creator of the puzzle must have erred and the pieces of the puzzle cannot be
arranged to form such a shape. If there were a solution, for the puzzle solver it would seem as if
a demon has cast a spell round this position and excluded it from our space (45).
But then the solution to the puzzle is provided:
Figure H
Cant we say: the figure which shews you the solution removes a blindness, or again
changes your geometry? It as it were shews you a new dimension of space (44). Before, in the
case of 42, one may not have seen a rectangle as a combination of two pairs of congruent right
triangles. But now one does. This can be understood as a change of aspect, similar to how ones
perception of the duck-rabbit changes.
Mathematical proofs can provide a new aspect in a manner similar to that of the solution
to a mathematical puzzle. For example, someone could consider the geometric proposition, A
straight line is an infinitely large circle, complete nonsense. It could be said that this person
does not see straight lines in this way. But after following and understanding the relevant proof
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in GalileosDiscourses and Mathematical Demonstrations Relating to Two New Sciences, this
person would likely begin to see them in this way.
The parallel between mathematical puzzles and mathematical proofs extends further. In
the case of thepuzzle, imagine that one of the pieces is lying so as to be the mirror-image of
the corresponding part of the pattern (49). The solution, previously impossible, becomes
available through the employment of a new technique: turning over the piece. Who could
determine whether turning a piece over should be deemed a valid move in solving this puzzle?
And on what basis could one make this judgment? If someone asserted that it were not a valid
move but it would solve the puzzle (and no other method could), it would be difficult to take this
objection seriously. This situation is similar to the one previously mentioned regarding Euclids
use of coincidence. Propositions, which formerly could not be proved, now can be proved with a
new technique. And new techniques in mathematical proof can yield interesting and fruitful
results.
Additionally, certain mathematical puzzles we find terribly uninteresting just as we do
many mathematical proofs. Consider the tangram presented in 70:
Figure I
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This shows us that a rectangle can be made out of certain non-rectangular shapes, just
like the puzzle in 45. But this puzzle is boring. One should be able to solve it without much
difficultly at all. Only in very rare circumstances would someone find this interesting. (For
instance, a man who owns two separate plots of land shaped like each of the pieces might be
excited by this puzzle.) Similarly, there are plenty of trivial geometric and algebraic proofs. They
too are easy to generate and understand but they do not offer a new aspect with which one can
understand the entities involved.
One may insist that, in all of these examples, these aspects are revealed (i.e. discovered),
rather than created or invented by the mathematician. This is not without good reason. When
experiencing a change of aspect, it certainly seems as if the new aspect were somehow already
there. When one experiences a change of aspect with the duck-rabbit, for example, the ink on the
page obviously does not change. And one could imagine experiencing that particular change of
aspect any time in the past (at least any time after one were made familiar with what rabbits look
like). But it would be nonsensical to say that the aspect, this perception, was there before it was
perceived. In the mathematical puzzle in 42, the individual unable to solve the puzzle simply
did not see the rectangle in such a way. When the solution is demonstrated, The new position
has as it were come out of nothingness. Where there was nothing, now there suddenly is
something (46).
The mathematical formalists of the twentieth century would likely acknowledge that this
psychological phenomenon occurs, but it would not carry the same significance that it does for
Wittgenstein. Likewise, practical application is often considered ancillary to the task of
mathematicians. But, from Wittgensteins perspective (which emphasizes the activity implicit in
mathematical proof), the use of mathematics should be considered more important. In fact, many
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of the propositions ofEuclidsElements show the reader how to do many things (e.g. Create an
equilateral triangle with only a compass and straight-edge; find the length of a right triangles
hypotenuse from the length of its legs; etc.).
The solution to the mathematical puzzle, for example, does more than provide a new
aspect. Before, you could not[solve] itand now perhaps you can (47). So it is with the hand
and pentacle demonstration as well: The proposition proved by [the diagram] now serves as a
new prescription for ascertaining numerical equality (30). That is, one can now arrange one set
of objects into a pentacle and another set of objects into a hand and know that they are like
numbered, without performing any correlation.
It should be emphasized that proofs do not need to provide a new ability (though they
often do). There is not one simple motivation underlying all mathematical innovation.
Wittgenstein suggests a metaphor:
a mathematician is always inventing new forms of description. Some, stimulated bypractical needs, others, from aesthetic needs, - and yet others in a variety of ways. And
here imagine a landscape gardener designing paths for the layout of a garden; it may well
be that he draws them on a drawing-board merely as ornamental strips without the
slightest thought of someones sometime walking on them (167).
Note that even the work of mathematical formalists can be understood in these terms. Hilbert, for
instance, made a series of aesthetic judgments in writing and organizing his Foundations of
Geometry. He was not necessarily concerned with any practical consequences of his work.
VII. Example
Consider thisproposition from EuclidsElements: If in a right-angled triangle a
perpendicular is drawn from the right angle to the base, then the triangles adjoining the
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perpendicular are similar both to the whole and to one another (EuclidsElements, VI.8). This
proposition serves to provide both a new ability and a new aspect.
The proof demonstrates how one can generate two similar right triangles from any given
right triangle assuming one has the ability to draw a line perpendicular to another from a given
point. Compared to the innovations made possible by calculus, this ability is not particularly
exciting. Rather, most find this proposition interesting for the aspect change it provides. As with
the mathematical puzzle, one may not have thought that this shape could be divided into these
two other shapes. This process could even be performed indefinitely on newly generated
triangles. It might be said that, after reading this proposition, one sees all right-angled triangles
in a new way.
It may be tempting to say that this account does not draw out the importance of the rigor
in the proof of the proposition. One could gain this new ability and aspect through simply
looking at its corresponding diagram, but the proposition requires proof. One may very well be
compelled to assert this proposition by applying laws of inference to the axioms of geometry.
But this compulsion occurs in the sense discussed before. If one were to stray from this
conclusion (whether it were presented on the grounds of a rigorous proof or merely a diagram),
one could run up against various practical difficulties. The rigor of the proof may offer its reader
greater certainty than just the diagram and enunciation. But it need not necessarily. For instance,
a layman who is unfamiliar with logical and geometric language would likely become more
certain of this proposition if given the enunciation with a diagram rather than a rigorous
geometric proof.
Regardless of the certainty this proof grants, one may be surprised by this proof. But
again, it is not that any particular step in the proof is shocking. It is what the proof made [one]
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realize (69) which surprises: before, perhaps one had not expected two similar triangles to fit
in a large one. Through reading the proof, the unexpected is realized. This is to be contrasted
with the opinion that this proposition surprises by drawing out the depths to which
mathematical investigation penetrates (Appendix II, 1), as if this proof has unearthed an
essence of the right triangle.
Also after reading through the proof, it may be tempting to say of right triangles , It is as
if God constructed them like that (72). It is important that one not lose sight of the fact that this
is a simile. One must consider in what scenario (if any) it would make sense to say that all right
triangles could be constructed. This is drawn out by imagining someones saying The shape
is made up of these parts; who made it? You? (72). The sense in which one could speak of
triangles (and not a particular one) being made or constructed is an interesting one. Because one
might be able to imagine Euclids proposition being false, it feels as if some sort of decision in a
process of construction could have made it the case. This is what may have been meant by the
simile employed in 72.
When one says: This shape consists of these shapes one is thinking of the shape as a
fine drawing, a fine frame of this shape, on which, as it were, things which have this shape are
stretched (71). In the case of this proposition, as mentioned before, one can imagine a right
triangle-shaped frame. If one were to lay two similar triangles of the appropriate size upon this
frame, they would fit. This is the sense in which a shape consists of other shapes. One need not
suppose some sort of metaphysical composition of mathematical entities which has been
discovered by employing this proof.
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VIII. Conclusion
This account of mathematician as inventor, rather than discoverer, frees one of the
metaphysical or ontological concerns which surround mathematics. For example, the question
What are the essential properties of a straight line? is still interesting, as straight lines are very
common in our experience of the world and our imagination. But the method of inquiry one
should employ in addressing this question should consist in considering the various situations in
which one calls something a straight line. The discussion would become most interesting in
situations where people may find disagreement (e.g. in non-Euclidean geometry). But in these
discussions, which have a philosophical character to them, one must keep in mind that the topic
of discussion is how one uses the term straight lineone is not discussing properties of some
sort of metaphysical mathematical object, about which there can be verification apart from
consensus. For Wittgenstein, Philosophy must not interfere in any with the actual use of
language, so it can in the end only describe it. For it cannot justify it either. It leaves everything
as it is. It also leaves mathematics as it is (PI, 124).
As mentioned previously, the mathematics that society accepts and appreciates (for
practical reasons, aesthetic reasons, or otherwise) is not arbitrary. Experience dictates which
mathematical propositions prove to be useful or interesting. But, for Wittgenstein, the
widespread empirical verification of mathematics (e.g. calculation) is not the only reason
mathematical propositions have the dignity of a rule (165). The success of mathematical
formalists, such as Russell, in describing basic mathematical propositions in terms of logic draws
out the close kinship mathematics has to grammar. Somuch is true when its said that
mathematics is logic: its moves are from rules of our language to other rules of our language.
And this gives it its peculiar solidity, its unassailable position, set apart (165).
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So the perspective from which mathematical formalists viewed their task and that of
other mathematicians may not be quite right. One need not understand mathematicians as
uncovering the propositions (foundational or otherwise) which constitute reality. Rather, one can
understand mathematicians as forming new rules, metaphorically building new roads for traffic;
by extending the network of old ones (166). Mathematicians throughout history have allowed
us to make groundbreaking achievements in the sciences and to see things in new ways. And for
this, we should not only commend their great analytic minds, but their creativity as well.
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