creativity in mathematics - noah litvin

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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.

creativity in mathematics - noah litvin

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