I. Notes for the presentation: ‘Evidence in Mathematics’

1. Preparation

For this presentation, little mathematical background will be required beyond highschool geometry and algebra (and I’ll try to keep to minimal use of those prerequisites).Nevertheless, we will at times make mention of somewhat advanced topics—e.g., the in-finitesimal calculus: its virtues and logical traps—but no technical knowledge about thesetopics will be required1.

In preparation for this session of our seminar please do these things:

(1) Read these notes.(2) Think about the two “exercises” given in these notes.(3) Read the first four pages of David Hilbert’s classic essay “On the Infinite” (copied

at the end of these notes).

2. The plan

People in mathematics, or in the sciences related to mathematics, know well that theissue of evidence—as it pertains to how one studies math, or engages in its practice, or doesresearch in it—is not a simple matter. It is not “a result is proved or it isn’t” and that’sthe end of it. The very complexity of different types, and different moods, of evidence, allintermingling in a subject which is as exact as mathematics undeniably is, has intriguedme for years.

One of my goals is to get a slightly better sense of the way various ‘faces of evidence’ playa role in my mathematical work. But much more importantly, I hope that this seminar willenable each of us to become aware of the various distinct profiles of evidence in a numberof disciplines, how these profiles change in time, how they shape and delimit and definethose disciplines, and how they influence the interaction between disciplines. I trust thatthis will enable all of us to appreciate (and view) our own work in a deeper way.

Although we’ll review, below, an annotated vocabulary list of terms related to evidence,I won’t give an all-encompassing definition of the phrase mathematical evidence itself. I’llonly mention here that I want to construe it as meaning something much more extensivethan numerical or statistical evidence, or mathematical proof, or even evidence related to

1The references for background reading that I have put within the body of the text are directly germaneto the issues that may be raised in our class session; and they are on a level that should be accessible topeople with little formal upper level mathematics. In contrast, the references that I put in footnotes may beless germane for our general discussion, and more directed to people with specific mathematical interests.

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the enterprise of producing appropriate mathematical models meant for application toother branches of knowledge2.

We will get into the issues of conjecture, proof, rigor and the more well-recognized waysof accumulating and labeling evidence, but to convince you that there are yet other typesof evidence playing their roles in pure mathematics, here are three simple examples.

• ‘Visual’ evidence:

Consider the rise of the fractals of Benoit Mandelbrot.

Up to the end of the First World War, the theory that was the progenitor of‘fractals’ was called Fatou-Julia theory. That theory studied the structure of certainregions in the plane (very) important for issues related to dynamics. Surely, Fatouor Julia would not have been able to make too exact a numerical plot of theseregions. And unless you plotted them very accurately, they would show up as blobsin the plane with nothing particularly interesting about their perimeters–somethinglike this:

Partly due to the ravages of the first world war, and partly from the generalconsensus that the problems in this field were essentially understood, there was alull, of half a century, in the study of such planar regions—now often called Juliasets.

But in the early 1980s Mandelbrot made (as he described it)“a respectful exam-ination of mounds of computer-generated graphics.” His pictures of such Julia setsand related planar regions were significantly more accurate, and tended to look likethe figure below (which is a more modern version of the ones Mandelbrot produced)

2For one of the many applications of mathematics to provide evidence in Law, see Poincare et al on theDreyfus case: http://www.maths.ed.ac.uk/~aar/dreyfus.htm.

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From such pictures alone it became evident that there is an immense amount ofstructure to the regions drawn and to their perimeters. This almost immediatelyre-energized and broadened the field of research, making it clear that very littleof the basic structure inherent in these Julia sets had been perceived, let aloneunderstood. It also suggested new applications. Mandelbrot proclaimed—withsome justification—that “Fatou-Julia theory ‘officially’ came back to life” on theday when, in a seminar in Paris he displayed his illustrations.

Computers nowadays (as we all know) can accumulate and manipulate massivedata sets. But they also play the role of microscope for pure mathematics, allowingfor a type of extreme visual acuity that is, itself, a powerful kind of evidence.

• The evidence of coincidence:

In the early seventies, the mathematician John McKay made a simple observa-tion. He remarked that

196884 = 1 + 196883.

What is peculiar about this formula is that the left-hand-side of the equation,i.e., the number 196884, is well-known to most practitioners of a certain branch

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of mathematics (complex analysis, and the theory of modular forms)3 while 196883which appears on the right is well-known to most practitioners of (what was inthe 1970s) quite a different branch of mathematics (The theory of finite simplegroups)4. Mckay took this ‘coincidence”—the closeness of those two numbers5—asevidence that there had to be a very close relationship between these two disparatebranches of pure mathematics; and he was right! Sheer coincidences in math areoften not merely sheer; they’re often clues—evidence for something missing, yet tobe discovered.

• Evidence coming as a new clue, midstream:

The “Monty Hall Probability Puzzle” is a problem extracted from a famous gameof an old TV show (’Let’s Make a Deal’) where—at least as I’ll recount it—if youwin, you win a goat. The relevant issue, for us, is given by the following generalground-rules of the game: you are faced with three closed doors (in a row, on-stage)and are told that the goat is behind one of those doors, there being nothing behindthe other two. All you have to do is open the right door, i.e., the door with thegoat; and you win. But you are asked, first, to indicate what door you intend toopen, without actually opening it. At this point, Monty Hall, the generous host ofthe show, will offer to help you make your final choice by opening one of the othertwo doors (behind which there is nothing) and then he will say: “you are free torevise your original choice.”

500px-Monty_open_door.svg.png (PNG Image, 500 ! 278 pixels) http://upload.wikimedia.org/wikipedia/commons/thumb/3/3f/M...

1 of 1 8/13/12 2:13 PM

What should you do (assuming, of course, that you actually want that goat)? Areyour chances of winning independent of whether or not you change your preliminary

3196884 is the first interesting coefficient of a basic function in that branch of mathematics: the ellipticmodular function.

4196883 is the smallest dimension of a Euclidean space that has the largest sporadic simple group (themonster group) as a subgroup of its symmetries.

5McKay gave a convincing interpretation of the “1” in the formula as well

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choice? Or is there a clear preference for one of the two strategies: sticking to yourguns, or switching?

Here is an exercise: come up with an answer to the above question and givean argument defending it. Note that this problem is very extensively covered anddiscussed on the internet, but—if you haven’t encountered it before—there is somebenefit to thinking it through independently. Only then is it fun, if you want, tosee what Google has to say about it.

The point for us is that evidence in mathematical reasoning sometimes arisesin slant ways. The structure of this simple game reminds me of a certain com-putational strategy that makes judicious choices from an inventory of differentalgorithms, and switches from one algorithm to another depending on new cluesthat come up mid-computation.

But these are only three of the many forms in which ‘evidence’ presents itself in puremathematical research.

II. A short annotated vocabulary list

• self-evident,• axioms, hypotheses, common notions• well-known,• defined, well-defined, unambiguous,• plausible inference, heuristics.

3. Self-evidence

In one of our founding documents, the assertion all men are created equal is proclaimedto be a self-evident truth, but 87 years later it was demoted to a mere proposition, i.e.,something that needs proof.

A tricky notion, self-evident: I find that I’m insouciantly generous in claiming self-evident status for lots of my own private opinions, my interpretations of fact, and of amotley assortment of other sentiments. It goes without saying that I do all this withoutany justification; no need!

The term itself, self-evident, is not often found in the mathematical literature. Never-theless, the various starting points of mathematical work—for example:

(1) the technology of basic logic (which incorporates the syllogism, the law of theexcluded middle, etc.)

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(2) the axioms in Euclid’s Elements

were taken at one time to be self-evident—or at least were taken as requiring only informal,conversational, justification, if any. . . until in later mathematical developments, it wasdiscovered that these ‘starting points’ had to be better understood, and had to be quitesubstantially constrained or reconfigured.

In the first case above the (so-called) crisis in the foundations of mathematics forcesmathematicians to take a far less relaxed view of the basic technology of logic—we’ll see abit of this.

In the second case listed above, i.e., in Euclid’s text, the starting presuppositions6 arecalled “common notions,” not “self-evident axioms.” We’ll also examine this, including thestory of the parallel postulate.

One big difference between the two labels “self-evident axioms” and “common notions,”is, of course, that the “self” of the first label emphasizes a certain “self-sufficiency”: youalone are the judge of how evident it is (to you). The “common” of “common notions”alludes to either a pre-established consensus, or some kind of tacit accord, possibly ofthe universally subjective or allgemeine Stimme status, that Kant tried to describe in TheCritique of Judgment7.

Here is a little thought-exercise in mathematical framing.

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6The question of which direction you pursue from the “starting point” is a theme of Book 6 of Plato’sdialogue The Republic where–as Socrates would have it–the mathematician builds constructions on unex-amined assumptions, while for the dialectician (i.e., the philosopher) assumptions are not

absolute beginnings but are literally hypotheses, i.e., underpinnings, footings, and spring-boards so to speak, to enable [the conversation] to rise to that which requires no assump-tion and is the starting-point of all.

7One can gauge how important implicit consensus is, as an ever-present concern and armature in math-ematical argument, just by toting up the number of times phrases like “well-known” or the passive “It isclear that” are used in mathematical treatises.

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Construction: On a piece of paper (meant to stand for the Euclidean plane) draw—with one stroke of the pen—a circle, or any closed loop. Now try to connect a point insidethe drawn circle to a point outside the circle by some continuous curve.

Claim: Your ‘continuous curve’ will, at some point, intersect the circle.

Now the exercise is to think about the possible status of this simple construction andthe claim following it. First, do you think that “it”—i.e., the assertion—-is true? Thequotation-marks around the “it” is simply to note that the claim itself and the constructionare hardly mathematically well-defined. To pin them down, we must put them into somekind of formal context. In a word, we have to “model” the construction-and-claim in someway8. Second, do you think that the claim should be framed in a way that makes it self-evident, or as requiring proof? I might mention here that the substance and status of thisvery claim9 was the topic of heated debate about the foundations of mathematics at thebeginning of the 20th century (that we’ll later discuss).

4. Well-defined

This notion gets to an essential difference between mathematics and other realms ofthought. Among all the other things that it is, mathematics is the art of the unambiguous.

It is almost uncanny the way in which mathematics has the capability of achieving sucha high level of exactitude in its definitions and assertions. Thanks to this, we can be“originalists” with regard to 5th century BC mathematical texts at a level that would beludicrous to expect, when puzzling over the US constitution.

Often much is made—in mathematical logical circles–of how crucial it is for theoriesto be consistent. This is true enough, but it is a piece—admittedly, the essential piece—of a larger day-to-day issue that arises when thinking about mathematics; namely, theimportance of being sensitive to ambiguities of all sorts (and not only the crisis that wouldoccur if one and the same proposition were to be provably both true and false).

Here is an easy mathematical example that might give you a sense of how finely tunedmathematicians are to the question of ambiguity.

Consider a pair of scalene triangles10 ABC and DEF that are congruent. Compare thatsituation with a pair of triangles GHI and JKL that are congruent, and one of the triangles(hence the other) is isosceles11.

8One natural way, for people who know some calculus—or more specifically, who are happy with theconcept continuous—is to cast the problem on the Cartesian plane. Even with that, special care must betaken in defining what it means for points to be inside or outside the closed loop. If you define these notionsinside, outside in a certain way, the “claim” will be true by definition; in some other way, it will requireproof.

9I’ve formulated it slightly more generally than as it occurs in the classical literature (where it is calledthe intermediate value theorem)

10a triangle is scalene if no two sides are equal11meaning that at least two of its sides are equal

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Now there is a structural difference between these two situations; modern algebra stronglypresses home to its practitioners how important it often is to keep such differences in mind.Namely, in the first, scalene, case the congruence itself between ABC and DEF is uniquein the sense that there is only one way of pairing the vertices of the first with the verticesof the second so as to achieve a perfect overlap of the two triangles12. While in the second,isosceles, case there are at least two distinct congruences.

To rephrase our example a bit more generally and succinctly: it can be important todistinguish between knowing that two objects are merely equivalent, or more firmly, areequivalent via a unique equivalence.

5. Plausible inference

I imagine that the legal profession has lots to teach scientists and mathematicians aboutthe subtly different levels of plausibility in inferential arguments.

Mathematicians may be known for the proofs they end up discovering, but they spendmuch of their time living with mistakes, misconceptions, analogies, inferences, partial pat-terns that hint at more substantial ones, rules-of-thumb, and somewhat-systematic heuris-tics that allow them to do their work. How can one assess the value of any of this,mid-process? Here are three important modes of plausible reasoning, each of which I willdiscuss in class with simple examples.

• reasoning from consequence,

• reasoning from randomness, and

• reasoning from analogy.

In this taxonomy, the first of these modes is largely non-heuristic while the other two areheuristic, my distinction being:

• A heuristic method is one that helps us actually come up with (possibly true, andinteresting) statements, and gives us reasons to think that they are plausible.

• A non-heuristic method is one that may be of great use in shoring up our sense thata statement is plausible once we have the statement in mind, but is not particularlygood at discovering such statements for us.

I recommend that (for fun) you take a look at a few pages of this classical expositorytreatise on plausible inference in mathematics:

12or if you prefer, in the jargon of plane geometry, to achieve “SAS”

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Polya, G., Mathematics and Plausible Reasoning, Volume 1: Induction and Analogy inMathematics, Princeton University Press (1956); Volume II Patterns of Plausible Inference,Princeton University Press (1968)

III. Two mathematical issues

6. The construction and reconstruction of the foundations of mathematics

The problems involved in the so-called ‘crisis in foundations’ are related to the use ofthe two words all and exist.

All: Now it is perfectly OK to make statements about all people since, although “cor-porations” may or may not be included, the universe (of people) referred to is fairly well-defined. If however, you begin a thought with the phrase ”All thoughts...” you are quan-tifying over a moving terrain, since your very new thought is—after all—a “thought,” sochanges the range over which you are making your assertion, a sort of Winnie-The-Poohconundrum. This may have seemed harmless until Bertrand Russell showed that such kindsof insouciant universal quantification led to contradictions.

Exist: When you use indirect argument to show that some “thing”—call it X—exists,you have in your hands a powerful tool. Its subtleness is that you aren’t required to putyour finger on that “X” you are trying to show exists; all you need do is to discredit theassertion“X does not exist.” The issue of whether or not discrediting such an assertion isenough to confer “ontological status to X” is what is hotly disputed.

If you questioned Hilbert, Kronecker, Brouwer, Frege, Russell, and Godel regardingtheir stance on the issues related to the two bullets above, you’ll get significantly differentanswers. We might discuss this in class.

For background, please read the first four pages of David Hilbert’s classic essay “On theInfinite” (copied at the end of these notes)13.

7. Euclid’s parallel postulate and its ‘evolution’

If a line segment intersects two straight lines forming two interior angles onthe same side that sum to less than two right angles, then the two lines,

13 For further reading, you might look at the complete essay “On the Infinite”–and related essays ofL.E.J. Brouwer, Frege, Russell, and Godel—all reprinted in Jean van Heijenoort’s From Frege to Godel: ASource Book in Mathematical Logic, 1879-1931 Harvard University Press (1967).

For very a different (and interesting!) take on the same subject matter, see Logicomix: An Epic Searchfor Truth by Apostolos Doxiadis, Christos H. Papadimitriou, Alecos Papadatos and Annie Di Donna,Bloomsbury, (2009).

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if extended indefinitely, meet on that side on which the angles sum to lessthan two right angles.

For a very readable introduction to the history of “reactions” to this postulate, includingattempts to actually prove it, using only the other Euclidean axioms, I recommend visitingthe on-line site Cut-the-knot run by Alexander Bogomolny. Specifically, see:

• http://www.cut-the-knot.org/triangle/pythpar/Fifth.shtml for a prelimi-nary discussion of the fifth postulate, and

• http://www.cut-the-knot.org/triangle/pythpar/Attempts.shtml for a run-down of attempts to prove it.

We’ll discuss in class, very briefly, the change of status of this postulate, and its connectionwith ‘models’ in pure mathematics. Regarding “models,” and the strain on models posedby experimental mathematics ands massive computation, Peter Norvig, Google’s researchdirector, said “All models are wrong, and increasingly you can succeed without them.”This sentiment was taken up by Chris Anderson in a recent issue of Wired Magazine andextensively commented on by the mathematician George Andrews14. Anderson noted thattraditional science depended on model-formation. He went on to say:

The models are then tested, and experiments confirm or falsify theoreticalmodels of how the world works. This is the way science has worked forhundreds of years. Scientists are trained to recognize that correlation isnot causation, that no conclusions should be drawn simply on the basis ofcorrelation between X and Y (it could just be a coincidence). Instead, youmust understand the underlying mechanisms that connect the two. Onceyou have a model, you can connect the data sets with confidence. Datawithout a model is just noise. But faced with massive data, this approach

14G. Andrews Drowning in the Data Deluge, Notices of the American Mathematical Society, 59 August2012 (933-941).

See also:

• G.E. Andrews, The Death of Proof? Semi-Rigorous Mathematics? Youve got to be kidding! Math-ematical Intelligencer 16 (1994) 16-18.

• J. Borwein, J., P. Borwein, R. Girgensohn, S. Parnes, Making Sense of Experimental Mathematics,Mathematical Intelligencer 18 (1996) 12-18.

and the two books:

• J. Borwein and D. Bailey, Mathematics by experiment: plausible reasoning in the 21st century, AKPeters (2003)

• J. Borwein, D. Bailey, R. Girgensohn, Experimentation in mathematics: computational paths todiscovery, AK Peters (2004)

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to science hypothesize, model, test is becoming obsolete. Consider physics:Newtonian models were crude approximations of the truth (wrong at theatomic level, but still useful). A hundred years ago, statistically basedquantum mechanics offered a better picture but quantum mechanics is yetanother model, and as such it, too, is flawed, no doubt a caricature ofa more complex underlying reality. The reason physics has drifted intotheoretical speculation about n-dimensional grand unified models over thepast few decades (the “beautiful story” phase of a discipline starved ofdata) is that we don’t know how to run the experiments that would falsifythe hypotheses: the energies are too high, the accelerators too expensive,and so on. Now biology is heading in the same direction. . . In short, themore we learn about biology, the further we find ourselves from a modelthat can explain it. There is now a better way. Petabytes allow us to say:“Correlation is enough.” We can stop looking for models. We can analyzethe data without hypotheses about what it might show. We can throw thenumbers into the biggest computing clusters the world has ever seen andlet statistical algorithms find patterns where science cannot.

.This may deserve discussion15.. . .

15 I want to thank Stephanie Dick who provided me with the following three references regarding arelated topic; namely automatized proofs in mathematics.

• Donald MacKenzie, Computing and the culture of proving, Philosophical Transactions of the RoyalSociety 363 (2005) 2335-2350.

• Donald MacKenzie, Slaying the Kraken: The sociohistory of mathematical proof, in Social Studiesof Science 29 (199) 7-60.

• Hao Wang, Toward mechanical Mathematics, IBM Journal (January, 1960) 2-22.

For discussions of similar issues, but with attention paid to changes in attitudes towards foundations, see

• Leo Corry’s The Origins of Eternal Truth in Modern Mathematics: Hilbert to Bourbaki and Be-yond, in Science in Context, 12 (1998) 137-183.

• D. A. Edwards and S. Wilcox’s Unity, Disunity and Pluralism in Science (1980)arxiv.org/pdf/1110.6545

• Arthur Jaffe’s Proof and the Evolution of Mathematics, in Synthese 111 (2) (1997) 133-146.

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IV. Excerpt of “On the Infinite”

What follows is an English translation16 of the first four pages of the essay “On the Infinite” by

David Hilbert, delivered June 4, 1925, before a congress of the Westphalian Mathematical Society

in Munster, in honor of the mathematician Karl Weierstrass. Hilbert devotes the end of his essay

to what he thought was a proof of the Continuum Hypothesis. Only later in the development of the

subject did one learn that, given the starting point that Hilbert takes, there is no correct proof!

As a result of his penetrating critique, Weierstrass has provided a solid foundation formathematical analysis. By elucidating many notions, in particular those of minimum,function, and differential quotient, he removed the defects which were still found in theinfinitesimal calculus, rid it of all confused notions about the infinitesimal, and therebycompletely removed the difficulties which stem from that concept. If in analysis todaythere is complete agreement and certitude in employing the deductive methods whichare based on the concepts of irrational number and limit, and if in even the most complexquestions of the theory of differential and integral equations, notwithstanding the use of themost ingenious and varied combinations of the different kinds of limits, there neverthelessis unanimity with respect to the results obtained, then this happy state of affairs is dueprimarily to Weierstrass’s scientific work. And yet in spite of the foundation Weierstrasshas provided for the infinitesimal calculus, disputes about the foundations of analysis stillgo on. These disputes have not terminated because the meaning of the infinite, as thatconcept is used in mathematics, has never been completely clarified. Weierstrass’s analysisdid indeed eliminate the infinitely large and the infinitely small by reducing statementsabout them to [statements about] relations between finite magnitudes. Nevertheless theinfinite still appears in the infinite numerical series which defines the real numbers and inthe concept of the real number system which is thought of as a completed totality existingall at once. In his foundation for analysis, Weierstrass accepted unreservedly and usedrepeatedly those forms of logical deduction in which the concept of the infinite comes intoplay, as when one treats of all real numbers with a certain property or when one argues thatthere exist real numbers with a certain property. Hence the infinite can reappear in anotherguise in Weierstrass’s theory and thus escape the precision imposed by his critique. It is,therefore, the problem of the infinite in the sense just indicated which we need to resolveonce and for all. Just as in the limit processes of the infinitesimal calculus, the infinite in thesense of the infinitely large and the infinitely small proved to be merely a figure of speech, sotoo we must realize that the infinite in the sense of an infinite totality, where we still find itused in deductive methods, is an illusion. Just as operations with the infinitely small werereplaced by operations with the finite which yielded exactly the same results and led toexactly the same elegant formal relationships, so in general must deductive methods based

16Translators: Erna Putnam and Gerald J. Massey; taken from Mathematische Annalen (Berlin) 95(1926), pp. 161-90.

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on the infinite be replaced by finite procedures which yield exactly the same results; i.e.,which make possible the same chains of proofs and the same methods of getting formulasand theorems. The goal of my theory is to establish once and for all the certitude ofmathematical methods. This is a task which was not accomplished even during the criticalperiod of the infinitesimal calculus. This theory should thus complete what Weierstrasshoped to achieve by his foundation for analysis and toward the accomplishment of which hehas taken a necessary and important step. But a still more general perspective is relevantfor clarifying the concept of the infinite. A careful reader will find that the literature ofmathematics is glutted with inanities and absurdities which have had their source in theinfinite. For example, we find writers insisting, as though it were a restrictive condition,that in rigorous mathematics only a finite number of deductions are admissible in a proof asif someone had succeeded in making an infinite number of them. Also old objections whichwe supposed long abandoned still reappear in different forms. For example, the followingrecently appeared: Although it may be possible to introduce a concept without risk, i.e.,without getting contradictions, and even though one can prove that its introduction causesno contradictions to arise, still the introduction of the concept is not thereby justified. Isnot this exactly the same objection which was once brought against complex-imaginarynumbers when it was said: “True, their use doesn’t lead to contradictions. Neverthelesstheir introduction is unwarranted, for imaginary magnitudes do not exist”? If, apart fromproving consistency, the question of the justification of a measure is to have any meaning,it can consist only in ascertaining whether the measure is accompanied by commensuratesuccess. Such success is in fact essential, for in mathematics as elsewhere success is thesupreme court to whose decisions everyone submits. As some people see ghosts, anotherwriter seems to see contradictions even where no statements whatsoever have been made,viz., in the concrete world of sensation, the “consistent functioning” of which he takes asspecial assumption. I myself have always supposed that only statements, and hypothesesinsofar as they lead through deductions to statements, could contradict one another. Theview that facts and events could themselves be in contradiction seems to me to be a primeexample of careless thinking. The foregoing remarks are intended only to establish thefact that the definitive clarification of the nature of the infinite, instead of pertaining justto the sphere of specialized scientific interests, is needed for the dignity of the humanintellect itself. From time immemorial, the infinite has stirred men’s emotions more thanany other question. Hardly any other idea has stimulated the mind so fruitfully. Yet, noother concept needs clarification more than it does. Before turning to the task of clarifyingthe nature of the infinite, we should first note briefly what meaning is actually given tothe infinite. First let us see what we can learn from physics. One’s first nave impressionof natural events and of matter is one of permanency, of continuity. When we considera piece of metal or a volume of liquid, we get the impression that they are unlimitedlydivisible, that their smallest parts exhibit the same properties that the whole does. Butwherever the methods of investigating the physics of matter have been sufficiently refined,scientists have met divisibility boundaries which do not result from the shortcomings oftheir efforts but from the very nature of things. Consequently we could even interpret thetendency of modern science as emancipation from the infinitely small. Instead of the old

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principle natura non facit saltus, we might even assert the opposite, viz., “nature makesjumps.” It is common knowledge that all matter is composed of tiny building blocks called“atoms,” the combinations and connections of which produce all the variety of macroscopicobjects. Still physics did not stop at the atomism of matter. At the end of the last centurythere appeared the atomism of electricity which seems much more bizarre at first sight.Electricity, which until then had been thought of as a fluid and was considered the modelof a continuously active agent, was then shown to be built up of positive and negativeelectrons. In addition to matter and electricity, there is one other entity in physics forwhich the law of conservation holds, viz., energy. But it has been established that evenenergy does not unconditionally admit of infinite divisibility. Planck has discovered quantaof energy. Hence, a homogeneous continuum which admits of the sort of divisibility neededto realize the infinitely small is nowhere to be found in reality. The infinite divisibility ofa continuum is an operation which exists only in thought. It is merely an idea which is infact impugned by the results of our observations of nature and of our physical and chemicalexperiments. The second place where we encounter the question of whether the infinite isfound in nature is in the consideration of the universe as a whole. Here we must considerthe expanse of the universe to determine whether it embraces anything infinitely large.But here again modern science, in particular astronomy, has reopened the question andis endeavoring to solve it, not by the defective means of metaphysical speculation, but byreasons which are based on experiment and on the application of the laws of nature. Here,too, serious objections against infinity have been found. Euclidean geometry necessarilyleads to the postulate that space is infinite. Although euclidean geometry is indeed aconsistent conceptual system, it does not thereby follow that euclidean geometry actuallyholds in reality. Whether or not real space is euclidean can be determined only throughobservation and experiment. The attempt to prove the infinity of space by pure speculationcontains gross errors. From the fact that outside a certain portion of space there is alwaysmore space, it follows only that space is unbounded, not that it is infinite. Unboundednessand finiteness are compatible. In so-called elliptical geometry, mathematical investigationfurnishes the natural model of a finite universe. Today the abandonment of euclideangeometry is no longer merely a mathematical or philosophical speculation but is suggestedby considerations which originally had nothing to do with the question of the finitenessof the universe. Einstein has shown that euclidean geometry must be abandoned. On thebasis of his gravitational theory, he deals with cosmological questions and shows that afinite universe is possible. Moreover, all the results of astronomy are perfectly compatiblewith the postulate that the universe is elliptical. We have established that the universeis finite in two respects, i.e., as regards the infinitely small and the infinitely large. Butit may still be the case that the infinite occupies a justified place in our thinking, that itplays the role of an indispensable concept. Let us see what the situation is in mathematics.Let us first interrogate that purest and simplest offspring of the human mind, viz., numbertheory. Consider one formula out of the rich variety of elementary formulas of numbertheory, e.g., the formula

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12 + 22 + 32 · · · + n2 =n(n + 1)(2n + 1)

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Since we may substitute any integer whatsoever for n, for example n = 2 or n = 5, thisformula implicitly contains infinitely many propositions. This characteristic is essentialto a formula. It enables the formula to represent the solution of an arithmetical problemand necessitates a special idea for its proof. On the other hand, the individual numericalequations

12 + 22 =2 · 3 · 5

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12 + 22 + 32 =5 · 6 · 11

6.

can be verified simply by calculation and hence individually are of no especial interest. Weencounter a completely different and quite unique conception of the notion of infinity in theimportant and fruitful method of ideal elements. The method of ideal elements is used evenin elementary plane geometry. The points and straight lines of the plane originally are real,actually existent objects. One of the axioms that hold for them is the axiom of connection:one and only one straight line passes through two points. It follows from this axiom thattwo straight lines intersect at most at one point. There is no theorem that two straight linesalways intersect at some point, however, for the two straight lines might well be parallel.Still we know that by introducing ideal elements, viz., infinitely long lines and points atinfinity, we can make the theorem that two straight lines always intersect at one and onlyone point come out universally true. These ideal ”infinite” elements have the advantageof making the system of connection laws as simple and perspicuous as possible. Moreover,because of the symmetry between a point and a straight line, there results the very fruitfulprinciple of duality for geometry. Another example of the use of ideal elements are thefamiliar complex-imaginary magnitudes of algebra which serve to simplify theorems aboutthe existence and number of the roots of an equation. Just as infinitely many straight lines,viz., those parallel to each other, are used to define an ideal point in geometry, so certainsystems of infinitely many numbers are used to define an ideal number. This applicationof the principle of ideal elements is the most ingenious of all. If we apply this principlesystematically throughout an algebra, we obtain exactly the same simple and familiar lawsof division which hold for the familiar whole numbers 1, 2, 3, 4, . . . We are already in thedomain of higher arithmetic.