To illustrate how our “intuitions” can insidiously lead us astray, consider the frequent use of sketches and diagrams to try to make our mathematical abstractions more concrete. These concretizations are almost unavoidable, even to the most mathematically acute minds. For example, David Hilbert, who as we will soon see tried as hard as anyone to impose the strictest rules on mathematics, wrote:
So the geometrical figures are signs or mnemonic symbols of space intuition and are used as such by all mathematicians. Who does not always use along with the double inequality a > b > c the picture of three points following one another on a straight line as the geometrical picture of the idea “between”? Who does not make use of drawings of segments and rectangles enclosed in one another, when it is required to prove with perfect rigor a difficult theorem on the continuity of functions or the existence of points of condensation? Who could dispense with the figure of the triangle, the circle with its center, or with the cross of three perpendicular axes?
Given that even the most rigorous of mathematicians rely on these aids to pure reason, it can happen that some entirely accidental feature of our sketch gets appealed to in the proof; or that the sketches make it appear that something is just plain obvious when it’s not.
Say, for example, you want to prove that the base angles of an isosceles triangle are equal, and by appealing to Sketch 1 below this seems simply obvious, not requiring a proof. Or say you want to prove that all the angles of triangles are acute (less than 90 degrees) and because you have this “truth” in mind you only draw sketches of triangles (Sketch 2) that conform to it. Those are just the only triangles that occur to you.
In other fields of thought as well—for example, and notoriously, ethics—people can have the illusion of “intuitions.” In ethics these illusory intuitions can create a great deal of havoc in the real world. Einstein and Gödel were on that quiet little back road in Princeton precisely because it seemed so intuitively obvious to a great many people in Einstein’s Germany and Gödel’s Austria that the right thing to do was to purify the Aryan nations of the world of their non-Aryan elements. Mathematics is not the only area where things can seem intuitively obvious and be quite, quite false. But mathematics has seemed to be unique because it, and it alone, seems to offer a method for truth-purification: the axiomatic system. No wonder that a rationalist like the seventeenth century’s Spinoza, whose Portuguese-Jewish family was living in Amsterdam for reasons similar to those that had brought Einstein and Gödel to Princeton, had wanted to appropriate the axiomatic methods of mathematics and apply it to human ethics. The desire to universalize the truth-purifying rigor of the mathematical method is precisely what the epistemological movement known as rationalism is all about. But the prior question that must be addressed, before one can begin to think about the possibility of generalizing the mathematical method, is: what exactly is the axiomatic system, and how does it attain its enviable rigor?
The idea behind the axiomatic (or “axiomatized”—I’ll use the two terms interchangeably) system is that the manifold truths of some particular branch of mathematics, say geometry or arithmetic, can be organized into axioms, rules of inference, and theorems. The axioms are the basic truths of the system, intuitively obvious. We understand what they mean and that appears sufficient for knowing that they are true. They do not require any proof beyond this. We then use the truth-preserving rules of inference1 to obtain other, nonobvious truths, the theorems, which flow from these givens.
Sketch 1
Sketch 2
For example, consider arithmetic, the simplest branch of all mathematics. Arithmetic concerns the structure of the natural numbers—again, the regular old counting numbers together with 0—and the relationships between them as given by the operations of addition, multiplication, and the successor relation, which takes you from any number n to the number immediately following it in the natural order (i.e., n + 1). All other arithmetical operations, such as subtraction and division, can be defined in terms of these three.
In 1889 Giuseppe Peano (1858–1932) reduced arithmetic to five axioms. Here are the first three: 0 is a number. The successor of any number is a number. No two numbers have the same successor. All three appear trivial, which is exactly how we want our axioms. The axioms are so trivial that we can assume that they are true without proving them, with all else following from them, like a huge twisting plant growing out of a simple seed. If we want the whole luxuriant growth to be certain then we want there to be no question possible about the truth of the axioms—and this “no question possible” is basically what we mean by “intuitively obvious” or “given” or “trivial” or “self-evident.”
The theorems of an axiomatic system, on the other hand, are only accepted as true once they are proved, derived from the axioms or derived from other theorems, using truth-preserving rules of inference. Think of it this way, if you care to: Axioms are like the classic first-borns in families: adored simply for being. Theorems are the children that come after, those who have to prove themselves worthy to gain acceptance. (First-borns can ignore the analogy. To me, a third-born, the metaphor has a certain appeal.)
So in an axiomatic system (first devised by the ancient Greeks, in particular, Euclid), we begin with a few (the fewer the better) axioms, which are supposed to be intuitively obvious, and then proceed onward to prove whatever follows from these axioms. (The fewer the better, because we want to keep our appeals to intuition to a minimum to maximize certainty.) In place of a libertarian policy of “let’s-just-depend-on-the-good-intentions-(intuitions)-of-citizens-(mathematicians)-to- do-the-right-thing,” the axiomatic system imposes some strict governmental controls. In place of random appeals to intuitions, there is to be general consensus on what is directly given, the bedrock, with everything else subjected to systematic rule-regulation. You can think of axiomatization as sort of “big government mathematics.” The motive behind the axiomatic system is to maximize certainty by minimizing appeals to intuitions, restricting them to the few ineliminable axioms. But the latter are crucial because, after all, we do have to start somewhere.
For much of the history of Western thought, at least since the time of Euclid, the axiomatized system was generally deemed to represent mathematics—and thus knowledge itself—in its most perfect form. Gottlob Frege, who further simplified Peano’s axiomatic system for arithmetic by deriving Peano’s five axioms from a single axiom, said: “In mathematics we must always strive after a system that is complete in itself.” It is this system-building that accounts, Frege said, for the unique certainty of mathematics and “no science can be so enveloped in obscurity as mathematics, if it fails to construct a system.”
The drive for limiting our intuitions went even further. The aim became to eliminate intuitions altogether. This aim is what brings us, at long last, to the notion of a formal system. A formal system is an axiomatic system divested of all appeals to intuition.
Why take the drastic step of intuition-divestment? Well, intuitions, as we said before, are a tricky business. Though genuine intuitions are true, how can we tell when we are in possession of the genuine article? Maybe we can’t. Maybe the feel, the urgent cogency that compels belief, is exactly the same whether the intuition is for real or is not. Then what good is the appeal to intuition? So, all things being equal, it would seem a good thing to rid ourselves of these appeals, especially when pursuing the “severest of all disciplines.”
In fact, all things were not equal, and the inequality was such as to give added impetus to the drive to ruthlessly eliminate intuitions from mathematics. The nineteenth century gave us mathematical developments that subverted our confidence in those intuitively obvious givens of our axiomatic systems. (First-borns can go terribly wrong.) The most dramatic of these undermining events was the discovery of non-Euclidean geometry. This unanticipated mathematical development demonstrated that one of the givens of Euclid’s geometry, the notorious parallels postulate, is not so axiomatic after all; in fact it
is possible to construct self-consistent geometries in which it isn’t even true!2 Then set theory, too, delivered us some nasty news about our putative intuitions. The givens of set theory, again so intuitively obvious, lead to the formation of such paradox-infected sets as the set of all sets that are not members of themselves.
Clearly the bedrock consisting of our mathematical intuitions was not much of a bedrock after all. If it was possible to purge our axiomatic systems of appeals to intuitions altogether then that was the way to go.
The elimination of intuitions is accomplished by draining the axiomatic system of all meanings, except those that can be defined in terms of the stipulated rules of the system. The rules, in terms of which everything else is defined, make no claim to being anything other than stipulated. They make no pretense of being descriptive of some objective reality, of independent objects like numbers and sets. A formal system is precisely what we are left with after this meaning-drainage. This deprivation constitutes further “governmental controls,” the most stringent that mathematicians could think of, so that no appeals to intuition come sneaking in. You can think of it as the Communist takeover of mathematics, abolishing private property (meanings), everything taken over by public rules.
A formal system, then, is an axiomatic system—with its primitive givens (the axioms), its rules of inference, and its proved theorems—except that instead of being constructed of meaningful symbols—such as terms referring to the number 0 or to the successor function—it is constructed entirely of meaningless signs, marks on paper whose only significance is defined in terms of the relations of each to one another as set forth by the rules. While pre-purged axiomatic systems were understood as being about, say, numbers (arithmetic) or sets (set theory) or space (geometry), a formal system is an axiomatic system that is not, in itself, about anything. We don’t have to appeal to our intuitions about numbers or sets or space in laying down the givens of the formal system. A formal system is constituted of stipulated rules: that specify the symbols (“alphabet”) of the system; that tell us how we may combine the symbols with one another to produce grammatical configurations (wffs); and that tell us how we may proceed to deduce wffs from other wffs (the rules of inference).
The formalization of axiomatic systems was meant to offer the highest standard of certainty so that we don’t have to depend on our intuitions as to what is mathematically obvious and what is not. It was meant to obviate our reliance on mathematical intuition altogether, to turn our mathematical activity into processes so completely determined by clearly specified rules as to be purely mechanical, requiring no imagination or ingenuity, not even a grasp as to what the symbols mean. To follow the rules of a formal system—and a formal system consists of nothing but rules—is to engage in a combinatorial activity that, consisting purely of recursive functions (roughly speaking, functions that tell you how to arrive at a result by taking the result of another recursive function, or of a really simple basic function3), could be programmed into a computer, that is, that is computable. This activity amounts to figuring things out by using an algorithm,4 a sequence of operations that tells you what to do at each step, depending on what the outcome of the previous step was.
As the previous paragraph aimed to indicate, a whole family of interrelated mathematical concepts makes its appearance with the move toward formalization. The concepts of a mechanical or an effective procedure, of recursive and computable functions, of combinatorial processes and of an algorithm: this family of concepts all mean pretty much the same thing, revolving around the idea of rules that are applied to the results of prior applications of rules, all with no regard to any meanings except for what can be captured in the rules themselves.
Intuition can get no dangerous foothold in a formal system. Intuitions tell us what to think about actual things—about space, about numbers, about sets. We don’t have intuitions about made-up, meaningless symbols and the rigid rules we have set down for manipulating these. We don’t need them. Everything our a priori reason needs to do in a formal system is specified by the rules, which is why the idea of a formal system is so closely connected with the idea of the computer, with what it is that computers can do and how it is that they do it. This is why the concept of the computable is part of the tangle of concepts surrounding the notion of formal systems.
Formal systems, even if their implications are sufficiently convoluted so that real mathematical cunning is required to get to them, have a transparency that precludes intuition. Intuition is (reputedly) a way of trying to circumnavigate the essential opacity of actual things, a way of making contact with them, and it’s a way that had proved itself to be eminently unreliable in mathematics just as elsewhere. A mathematics done formally is a mathematics purged of any “given” truths—those claiming an unquestionable source in the “true nature of things,” in and of themselves.
If it could be shown that logically consistent formal systems are adequate for proving all the truths of mathematics, then we would have successfully banished intuitions from mathematics. (The proviso of “logically consistent” is of course necessary, since from inconsistent systems one can prove anything at all.) We would have shown, too, that mathematics should not be considered as inherently about anything. By banishing intuitions we would be dissolving away the putative objects of mathematical descriptions. We would be showing mathematics not to be descriptive at all.
The assertion of the possibility and desirability of banishing intuitions by showing formal systems to be entirely adequate to the business of mathematics is the metamathematical view known as formalism.
In formalism’s retelling, mathematics becomes chess raised to a higher order of intricacy. There is, we can all agree, no objective chess reality that the system of chess captures. The stipulated rules constitute the whole truth of chess. Similarly, according to formalism, the stipulated rules constitute the whole truth of mathematics. We win in mathematics by proving theorems—that is, by showing some uninterpreted string of symbols to follow from other uninterpreted strings of symbols, using the agreed-upon rules of inference. There is no external truth against which mathematics has to measure itself.
Gödel’s first incompleteness theorem states the incompleteness of any formal system rich enough to express arithmetic. So Gödel’s conclusion, you might suspect, has something to say about the feasibility (or lack thereof) of eliminating all intuitions from mathematics. The most straightforward way of understanding intuitions is that they are given to us by the nature of things; again intuition is seen as the a priori analogue to sense perception, a direct form of apprehension. So Gödel’s conclusion, in having something to say about the feasibility (or lack thereof) of eliminating all appeals to intuitions from mathematics might also have a thing or two to say about the actual existence of mathematical objects, like numbers and sets. In other words, the adequacy of formal systems—their consistency and completeness—is linked with the question of the ultimate eliminability of intuitions, which is linked with the question of the ultimate eliminability of a mathematical reality, which is the defining question of mathematical realism, or Platonism. It is because of these linkages that Gödel’s conclusions about the limits of formal systems have so much to say. This is how they got to be the most verbose theorems in the history of mathematics and how they were understood, at least by their author, to assert the metamathematical position to which he had given his heart and soul. The young student had found a proof for a theorem, the first incompleteness theorem, that had the rigor of mathematics and the reach of philosophy.
Mathematical verbosity, as opposed to verbosity of any other sort, could not have better suited the personal eccentricities of Kurt Gödel, a man who had so much to say on the nature of mathematical truth and knowledge and certainty, but wanted to be able to say it using only the rigorous methodology of mathematics. With a proof in hand, he would not have to involve himself in the sorts of combative human conversations he regarded with distaste, maybe even with horror. Th
ere never was a man, I’ll wager, who combined so much conviction with so little inclination to argue his convictions by the normal means given to us, viz. human speech.
The irony of course is that while his theorems were accepted as of paramount importance, others did not always hear what he was attempting to say in them. They heard—and continue to hear—the voice of the Vienna Circle or of existentialism or postmodernism or of any other of the various fashionable outlooks of the twentieth century. They heard everything except what Gödel was trying to say.
Math Goes Formal
The leading advocate of formalism was David Hilbert, who was the most important mathematician of his day. “Mathematics,” wrote Hilbert, “is a game played according to certain simple rules with meaningless marks on paper.” His proposal to formalize one branch of mathematics after the other, starting with the most basic branch of all, arithmetic, came to be called the Hilbert program. The successful completion of the Hilbert program would offer significant vindication of formalism, explaining the sui generis aprioricity of mathematics as derivative from the stipulation of rules.
Mathematicians, according to formalism, are not in the business of discovering descriptive truths, whether of the real world of things in physical space or the trans-empirical world of numbers and sets. They were never really meant to discover, for example, how many lines parallel to a given line can run through a given point in space that isn’t on that line. They are simply in the business of manipulating the mechanical rules of self-enclosed formal systems that are complex enough to test the deductive skills of mathematicians.
Incompleteness Page 11
