14 DAY TRIAL // We are accustomed with the fact that people can use our everyday language to say all sorts of nonsensical stuff. It is also possible to say things that are false. A story of a flying horse can be understood but it is false; the claim that "our faces are formless like the stars" (Tristan Tzara) doesn't even have meaning (at least not in any usual manner). In other words, the linguistic rules that we use to create our speech acts allow the production of both false and meaningless statements.
Similarly, the rules of drawing and of understanding drawings allow the creation of meaningless pictures ? as the impossible figures show. But for a long time, it was thought that mathematics is a special language immune to the possibility of creating either false or meaningless statements. But G?del's theorems have basically showed that this is not the case. Math is much more similar to usual languages than once thought. (Math is more rigorous than natural languages, with the emphasis on "more" and not on "rigorous" ? i.e. there is a gradual transition from everyday language to the high school math to the hard-core formal math.)
Formal stuff
In the formal description of mathematics, one has a set of axioms and a set of formal rules that allow the generation of theorems ? other propositions. When one has a set of axioms and a set of rules, one can ask two questions:
Are these axioms and rules coherent? There is always the possibility that the set of axioms is faulty ? that it allows one to generate both the statement that A is true and that A is false. This is no good because usually such contradictions don't stay contained ? they spread throughout the entire formal system and allow the demonstration that any statement is both true and false. This makes the entire formal system useless.
Are these axioms and rules sufficient, do they make a complete set? Apart from the dire possibility of incoherence there also exists the possibility that some of the statements generated by using the formal rules are neither true nor false ? in other words, you can formulate the statements but you cannot prove them to be either true nor false.
In 1921, the mathematician David Hilbert proposed that all mathematics should be placed on a very clear formal basis and that the issues of coherence and completeness should be settled. At that point he had already done that for the Euclidean geometry, revealing a large number of hidden axioms and proving that, when all these axioms are taken into account, the Euclidean geometry is both complete and coherent. The next step was doing the same thing for arithmetic.
But, to everybody's surprise, Kurt G?del has proven in 1930 that one cannot prove arithmetic to be either coherent or complete (if one assumes it to be coherent it isn't complete). Moreover, he has showed that any formal system that uses arithmetic is also incomplete. As you can imagine, this has quite a few of consequences because virtually every theory involves numbers and counting! This means that any such mathematical theory would generate unprovable statements, much like the way in which usual language can generate meaningless statements. This has left the entire Hilbert's program in shatters. The happy case of Euclidean geometry is the exception rather than the rule.
What is meaning?
There are many theories on meaning. In the same way as the word "truth" is used in many ways (see article), the word "meaning" has a similar fate. There are three basic meanings of the word "meaning":
Meaning as a relation to something external, in the real world. According to this view, the meaning of some statement (or its "significance") is the thing to which it refers to. The meaning of the word "chair" for instance is the set of all actual chairs (the word "chair" somehow metaphorically "points" to these actual chairs).
A more general way to express this perspective is to say that the significance of a statement is defined by the conditions that have to happen in the real world for the statement to be true. This is a somewhat weird perspective because it sees the significance as being determined by the truth, rather than the other way around (the intuitive perspective is that you first understand what a statement is saying and then you decide whether or not it is true). But when one thinks about what something "refers to" this reverse relation exists.
Meaning as the psychological effect. According to this view, the meaning of a sentence (or its "sense") is the set of all thoughts and feelings triggered in your head by the statement. For example the meaning of the word "chair" is what you think about when you hear or read or think about that word. If you are a chair designer the word "chair" might "point" to some possible chair that hasn't yet been created ? this "pointing" happens in your head and you might do something with it.
There is some relation between the significance and the sense of a sentence, between the external and the internal meanings. The significance is what makes the connection between what two different people understand of the same statement. Without significance there would be no communication, no mutual understanding. Without the sense there would be nothing to communicate about.
Meaning as the set of consequences. This is a more general perspective, that tries to incorporate both the significance, the sense and the relation between them. Moreover, this view takes into consideration explicitly the fact that the meaning of something depends on the context. The meaning of the word "chair" is different according to whether the conversation is held in a restaurant, in a dentist's office, or prior to an execution on the electric chair.
According to this view, the meaning of a sentence is the type of influence the sentence has on the behavior of the subject (who listens to it, reads it, thinks of it etc.). If that sentence wouldn't have been uttered, the subject would have done something else than he had actually done. The difference between what one would have done and what one has done is the meaning. This meaning is obviously rather speculative, but this is hard to avoid ? meaning isn't something that one can observe in the same way as one can observe a thing (say, a flower). (Read more on the concept of information.)
What meaning could a formal statement have?
What is the meaning of a formal, mathematical sentence? Such a statement has no significance ? it does not "point" to any real, actual, thing. This is what one understands by "formal" ? something is formal precisely because it doesn't talk about something. It just talks in a very rigorous fashion! (About nothing, about anything.) Still, the statements do have certain sense for various mathematicians, as they provoke various thoughts in these guys' heads. But how can the mathematicians agree with each other?
In usual conversations we can agree to each other because what we say doesn't just impact our minds but also refers to stuff in the external world. But mathematicians don't have that! What they do have are the mathematical consequences, derived formally. So the meaning of such formal statements resides in their formal consequences.
What formal consequences does a statement which is neither true nor false have? Answer: None. Unlike statements in physics that can be put against observable facts for determining their truth, mathematical statements derive their truth from the fact that they can be proven true within that formal system. So, if they cannot be proven, they are neither true nor false. But if they are neither true nor false than one cannot use them in any way ? they have no formal consequences.
So, what G?del had proved was that formal systems sufficiently complex to incorporate arithmetic were capable of generating meaningless statements. Just like everyday language!
Many mathematicians don't see it this way because they are so keen on formal stuff that they have difficulties noticing subtleties regarding the meaning of various words ? such as the word "truth". For example, Gregory Chaitin wrote: "Begin by considering the liar's paradox: 'This statement is false.' This statement is true if and only if it is false, and therefore it is neither true nor false. Now let's consider 'This statement is unprovable.' If it is provable, then we are proving a falsehood, which is extremely unpleasant and is generally assumed to be impossible. The only alternative left is that this statement is unprovable. Therefore, it is in fact both true and unprovable. Our system of reasoning is incomplete, because some truths are unprovable."
What's wrong with this presentation is his use of the word "true" ? it is isn't very clear what he means by it. It seems that he uses the word in a physicist's sense, but that sense doesn't apply! The proposition he refers to ("This statement is unprovable.") isn't both true and unprovable, it is unprovable and thus neither mathematically true nor false, and thus mathematically meaningless. His presentation makes us believe that we have in our heads a "system of reasoning" on one hand and some additional "system of truth detection" on other hand. But we only have the "system of reasoning" and the task of this system is to produce "truth" ? there's no additional system.
The issue isn't that "the system of reasoning" is supplemented by some other system, but that this system is different from what mathematicians and logicians assumed it to be ? a purely mechanical, deduction machine that has nothing to do with making choices or gathering information.
"When G?del's theorem first appeared, with its more general conclusion that a mathematical system may contain certain propositions that are undecidable within that system, it seems to have been a great psychological blow to logicians, who saw it at first as a devastating obstacle to what they were trying to achieve," Edwin Jaynes wrote. "Yet a moment's thought shows us that many quite simple questions are undecidable by deductive logic. [...] For example, two persons are the sole witnesses to an event; they give opposite testimony about it and then both die. Then we know that one of them was lying, but it is impossible to determine which one. In this example, the undecidability is not an inherent property of the proposition or the event; it signifies only the incompleteness of our own information.
"But this is equally true of abstract mathematical systems; when a proposition is undecidable in such a system, that means only that its axioms do not provide enough information to decide it. But new axioms, external to the original set, might supply the missing information and make the proposition decidable after all. In the future, as science becomes more and more oriented to thinking in terms of information content, G?del's result will be seen as more of a platitude than a paradox."
The myth of the "both true and unprovable" (in conjunction with the mistaken, and actually na?ve, assumption that our "system of reasoning" is purely deductive) has been abused in incredible ways, most famously by Roger Penrose. He tries to use G?del's theorems for "proving" that our minds have all sorts of mysterious abilities. He then tries to discover where exactly that supplementary "system of truth detection" is and what it is made of!
"I sat in on a course when I was a research student at Cambridge, given by a logician who made the point about G?del's theorem that the very way in which you show the formal unprovability of a certain proposition also exhibits the fact that it's true," Penrose recalls. "I'd vaguely heard about G?del's theorem ? that you can produce statements that you can't prove using any system of rules you've laid down ahead of time. But what was now being made clear to me was that as long as you believe in the rules you're using in the first place, then you must also believe in the truth of this proposition whose truth lies beyond those rules. This makes it clear that mathematical understanding is something you can't formulate in terms of rules."
In other words, it is supposed that G?del has shown us that we can determine with our mysterious mental powers that a highly abstract formula is true, although that formula is impossible to prove true within that formal system. How the hell does out mind pull off such a feat?! Answer: It doesn't. Penrose's argument is simply based on an erroneous claim: "the very way in which you show the formal unprovability of a certain proposition also exhibits the fact that it's true," he said. Let me show you that this is not so.
Logic and arithmetic are more closely related than you might think
In a previous article I tried to describe the intuitive processes that allow us to have numbers. Numbers exist because we measure things (and not the other way around!). We can measure things in two ways ? either by comparing them with some arbitrary unit of measurement (and say "this thing is so many times greater than the unit") or by equilibrium (saying that this thing can be balanced by these other things which add up together to account for the measured thing).
Physicists call the properties that are measured with the first method extensive properties, and the ones that are measured with the equilibrium method intensive properties. For example volume is extensive while pressure is intensive (it has a certain "intensity" all over the place).
The first method of measuring stuff gives rise to the mathematical idea of multiplication, while the second method to the mathematical idea of summation. One can describe these processes in a formal manner ? we can formulate a bunch of axioms that describe these two processes.
These axioms are the following (you can think about how you get them by thinking about actually measuring various things): both summation and multiplication are: commutative [x + y = y + y], associative [x + (y + z) = (x + y) + z], have a neutral element [if you add nothing to something you get the same something, x + 0 = x, if you multiply something one time you get the same something, x * 1 = x], and, the most subtle property, they are symmetric [you can go in both directions, for example from 1 to 2 to 3 and so on, by adding 1, as well as from 3 to 2 to 1, by adding -1, the symmetric of 1].
The idea that the operations are symmetric creates the negative numbers and the fractional numbers. More over, the fact that the operations are symmetric translates into the fact that each number has two symmetric brothers (or sisters if you prefer) ? one relative to summation and another relative to multiplication. The symmetric is the number that added (or multiplied) to x gives you the neutral element (0 or 1).
There is one more axiom that can be uncovered ? the distributivity axiom. This axiom describes how summation and multiplication are related to each other. It simply says that adding up two things (x and y) and then multiplying the result by a certain factor n, gives you the same end result as if you would multiply x and y by a factor of n separately and only afterwards adding up the partial results [n * (x + y) = (n * x) + (n * y)]. You can imagine, for example, that this simply represents two different groupings of the same weights on the two scales of a balance ? if there are the same weights on both scales the balance is in equilibrium no matter how you arrange the weights.
Now, what's interesting about this set of axioms (by the way, such a formal structure is called a "field") is that it isn't complete. You may want to use these axioms to generate the numbers. The axioms tell you that you have at least two numbers, 1 and 0. How about creating more? You can prove stuff: For instance you can prove that 1 + 0 = 1, that any number multiplied by 0 is 0 etc. You can generate the number -1, the symmetric brother of 1. But there are some things you cannot prove: for example that 1 + 1 is different from 0! Or that (-1) + (-1) is different from 0.
You have two options here: You either assume that 1 + 1 = 0 or that 1 + 1 is different from 0. The funny thing is that if you make the first assumption you obtain the classical bivalent logic. The formula 1 + 1 = 0, once accepted, allows you to demonstrate that 1 is its own symmetric brother (1 = -1)! In this case + is usually called "exclusive or" and * is called the "logical and". This is all the math computers use ? they're fine with the first choice.
However, if you make the second choice, you are allowed to invent a new number ? let's call it "2". Then, you can generate its symmetrics: the number 1/2, and the number -2. However, you now cannot prove that 1 + 2 is different from 0. It's the same story again. If you assume that 1 + 2 is 0 you would get a somewhat weird and useless "polyvalent" logic.
To make the long story short, you can always assume that the unprovable formula is true, that x + 1 is always different from 0, and this allows you to generate all fractional numbers. But you are faced with a choice at every step. Penrose is not at all right to believe that the choice is obvious and that we have some sort of mental capacity to "see" what choice has to be made. We don't have such a capacity, and there isn't any "natural" choice ? that's why we have both bivalent logic and arithmetic! (As well as all sorts of polyvalent logics.)
You could adopt the axiom "x + 1 is always different from 0", which would be a meta-axiom because it doesn't refer to the numbers directly but to statements about numbers, but this would only get you into more trouble. You simply cannot escape the unprovable stuff and, consequently, the need to make choices. On this I can quote Chaitin with no restraints:
"If Hilbert had been right, mathematics would be a closed system, without room for new ideas," Gregory Chaitin wrote. "There would be a static, closed theory of everything for all of mathematics, and this would be like a dictatorship. In fact, for mathematics to progress you actually need new ideas and plenty of room for creativity. It does not suffice to grind away, mechanically deducing all the possible consequences of a fixed number of basic principles. I much prefer an open system. I do not like rigid, authoritarian ways of thinking."
"You can have it, pal!" G?del would have replied.