A half year ago, I promised to try finishing my post on why there is something rather than nothing by the end of February. I had it outlined and ready to go, but I never went. Now I have to make up something to explain why.
As you probably know, I have a strong affinity for visiting Emergency Rooms. By the time I recovered, we were on the road again. Once we find a place to stay, the TV goes on, and the only time I get to make love with my laptop is in the wee hours of the morning. Once there, we spend the day seeing the sights, and turning on the TV as soon as we get back. Of course, we visit friends and family along the way, and no trip would be complete if I didn’t investigate another Emergency Room at a new hospital. (By the way, if you have a choice, I highly recommend the Sentara hospital in Williamsburg, VA.) Along with everything else, I decided it’s time to learn German.
I also decided to include what was going to be my dissertation (from 55 years ago) in the math book I’m writing. I had already delivered 3 lectures on it along with mimeographed notes when two catastrophes conspired against me. The first was a change in requirements in our department for a PhD, and worse yet was a disastrous marriage. I was earning more than enough money with both a fellowship and a teaching assistantship, but somehow it was always gone by mid month. I had to find a job that paid better than teaching, which ended any thoughts of finishing my degree.
Those notes are long gone, but I still remember everything I was going to do, and I decided to at least outline everything and fill it in when I actually start writing the chapters. If nothing else torpedoed finishing my work from February, this definitely did the job.
My dissertation was to show that mathematics is simply a type of linguistic grammar, which was (and probably still is) an approach no one else had taken. The underlying ideas were developed over a period of 10 years (I’m slow). Starting in 1958, I came up with the idea that axioms (or postulates) were nothing more than definitions expressed as equations. In 1962, I found simple definitions for mathematics and science. Again in 1966, I found a definition for languages, their grammars, and equality. By 1968, I found a definition for equality that worked in any language, and found that the only way it worked for mathematics is if the language was completely unambiguous (something I called a semasia). At this point, I started specifying the underlying grammars for math and logic.
Looking at these ideas in more detail, viewing axioms as definitions had roots as early as the the turn of the 20th century, where axioms were were the same as postulates, which were the assumptions needed to create a system of mathematics or logic. Originally, axioms were rules on how to handle numbers, and postulates were “self-evident” observations about geometry. Even into the 1950s, some mathematicians were still clinging to a distinction between the two. Edward Huntington, who made some major contributions to logic, considered that axioms “should only be applied to statements of fact [including] obviously true statements about certain definite operations on angles or distances”, and postulates are “conditions which a given system may or may not happen to satisfy”.
The concept I used was that there are two types of definitions: extrinsic and intrinsic. An extrinsic definition depends on stating the meaning of a term using previously defined terms. In such a definition, the term being defined appears by itself on the left side of a single equation, and previously defined terms are on the right. An intrinsic definition is an equation showing how new terms are to be used. When several terms are being defined simultaneously, the intrinsic equations are called codefinitions. Codefinitions are necessarily intrinsic. Of course this all depends on the meaning of equality, which I didn’t untangle until 8 years later. Axioms are just the codefinitions needed to develop a system of mathematics or logic.
Repeating, a definition is extrinsic if it stands by itself and is expressed in terms of what has already been defined, and it is intrinsic if it only shows how an idea is to be used.
As examples of what I’m trying to explain, let’s start with ‘P’, ‘Q’, ‘’, ‘’, and ‘~’ from logic. ‘P’ and ‘Q’ represent statements which must be either true or false (propositions). The symbols ‘’, ‘’, and ‘~’ stand for ‘or’, ‘and’, and ‘not’ respectively.
Consider the two statements
PQ = ~(~P~Q) and
~(~P~Q)~(~PQ) = P.
These probably don’t make any sense without understanding what they are saying, but the first statement is an example of an extrinsic definition of ‘P and Q’ using ‘or’ and ‘not’ which are assumed to have been previously defined. Note that ‘and’ occurs only once and on the left side of the equation.
The second statement shows an intrinsic definition of ‘~’, assuming that ‘or’ has already been defined. Here, ‘~’ occurs many times, and is allowed to be on both sides of the equation.
What many logicians don’t realize is that the second statement is actually a definition of ‘not’. The idea of intrinsic definitions is not well known, even among mathematicians. I hope that this didn’t muddy the waters too badly.
Now it’s time to ask exactly what math and science are, and is mathematics a branch of science? When I came up with definitions in 1962, I had no idea that Nobel prize winner, Eugene Wigner, had written his famous paper The Unusual Effectiveness of Mathematics in the Natural Sciences two years earlier. I didn’t discover it until the mid 1970s. This is precisely what my definitions were aimed at. It also counters Gauss’s description of math as the “Queen of Sciences”.
Although I knew I wasn’t somewhere out in left field, I didn’t find any indication that anyone else had thought this through. With the advent of Google in the mid 2000s, I learned that the mathematician G. H. Hardy had beaten me to the general definition of math by about 20 years. As far as I know, no one else has come up with similar definition for science.
It takes a while to understand why the definition of science works. It’s almost intuitive seeing the definitions side by side, why they answer Wigner’s question. Many other mathematicians have hit on related precepts, although no one else has followed through with how to use them. For me, it took understanding what language is to see how everything fits together.
In defining language, I looked at what lay behind the construction of a language – a grammar. From this point of view, it’s not only possible to analyze a language, but you can also use it to build a language. Many other approaches (Naom Chomsky, for instance) specify exactly how a language is used for a communication to be a language. The difference is that my definition is descriptive while many others are prescriptive. One result is that according to Chomsky, Pirahã is not a human language – a danger of the prescriptive approach.
In looking at a grammar, it becomes obvious how useful a concept like equality would be. It can be built directly into any grammar. The problem is that for a general language, equality isn’t quite what is needed for mathematics. To get there requires an unambiguous language to gain the full power of equality. Having that in place allows the construction of math, logic, and the sciences. The next problem is to show how to actually do it.
This was the heart of my dissertation. I needed to show that math, logic, and the sciences were completely independent of each other, and that math, in particular could be built from linguistics without involving logic. Depending on how you start, it can be done several different ways. I chose the traditional route using sets and classes.
Set theories are designed to prevent inherent contradictions like Russell’s paradox (which is a variation on the liar’s paradox). With some restrictions, sets can be generalized into much larger structures called classes without incurring additional paradoxes. Both sets and classes can be visualized as Venn diagrams.
What happens with the linguistic approach is that instead of starting with sets, classes are defined first and sets are treated as special types of classes. In doing this, elements of sets and classes are never intrinsically defined. Elements are indirect structures that depend on how sets are (intrinsically) defined.
This final step showing how math can be developed directly from linguistics is what I was looking at when I was distracted from finishing the post on the origin of matter in February.
Once you have classes, defining sets is pretty straight-forward. In defining classes, I used 5 intrinsic codefinitions, and was on my merry way when I discovered a much prettier way by accident. In 1933, Edward Huntington (remember that name?) published an intrinsic definition for negation in logic. He amended it to its present form in 1934. He didn’t interpret it as a definition, and neither has anyone else. It’s called Huntington’s axiom.
It turns out that it is easily reinterpreted linguistically and is precisely what I needed for class theory. I had looked at it earlier, and not being the brightest kid on the block, I discarded it because I couldn’t crack it open to use everything that it unlocks. Huntington did. In addition, he found an exceedingly clever proof for the two distributive properties from logic that also carry over into class theory.
To gain some insight on how brilliant his work was, consider that another axiom was proposed that same year. It was called Robbin’s axiom, and in spite of work by top mathematicians of the time, no one could prove that logic was derivable from it. It took over 60 years before the problem was solved, and even then, it wasn’t by a human. A specialized theorem-proving computer program finally did it.
Continuing this saga, in 2000, Stephen Wolfram started with a theorem-proving program and started looking for the shortest axiom that could produce a complete logic system. When he found it, it took several hundred steps to prove that it worked. That’s something a bit beyond what a normal human might do in a lifetime. His result is called the Wolfram axiom.
I decided to rewrite my section on class theory using that last system produced entirely by a human – the Huntington axiom. My decision was based on the lengths of the proofs and the sheer ingenuity it took to derive them.
Most of my time has gone to reworking his method, trying to make it as simple as possible. So there you are – my top-ranking excuses.
The Infrequent Atheist 