Disclaimer: I stole the title 'A Mathematician's Apology' from the exposition of G. H. Hardy. If you have the time, do read it. It can either be a 50-page rant by a mathematician, or a 50-page journey into the mind of a working mathematician (which has been unjustly presented in the popular culture).
Actually I am lying. Why does the article have to read like one or the other? I had merely constructed a false dilemma. The article could just as well have been as thrilling as a murder story. More likely, you will not even have trawled through the first paragraph, and therefore, do not have any opinion of the piece whatsoever. Or do you?
Recently, I had the pleasure of carrying out numerous discussions with old friends, who have all gone on different paths in their lives, and formed strong opinions about the world, influenced by their respective fields. The general consensus is that logical thinking and argumentation is the way to go. It must have come as a surprise, when I, of all people, expressed doubt about the infallibility of logic. After all, isn't the job of a mathematician or physicist to form, discover and analyse logical statements about the mathematical or physical world?
As I see it, most professionals, and even scientists, hold mathematics and physics in good faith: Physics is a pillar of human knowledge, holding unshakable truths about the universe that we inhabit. The language of physics is mathematics, which is the oracle of last resort, but presumably, always knows right from wrong.
This is where I begin my apology. (1) Physics is not logically consistent, (2) the language of physics is not always mathematics, and (3) mathematics is... not always logical.
(1) The orthodox presentation of modern physics gives two fundamental descriptions of nature: Quantum field theories, and general relativity. Both claim the status of a "fundamental theory", but neither can be derived from the other. This example should shatter the myth that physics is consistent. String theorists will readily tell you otherwise. They say that theirs is a mathematically consistent framework that can incorporate quantum mechanics and gravity --- a theory of everything! Of course, we don't really know how to falsify string theory, and one might argue that it's a matter of taste (or indoctrination!) to accept a physical theory only after intense empirical scrutiny.
I don't preach empiricism, nor do I dismiss it. It is a big question mark to me. Hence, my doubts about the widespread use of "evidence-based" thinking in the real world. Law demands evidence. Medical research is judged on "sample size" and quality of "control subjects". Economics and policy making is often obsessed with analyzing data, numbers, graphs and, God-forbid, "measures" of human desire and emotion. So, here is the confession of a student of physics: I am not entirely sure about the role of experimental evidence in the grand pursuit of the "truth".
(2) Is the language of physics, mathematics? Sometimes. I will not pretend that science is completely honest. In fact, honesty gets thrown out of the window when the really unsettling questions are asked. Consider the following remarks, all made by professors teaching me quantum mechanics:
What can we make of the meaning of "truth" and "falsehood", if in our physical laws, we sometimes reject the arbitration of mathematical logic? Do we implicitly concede ultimate judgement to a deity of sorts? Does "judgement" even make any sense in this case?
The problem runs even deeper. Even if physics were to be successfully axiomatized, there are still disturbing problems in the foundations of mathematics. Again, the orthodox treatment (actually I'm not too sure about this --- nowadays, it seems trendy to take a contrarian view), is based on axiomatic set theory. For instance, the numbers that we are used to playing with, have a rigorous (but not necessarily firm) foundation in set-theoretic language. So, discrete mathematics, combinatorics, and computer science have strong foundations in this approach. But even among mathematicians, there are vehement objections to set theory as the de-facto foundation of mathematics.
Set theory has a "constructivist" philosophy --- in order to prove the existence of an object, you have to "construct it". A huge problem arises when one wishes to admit the continuum (implicitly used, by the way, whenever you use calculus). The continuum is so huge, we could never construct it on paper, so we construct it with our mind! The axiom of choice is put in by hand as an axiom of set theory, and voila! We can now say that the real numbers exist. If that doesn't give you a tinge of discomfort, perhaps your new superpower would --- with the axiom of choice, you could chop a pea into little pieces and reassemble them into the Sun! But of course, physics ceases to be based on mathematics here, and elsewhere, where not needed...
(3) Perhaps we simply have not developed physics and mathematics sufficiently, but ultimately, should we not be able to refine all our knowledge into a coherent and logical framework? Godel's incompleteness theorems roughly says that there are inherent limitations to axiomatic systems (I will not butcher them with my inadequate understanding of Godel). Suffice to say, even for the arithmetic which we are all too familiar with, there are undecidable statements. If you are a very logical and observant person, you might have spotted my use of the "Liar's paradox" in the first sentence of the second paragraph. Was I lying? Or was I not? The prudent teacher should stop the inquisitive student at this moment.
Even this may not dull your optimism about human knowledge. Who cares about the undecidable statements anyway? As long as everything is logical... But what... is logic? From here, there are at least two possible paths. The first is that of a mathematician delving in mathematical logic. Theirs is a world of constructivist logic, intuitionistic logic, symbolic logic, fuzzy logic, higher-order logic, quantum logic... etc. Exotica that is far-removed from the everyday logic that we use in practically all human activity, and which we hold so exceedingly dear in our desire to tell fact-from-fiction, and to judge each other. The weapon which we wield so readily, but understand so little about. The basis of law, government and order in human society, and concurrently the basis for war, capitalist exploitation, and self-actualization.
The second path is, of course, insanity.
Actually I am lying. Why does the article have to read like one or the other? I had merely constructed a false dilemma. The article could just as well have been as thrilling as a murder story. More likely, you will not even have trawled through the first paragraph, and therefore, do not have any opinion of the piece whatsoever. Or do you?
Recently, I had the pleasure of carrying out numerous discussions with old friends, who have all gone on different paths in their lives, and formed strong opinions about the world, influenced by their respective fields. The general consensus is that logical thinking and argumentation is the way to go. It must have come as a surprise, when I, of all people, expressed doubt about the infallibility of logic. After all, isn't the job of a mathematician or physicist to form, discover and analyse logical statements about the mathematical or physical world?
As I see it, most professionals, and even scientists, hold mathematics and physics in good faith: Physics is a pillar of human knowledge, holding unshakable truths about the universe that we inhabit. The language of physics is mathematics, which is the oracle of last resort, but presumably, always knows right from wrong.
This is where I begin my apology. (1) Physics is not logically consistent, (2) the language of physics is not always mathematics, and (3) mathematics is... not always logical.
(1) The orthodox presentation of modern physics gives two fundamental descriptions of nature: Quantum field theories, and general relativity. Both claim the status of a "fundamental theory", but neither can be derived from the other. This example should shatter the myth that physics is consistent. String theorists will readily tell you otherwise. They say that theirs is a mathematically consistent framework that can incorporate quantum mechanics and gravity --- a theory of everything! Of course, we don't really know how to falsify string theory, and one might argue that it's a matter of taste (or indoctrination!) to accept a physical theory only after intense empirical scrutiny.
I don't preach empiricism, nor do I dismiss it. It is a big question mark to me. Hence, my doubts about the widespread use of "evidence-based" thinking in the real world. Law demands evidence. Medical research is judged on "sample size" and quality of "control subjects". Economics and policy making is often obsessed with analyzing data, numbers, graphs and, God-forbid, "measures" of human desire and emotion. So, here is the confession of a student of physics: I am not entirely sure about the role of experimental evidence in the grand pursuit of the "truth".
(2) Is the language of physics, mathematics? Sometimes. I will not pretend that science is completely honest. In fact, honesty gets thrown out of the window when the really unsettling questions are asked. Consider the following remarks, all made by professors teaching me quantum mechanics:
"There are no theorems in physics."
"Everything in quantum mechanics starts from these axioms: ..."
"Physics is not mathematics. Mathematics is just a tool."
"This is what a proof looks like in physics (handwaves and chuckles to himself)..." [meant as a form of dark humour I suppose]
"Mathematics can be invented... for physics." [I came up with this lousy one myself]David Hilbert immortalized this dilemma in his "Sixth Problem": Axiomatize all of physics. This is one of the few remaining problems and will earn you both a Nobel prize and a Fields medal, if you are interested in these sort of decorations.
What can we make of the meaning of "truth" and "falsehood", if in our physical laws, we sometimes reject the arbitration of mathematical logic? Do we implicitly concede ultimate judgement to a deity of sorts? Does "judgement" even make any sense in this case?
The problem runs even deeper. Even if physics were to be successfully axiomatized, there are still disturbing problems in the foundations of mathematics. Again, the orthodox treatment (actually I'm not too sure about this --- nowadays, it seems trendy to take a contrarian view), is based on axiomatic set theory. For instance, the numbers that we are used to playing with, have a rigorous (but not necessarily firm) foundation in set-theoretic language. So, discrete mathematics, combinatorics, and computer science have strong foundations in this approach. But even among mathematicians, there are vehement objections to set theory as the de-facto foundation of mathematics.
Set theory has a "constructivist" philosophy --- in order to prove the existence of an object, you have to "construct it". A huge problem arises when one wishes to admit the continuum (implicitly used, by the way, whenever you use calculus). The continuum is so huge, we could never construct it on paper, so we construct it with our mind! The axiom of choice is put in by hand as an axiom of set theory, and voila! We can now say that the real numbers exist. If that doesn't give you a tinge of discomfort, perhaps your new superpower would --- with the axiom of choice, you could chop a pea into little pieces and reassemble them into the Sun! But of course, physics ceases to be based on mathematics here, and elsewhere, where not needed...
(3) Perhaps we simply have not developed physics and mathematics sufficiently, but ultimately, should we not be able to refine all our knowledge into a coherent and logical framework? Godel's incompleteness theorems roughly says that there are inherent limitations to axiomatic systems (I will not butcher them with my inadequate understanding of Godel). Suffice to say, even for the arithmetic which we are all too familiar with, there are undecidable statements. If you are a very logical and observant person, you might have spotted my use of the "Liar's paradox" in the first sentence of the second paragraph. Was I lying? Or was I not? The prudent teacher should stop the inquisitive student at this moment.
Even this may not dull your optimism about human knowledge. Who cares about the undecidable statements anyway? As long as everything is logical... But what... is logic? From here, there are at least two possible paths. The first is that of a mathematician delving in mathematical logic. Theirs is a world of constructivist logic, intuitionistic logic, symbolic logic, fuzzy logic, higher-order logic, quantum logic... etc. Exotica that is far-removed from the everyday logic that we use in practically all human activity, and which we hold so exceedingly dear in our desire to tell fact-from-fiction, and to judge each other. The weapon which we wield so readily, but understand so little about. The basis of law, government and order in human society, and concurrently the basis for war, capitalist exploitation, and self-actualization.
The second path is, of course, insanity.
