I have been interested in science since before I can remember. I was reading popular science, I believe, before first grade. I was reading some undergraduate textbooks (and the occasional research monograph – not that I understood them) before high school. I did this in order to learn the truth about what the world is. Eventually I learned enough to realize that I wanted to really understand General Relativity. For that, I learned from a book I no longer recall, I needed to learn tensors.
I, therefore, set myself on the project of building my mathematical knowledge base sufficiently far to figure tensors out. By the time I started high school (or rather, its equivalent), I had switched from desiring mathematics as a tool (for understanding Relativity) to taking it as a goal in itself. My goal for the next few years was to learn high school math well enough to be able to study mathematics at university as a major subject. I succeeded.
During the high-school years I also continued reading about physics. My final-year project in my high-school equivalent was an essay (sourced from popular science books) on the history of physics, from the ancients to the early decades of the 20th Century, enough to introduce relativity and quantum mechanics, both at a very vague level (the essay is available on the web in Finnish). I am ashamed to note that I also included a brief section on a “theory” which I was too young and inexperienced then to recognize as total bunk.
At university, I started as a math major, and began also to study physics as a minor subject. Neither lasted, but the latter hit the wall almost immediately. It was by no means the only reason why I quit physics early, but it makes a good story, and it was a major contributor: I was disillusioned. Remember, I was into physics so that I could learn what the world is made of. The first weeks of freshman physics included a short course on the theory of measurement. The teacher gave us the following advice: if your measurements disagree with the theory, discard the measurements. What the hell?
I had understood from my popular-science and undergraduate textbook readings over many years that there is a Scientific Method that delivers truthful results. In broad outline, it starts from a scientist coming up, somehow (we are not told how), with a hypothesis. They would then construct an experiment designed to test this hypothesis. If it fails, the hypothesis fails. If it succeeds, further tests are conducted (also by other scientists). Eventually, a successfully tested hypothesis is inducted into the hall of fame and gets the title of a theory, which means it’s authoritatively considered the truth. I expected to be taught this method, only in further detail and with perhaps some corrections. I did not expect to be told something that directly contradicts the method’s central teaching: if the data and the theory disagree, the theory fails.
Now, of course, my teacher was teaching freshman students to survive their laboratory exercises. What he, instead, taught me was that physics is not a noble science in pursuit of the truth. I have, fortunately, not held fast to that teaching. The lasting lesson I have taken from him, though I’m sure he did not intend it, is that the scientific method is not, in fact, the way science operates. This lesson is, I later learned, well backed by powerful arguments offered by philosophers of science. (I may come back to those philosophical arguments in other posts, but for now, I’ll let them be.)
There are, of course, other sources for my freshman-year disillusionment. Over my teen years I had participated in many discussions over USENET (a very early online discussion network, now mostly obsolete thanks to various web forums and social media). Many of the topics I participated in concerned with the question of truth, particularly whether God or other spiritual beings and realms exist. I very dearly wanted to learn a definite answer I could accept as the truth; I never did. A very common argument used against religious beliefs was Occam’s razor: the idea that if something can be explained without some explanans (such as the existence of God), it should be so explained. Taken as a recipe for reasoning to “the truth”, it seems, however, to be lacking. Simpler explanations are more useful to the engineer, sure, but what a priori grounds there possibly can be for holding that explanatorily unnecessary things do not exist? For surely we can imagine a being that has the power to affect our lives but chooses to not wield it (at least in any way we cannot explain away by other means), and if we can imagine one, what falsifies the theory that one isn’t there?
Many scientists respond to this argument by invoking Karl Popper and his doctrine of falsification. Popper said, if I recall correctly (and I cannot right now be bothered to check), that the line separating a scientific theory from a nonscientific one is that the former can be submitted to an experiment which in principle could show the theory false, and for the latter there is no such experiment that could be imagined; in a word, a scientific theory is, by Popper’s definition, falsifiable. Certainly, my idea of a powerful being who chooses to be invisible is not falsifiable. While there are noteworthy philosophical arguments against Popper’s doctrine, I will not discuss them now. I will merely note that the main point of falsificationism is to say that my question is inappropriate; and therefore, from my 19-years-old perspective, falsificationism itself fails to deliver.
My conclusion at that young age was that science does not care about The Truth; it does not answer the question what is. It seemed to me, instead, that science seeks to answer what works: it seeks to uncover ways we can manipulate the world to satisfy our wants and needs.
My current, more mature conclusion is similar to the falsificationist’s, though not identical. The trouble is with the concept of truth. What is, in fact, truth?
In my teens, I did not have an articulated theory of truth; I took it as granted. I think what I meant by the term is roughly what is called the correspondence theory of truth by philosophers. It has two components. First, that there is a single universe, common to all sensing and thinking beings, that does not depend on anyone’s beliefs. Second, that a theory is true if for each thing the theory posits there is a corresponding thing in the universe possessing all the characteristics that the theory implies such a thing would have and not possessing any characteristics that the theory implies the thing would not have; and if the theory implies the nonexistence of some thing, there is no such thing in the universe. For example, if my theory states that there is a Sun, which shines upon the Earth in such a way that we see its light as a small circle on the sky, it is true if there actually is such a Sun having such a relationship with Earth.
Unfortunately, the correspondence theory must be abandoned. Even if one concedes the first point, the existence of objective reality, the second point proves too difficult. How can I decide the (correspondence-theory) truth of the simple theory “there is a chair that I sit upon as I write this”, a statement I expect any competent theory of truth to evaluate as true? Under the correspondence theory of truth, my theory says (among other things) that there is some single thing having chair-ness and located directly under me. For simplicity, I will assume arguendo that there are eight pieces of wood: four roughly cylindrical pieces placed upright; three placed horizontally between some of the upright ones (physically connected to prevent movement); and one flat horizontal piece placed upon the upright ones, physically connected to them to prevent movement, and located directly below my bottom. I have to assume these things, and cannot take them as established facts, because this same argument I am making applies to them as well, recursively. Now, given the existence and mutual relationships of these eight pieces of wood, how can I tell that there is a real thing they make up that has the chair-ness property, instead of the eight pieces merely cooperating but not making a real whole? This question is essentially, does there exist a whole, separate from its parts but consisting of them? Is there something more to this chair than just the wood it consists of? The fatal blow to the correspondence theory is that this question is empirically unanswerable (so long as we do not develop a way to talk to the chair and ask it point blank whether it has a self).
Scientists do not, I think, generally accept the correspondence theory. A common argument a scientist makes is that a theory is just a model: it does not try to grasp the reality in its entirety. To take a concrete example, most physicists are, I believe, happy to accept Newtonian physics so long as the phenomena under study satisfy certain preconditions so that Newtonian and modern physics do not disagree too much. Yet, it is logically impossible for both theory of Special Relativity and Newtonian physics to describe, in a correspondence theory sense, the same reality: if the theory of Special Relativity is correspondence-theory true, then Newtonian physics cannot be; and vice versa.
If not correspondence theory, then what? Philosophers of science have come up with a lot of answers, but there does not seem to be a consensus. The situation is bad enough that in the behavioural sciences there are competing schools that start with mutually incompatible answers to the question of what “truth” actually means, and end up with whole different ways of doing research. I hope to write in the future other blog posts looking at the various arguments and counterarguments.
For now, it is enough to say that it is naïve to assume that there is a “the” truth. Make no mistake – that does not mean that truth is illusory, or that anybody can claim anything as true and not be wrong. We can at least say that logical contradictions are not true; we may discover other categories of falsehoods, as well. The concept of “truth” is, however, more complicated that it first seems.
And, like physics, we may never be able to develop a unified theory of truth. Perhaps, all we can do is a patchwork, a set of theories of truth, each good for some things and not for others.
Going back to my first year or two at university, I found mathematics soothing, from this perspective. Mathematics, as it was taught to me, eschewed any pretense of correspondence truth; instead, truth in math was, I was taught, entirely based on the internal coherence of the mathematical theory. A theorem was true if it could be proven (using a semi-formal approach common to working mathematicians, which I later learned to be very informal compared to formal logic); sometimes we could prove one true, sometimes we could prove one false (which is to say, its logical negation was true), and sometimes we were not able to do either (which just meant that we don’t know – I could live with that).
I told my favourite math teacher of my issue with physics. He predicted I would have the same reaction in his about-to-start course on logic. I attended that course. I learned, among other things, Alfred Tarski’s definition of truth. It is a linguistic notion of truth, and depends on there being two languages: a metalanguage, in which the study itself is conducted and which is assumed to be known (and unproblematic); and an object language, the language under study and the language whose properties one is interested in. Tarski’s definition of truth is (and I simplify a bit here) to say that a phrase in the object language is assigned meaning based on its proffered translation. For example, if Finnish were the object language and English the metalanguage, the Tarskian definition of truth would contain the following sub-definition: “A ja B” is true in Finnish if and only if C is the translation of A, D is the translation of B, and “C and D” is true in English.
The Tarskian definition struck me initially as problematic. If you look up “ja” in a Finnish–English dictionary, you’ll find it translated as “and”. It now becomes obvious that Tarski’s definition does not add anything to our understanding on Finnish. And, indeed, it is one more link in the chain that says that mathematics is not concerned with correspondence truth. We cannot learn anything about the real world from studying mathematics. But I knew that already, and thus, in the end, Tarskian truth did not shatter my interest in mathematics.
I also learned in that course of Kurt Gödel’s famous incompleteness theorem. It states (and I simplify a lot) that a formal theory that is sufficiently powerful to express all of mathematics cannot prove its own coherence. This was the result my teacher was alluding to earlier, but it did not bother me. I had been taught from the beginning to regard mathematics as an abstract exercise in nonsense, valuable only for its beauty and its ability to satisfy mathematician’s intellectual lusts. What do I care that mathematics cannot prove itself sane?
What I did not then know is the history. You see, up until the late 19th Century, I believe mathematicians to have adhered to a correspondence theory of truth. That is, mathematics was, for them, a way to discover truths about the universe. For example, the number two can be seen as corresponding to the collection of all pairs that exist in the universe. This is why certain numbers, once they had been discovered, were labeled as “imaginary”; the mathematicians who first studied them could not come up with a corresponding thing in the universe for such a number. The numbers were imaginary, not really there, used only because they were convenient intermediate results in some calculations that end up with a real (existing!) number. This is also, I believe, one of the reasons why Georg Cantor’s late 19th Century set theory, which famously proved that infinities come in different sizes, was such a shock. How does one imagine the universe to contain such infinities?
But more devastating were the paradoxes. One consequence of Cantor’s work was that infinities can be compared for size; also, that we can design a new numbering system (called cardinal numbers by set theorists) to describe the various sizes of infinity, such that every size of infinity has a unique cardinal number. Each cardinal number itself was infinite, of the size of that cardinal number. It stands to reason that the collection of all cardinal numbers is itself infinite, and since it contains all cardinal numbers (each being its own size of infinite), it is of a size of infinity that is greater than all other sizes of infinity. Hence, the cardinal number of all infinities is the greatest such number that can exist. But it can be proven that there is no such thing; every cardinal number has cardinal numbers that are greater than it. If one were to imagine that Cantor’s theory of the infinities does describe the reality, that would imply that the universe itself is paradoxical. This Cantor’s paradox isn’t the only one; there are many others discovered about the same time. Something here is not right.
A new branch of mathematics emerged from this, termed metamathematics, whose intent was to study mathematics itself mathematically. The idea was that finite stuff is well understood, and since it corresponds to the reality, we can be sure it is free of paradoxes. Metamathematicians aimed to rebuild the mathematics of the infinite from ground up, using only finite means, to see what of the infinity actually is true and what is an artefact of misuing mathematical tools due to poor understanding of them. This work culminated in two key discoveries of the 1930s: Kurt Gödel’s incompleteness theorem, which basically said that metamathematics cannot vindicate mathematics, and Alan Turing’s result that said that mathematics cannot be automated. Of course, the technique Turing used is his famous Machine, which is one of the great theoretical foundations of computer science.
Fast forward sixty years, to the years when I studied mathematics in university. The people who taught me were taught by the people who had been taught by the people who were subjected to Gödel and Turing’s shocks. By the time I came into the scene, the shock had been absorbed. I was taught to regard mathematics as an intellectual game, of no relevance to reality.
I eventually switched to software engineering, but I always found my greatest interest to be at the intersection of theoretical computer science and software engineering, namely the languages that people use to program computers. In theoretical computer science, the tools are of mathematics, but they are made relevant to reality because we have built concrete things that are mathematics. Mathematics is not relevant to reality because it describes reality, but because it has become reality! And since the abstract work of the computers derive their meaning, Tarski-like, from mathematics, we have no problem with a correspondence theory. Truth is, again, uncomplicated, and it was, for me, for many years.
Until I realized that computers are used by people. In the last couple of years I have been reading up on behavioural research, as it is relevant to my own interest in language design. Again, the question of truth, and especially how we can learn it, becomes muddled.
Forgive me, gentle reader. This blog post has no satisfactory ending. It is partly because there is no completely satisfactory answer that I am aware of. I will be writing more in later posts, I hope; but I cannot promise further clarity to you. What I can promise is to try to show why the issue is so problematic, and perhaps I can also demonstrate why my preferred answer (which I did not discuss in this post) is comfortable, even if not quite satisfactory.
(Please note that this is a blog post, not a scholarly article. It is poorly sourced, I know. I also believe that, apart from recounting my personal experiences, it is not particularly original. Please also note that this is not a good source to cite in school papers, except if the paper is for some strange reason about me.)