The fatal flaw of mathematics
from Lars Syll
Gödel’s incompleteness theorems raise important questions about the foundations of mathematics.
The most important concerns the question of how to select the specific systems of axioms that mathematics are supposed to be founded on. Gödel’s theorems irrevocably show that no matter what system is chosen, there will always have to be other axioms to prove previously unproved truths.
This, of course, ought to be of paramount interest for those mainstream economists who still adhere to the dream of constructing a deductive-axiomatic economics with analytic truths that do not require empirical verification. Since Gödel showed that any complex axiomatic system is undecidable and incomplete, any such deductive-axiomatic economics will always consist of some undecidable statements. When not even being able to fulfil the dream of a complete and consistent axiomatic foundation for mathematics, it’s totally incomprehensible that some people still think that could be achieved for economics.
Separating questions of logic and empirical validity may — of course — help economists to focus on producing rigorous and elegant mathematical theorems that people like Lucas and Sargent consider “progress in economic thinking.” To most other people, not being concerned with empirical evidence and model validation is a sign of social science becoming totally useless and irrelevant. Economic theories building on known to be ridiculously artificial assumptions without an explicit relationship with the real world is a dead end. That’s probably also the reason why general equilibrium analysis today (at least outside Chicago) is considered a total waste of time. In the trade-off between relevance and rigour, priority should always be on the former when it comes to social science. The only thing followers of the Bourbaki tradition within economics — like Karl Menger, John von Neumann, Gerard Debreu, Robert Lucas, and Thomas Sargent — has given us are irrelevant model abstractions with no bridges to real-world economies. It’s difficult to find a more poignant example of an intellectual resource waste in science.
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Real-World Economics Review Blog 
One can always come up with new axioms to remedy the incompleteness or inconsistency, and carry on. So, Gödel’s proofs are hardly the final mathematical criticism to neo-classicals. A better way is to follow Turing and show that any mathematical formalism of economics is undecidable. For those interestec in this line of thinking, I strongly recommend works of late V K Vellupillai.
Regards
Vehap, could you recomend a couple good sources for V K Vellupillai for an introduction to his thinking and ideas?
Three article suggestions: 1-)K Vela Vellupillai, S Zambelli (2015) “Simulation, computation and dynamics in economics”,Journal of Economic Methodology, 22:1, 1-27 2-) K Vela Vellupillai (2013) “Computability theory in economics, frontiers and a retrospective”, ASSRU, Dept. of Economics, University of Trento 3-) K Vela Vellupillai (2009) “Algorithmic revolution in the social sciences: Mathematical economics, game theory and statistics”, Dept. of Economics, University of Trento. There is also a recent book by Vellupillai, “Models of Simon” (2017), Routledge Advances in Experimental and Computable Economics, which I have not read.
Lars Syll is misusing Gödel’s incompleteness theorem and the existence of undecidable problems after Turing. Since the time of Gödel and Turing (i.e. in the fist half of the twenties century) we know all that. There is a hole at the bottom of math. We will never know everything with certainty. But that does not imply that mathematics is useless. What did the narrator of the video finally recommend? He recommended courses on number theory and computer science fundamentals. They are mathematics.
It is true that we are not behaving after maximization principle. There are full of intractable problems if one wants strict (or even an approximate) solutions. We should accept that human beings are not behaving like that. It may help to criticize Arrow-Debreu general equilibrium theory. But it does not imply that we cannot do economics. Admitting that intractable problems are ubiquitous in mathematics and in our life, we can build economics and math is useful in such theory making. See, for example, chapter 1 of our book Microfoundations of Evolutionary Economics.
The following is what we (three authors of the book) have written in the Reply to the Symposium on our book which is recently made public in Metroeconomica:
Notes: (1) SMT is the abbreviation of our book. (2) Two footnotes are omitted.
One should not confuse axiomatic economics with computational economics. Gödel’s proofs say that there can be no formal axiomatic system that is complete and consistent. That does not mean that one cannot do mathematics or economics, of course not, but it does say that you cannot have axiomatic economics. That goes right into the heart of micro/macro approach of neo-classical economics.
Furthermore, a formal system is either closed, meaning that its truths are analytic, or open, in which case they are synthetic. All synthetic systems are open, meaning they involve empirical truths. Here the law of the excluded middle applies, that is, tertium non datur.
Dear Vahap,
are you a PhD student at Birmingham University? I found your name in the ResearchGate but unfortunately my PC has a trouble and cannot send you message through RG.
Would you send me an e-mail at y@shiozawa.net. I have some papers that I want you to read (including some chapters of SMT). After that, we can continue our discussion based on a more concrete example.
Dear Vahap,
I did not read your post on November 27, 2021 at 6:49 am when I posted mine on November 27, 2021 at 8:50 am, because I was hidden by some reasons. Even when I posted the above comment on November 27, 2021 at 11:48 pm, I did not know that you have posted your first comment.
This is my first reaction to your first post. I did not know V K Vellupillai and his works. I skipped search engines and found three of his papers, which seem to be interesting. I will read them soon. We may be able discuss them. By the way, the Wikipedia (eg.) article on him does not point that Vela has passed away. It is deplorable because he is 3 years younger than I.
There is a lot of common ground here. The economic system is complex and open; everyone agrees. By making ceteris paribus assumptions, in effect ignoring some elements, it is possible to treat a subdivision of the system as closed. We can then analyse it and deduction or mathematical computation can play a part. The assumptions we make to partition the subdivision have to be sensible. It might be sensible, for example, to abstract from population growth or technical innovation or political changes because there may be periods when these things play a minor role on the processes we are examining and the variables we are concerned with are then truly dominant. Of course in any real-world application of our model we shall have to try and take the other elements back into account as best we can. At least the model has helped us understand a subset of the processes we have to consider. How we test our model or understanding when in reality the ceteris paribus conditions never fully hold can be tricky but there are methods we can use.
The point I take Lars to be making is that the assumptions economists sometime make to analyse the subdivision are not sensible. If they assume perfect foresight, for example, that is a situation that never obtains and sane people do not behave as if it does. The analysis therefore is highly unlikely to be illuminating. Assumptions define a domain of the possible universe of discourse. The domain is a useful one if a situation could exist where that domain was real and almost closed. If the domain is one that can never ever exist or be even approximately closed, our assumptions are likely to lead us astray.
What I think Lars is not saying is that a model has to take everything into account. If a model is that concrete and comprehensive it becomes a one to one mapping of reality; the map becomes the terrain. Everything may be linked but fortunately, it is possible to compartmentalise reality. Bridges did not all collapse until we discovered quantum physics. You can be an excellent cook without understanding biochemistry. You can have serviceable theory of aggregate consumer spending without knowing all about the sociology of social class .
When Lars talks of “those mainstream economists who still adhere to the dream of constructing a deductive-axiomatic economics with analytic truths that do not require empirical verification”, we must be careful not to erect a straw man. Some economists dream of constructing highly abstract models that will never describe any existing situation. But because of their very abstraction they hope the models will represent a component or set of influences that will be present in very many situations. You study a limiting case to gain insight into how more general situations could play out. It is legitimate to doubt whether that is always a fruitful approach. However I do not know any economist who has ever maintained that the usefulness of a model can be ascertained without empirical verification. Everyone pays lip-service to economics being an empirical discipline. But as in many other disciplines people have a bad habit of ignoring findings that cast doubt on a favoured theory.
Dear Gerald,
all you wrote is right, but it must be useless, because Lars Syll never changes his opinion. He is always a convinced destroyer of economics. No constructive arguments work for him.
When I post my comments here, they do not aim to argue with him (in two or three years ago I tried many times to do so but he was deaf). What I try to do here is to diminish to some extent the negative effects of Lars Syll’s criticism against economics. It does not mean that I defend mainstream economics. I am also negative of mainstream economics, perhaps much more than Lars is. Lars is often right in his criticisms but loses right weight and unduly overgeneralize his observations (criticisms against mainstream economics are generalized against economics in general), probably because he cannot see any possibilities of reconstructing a new alternative economics. Lars’s arguments drive away all promising young students of economics away from economics. This is the most disastrous situation that we should avoid.
The message like impossibility of economics in general is definitely toxic and make no good for the reconstruction of economics, which I assume to be the main aim of Real-World Economics blog and Real World Economics Association.
Shiozawa repeatedly stoops to making ad hominem accusations that are ludicrous: Lars is destroying young people’s desire to study economics, he is only appealing to “math phobia”, and other such fallacious arguments. According to Shiozawa single handedly Lars is destroying the entire field of economics and driving away an entire generation of students from entering the field. Ironically, I suspect that if one did some simple econometrics (e.g., number of new students entering advanced degree programs in economics around the world) the sillyness of this rant becomes obvious. Such utter nonsense from someone who claims to be revolutionizing economic theory with his discovery of the new microfoundations of the entire world economy. ( See RWER, 1/16/2020)
Reblogged this on muunyayo .
The number of British graduate students in economics is sharply down in recent decades. Most graduate students of economics in the UK are foreign, often from China. However, Lars is not to blame for that! The state of economics and fact that finance, management and law are more lucrative are likelier culprits.
Of course, Lars is not responsible to the decline of trust in economics. He is only adding a small accelerating force in that direction. Your diagnostics is more sociological (or psychological) than requested for researchers in economics. Among the people there is a deep mistrust in the actual economics and we are not able to present an alternative one. On this point, we economists are all responsible.
Mathematical talk can be technical talk about mathematics. This sort of talk is based on the assumption that the secret of mathematical power lies in the formal relations among symbols. But technical talk about mathematics cannot, by itself, provide a complete understanding of mathematics. Technical talk, in fact, is not only incomplete-it is a fairy tale. At least, it is a fairy tale to the extent that it obscures and even denies the social dimension of technical talk. This dimension is highlighted in social talk about mathematics. Once we begin to talk about mathematics in this way, it becomes clear fairly quickly that technical talk is social talk!
We can talk about mathematics using terms such as “social power”, “social structure”, “social class”, “culture”, and “values”. Whereas “technical talk” isolates mathematics from other social practices (thereby “spiritualizing” the technical), social talk links mathematics to other social practices, and reveals the social nature of technical talk itself. Just as speech cannot be understood as a “parade of syntactic variations”, and myths are not merely sets of “structural transformations”, so mathematical objects are not simply “concatenations of pure form”. Thus, to study a mathematical form is to study a sensibility, a collective formation, a worldview. The foundations of mathematical forms -like the foundations of art, poetry, religion, and all other human activities and productions are as wide and as deep as social existence.
I have adopted this way of talking about mathematics from Clifford Geertz’s observations on “art as a cultural system”. Talking about mathematics this way gives us the sociology of mathematics. In broader terms, following Geertz, we could speak of “the natural history of signs and symbols”, “the ethnography of vehicles of meaning”, or “the social history of the imagination”. In order to engage in social talk about mathematics, we must study the social worlds in which mathematicians “look, name, listen, and make” (Geertz, 1983: 94-120). Like the concept of “art worlds” (Becker, 1982), the concept of “math worlds” draws us into a network of cooperating and conflicting human beings. Mathematical objects embody math worlds. They are produced in and by math worlds. That is, we could say that math worlds, not individual mathematicians, manufacture mathematics (cf Fleck, 1979/1935; Restivo, Van Bendegem, and Fischer, 1992). Mathematicians, like other workers, use tools, machines, techniques, and skills to transform raw materials into finished products. The products of mathematical work are mathematical objects (e.g., theorems, rational numbers, points, functions, the integers, numerals, etc.). There are two general classes of raw materials out of which mathematical objects can be fashioned. One is the class of all things in human experience which are outside mathematics but can be “mathematized”. The second is the class of all mathematical objects. Understanding mathematics as a social fact depends in part on recognizing that the classes of mathematical objects, raw materials of mathematical work, and tools and machines for mathematical work overlap. This is the perspective guiding the work reported here, but it is at the same time a perspective under construction”.
My objective in this book is to illustrate different, sociologically grounded, ways of thinking, writing, and speaking about mathematics. It is necessary to get used to “social talk” as opposed to “technical talk” about mathematics before a more systematic theoretical treatment of mathematics as a social product and social construct can be undertaken. At this stage, a story that mixes historical details and sociological theory without being clearly history or theory is an appropriate reflection of what we have achieved in the historical sociology of mathematics.
A word of caution is required regarding my apparently casual application of the term “mathematics” across cultural time and space. Splengler, as we will see shortly, suggests that different cultures are incommensurable. But if incommensurability were absolute, we would not only be mute in the face of other cultures; we would be mute in the face of each other -and even of ourselves! But it is social practice that in the end overcomes what we might call a Derridaian muteness; communication is possible because we have to do things together, not because language is a perfect or even a possible mode of communication. And in fact there are certain events in human experience-certain practices, if you will-such as birth and death -which provide a basis for practical if not perfect cross-cultural, cross-historical, and cross-( and auto-) biographical understanding. It is safe to assume that we can understand, compare, and explain cultures across time and space if we are aware of the limitations imposed by the incommensurability principle, and if we exercise care and caution in using our linguistic and other tools of inquiry. Care, caution, and awareness, however, should never reign absolutely in our always risky efforts to ask and answer questions about our world. In this exploratory work, I have not thought it necessary to try to solve the problem of studying something called “mathematics” in different times and places, except to remind myself and my readers of the problem. In general, I will refer to “mathematical” or “number” work rather than “mathematics” (in its occupational or professional sense), and use “mathematics” to refer to mathematical or number work by more or less full-time specialists (and, then, use the terms mathematical worker and mathematician, respectively).
Sal Restivo (Sociologist-Historian) (Mathematics In society and history, 1992) (first book written on sociology of mathematics)
In the same vein, mathematics are useful, not because they are perfect or complete. Mathematics often gives us an entity in the World Three (in Popper’s term), on which we can contemplates more deeply than without such entities. They are often called models but models are not useful by itself. Its usefulness depends on how deep we have understood the object we study by it.
Such attempts are not always successful. On the contrary, they more often fail than succeed. Some attempts are doomed to fail because of the very construction of the model. But, this fact does not imply that all examinations based on a mathematical model are necessarily useless and toxic.
Lars Syll put it in another recent post on December 1, 2021:
I agree with Lars Syll on this very part. This is what we are trying to do when we do economics, by using mathematical models. To use mathematics or not is not very relevant of whether an economic theory is good or not.
In another post of Lars Syll on December 1, 2021, Tony Lawson posed a question that includes this part:
In one of my responses, I contended that
Is it possible to make plausible economic analysis using totally or partly mathematics as means of analysis? I dare say yes.
and cited the results of our studies published in the book that I cited in my post above on November 27, 2021 at 8:50 am and explained how our theory is different from conventional neoclassical theories. I explicitly claimed that the theory stands on the same ontological view as Lawson repeatedly insists.
Lawson responded to my post, but refused to examine if my claims were right or not. How can you explain this reaction by your sociology? My interpretation is this:
There was a big chance that he had to admit my claims and that what Lawson thinks third kind of modern economists can go beyond the second kind of modern economists with good intention but without analytical tools. (See Jamie Morgan (ed.) What is Neoclassical Economics? Rotuledge, 2016, pp.63-64) Steve Keen dubbed them “the ugly” and “the good”. (ibid. p.239) In these terms, the ugly can go beyond the good.
Arguing that all of our thoughts and actions are sociocultural constructs including mathematics does not mean that they are somehow “arbitrary” or “random.” Rather, it means their reality is made by people.
The question that begs answering is why Shiozawa didn’t present explicitly his set of eighteen (plus auxiliary) “complicated postulates” rather than misleading Tony Lawson that his book is based upon “five sets of properties” he listed above?