Triangles and agential realism

Biking to work #

This year I moved to a new neighborhood. 20 km separate me from the university, and I bike to work. This means that I have a lot of time to listen to staff. So I looked for interesting podcasts and I stumbled upon an amazing show: “Opinionated history of mathematics” by Viktor Blåsjö. The anecdote of me biking and listening to Blåsjö’s podcast is not unimportant: it shows how ideas are always rooted in a particular material context - which is what I would like to underline in this post.

Another important preliminary consideration: the podcast is in English, this is why I am writing this post in English, which is not my usual working language - I beg your pardon for my mistakes and imprecisions.

Humanities and mathematics #

The podcast is now in its second season. The first one was dedicated to an enlightening analysis of Galileo’s role in the emergence of “modern science”. Blåsjö’s interpretation is indeed opinionated: he claims that Galileo was “a very poor mathematician” and that the importance we accord to him derives from the fact that humanists instead of mathematicians have written the history of science. In other words, Galileo has only the merit of explaining complex things (that he did not completely understand) in a simple way, using his good rhetoric abilities.

Throughout the podcast, Blåsjö develops a fierce critique of the humanities, based on a rich and extremely well presented documentation.

This critique continues in the second season where at one point he compares philosophical theories to casino coins: they have value inside the casino, but they have no value outside. The relationship to reality is a serious problem in the humanities, especially today - I have talked about if often in this blog.

As a humanist, I agree with many of Blåsjö’s remarks and I think that, especially today, they should be taken very seriously. One observation, for example, is that almost all the talk of humanities researchers about “artificial intelligence” is worthless and unsubstantiated because these researchers have only the vaguest understanding of the algorithms and computational approaches they talk about. There is enough to another Sokal affair: we should stop talking about things we do not understand!

Being a humanist should not mean being lazy about understanding mathematics (and other technical or scientific issues). And also, being a humanist should not mean avoiding rigorous methodologies and definitions. On the contrary, a humanist can be nothing but a polymath: if there is anything like a humanist, this individual must be a concentrate of precise and exhaustive knowledge in different disciplines. I intend to return to this question in the future. I would like to develop a series of posts on the place of humanities today.

The foundation of geometry #

Here I want to address one question raised by Blåsjö in the second season. The question of the foundation of geometry - yes I know, it is a bit too ambitious for a blog post… so let us say: I just want to gather some ideas about this subject starting from the way Blåsjö presents it.

Blåsjö offers a profound analysis of Euclid’s Elements and tries to understand what was the conception of geometry of the ancient Greeks - or to be more precise of the ancient Greek geometers. He shows that the platonic paradigm cannot be consider as the common way to conceive geometry in Greece. According to Plato, geometry is rooted in ontology and it is unrelated to sensible experience. Plato has a realistic idea of mathematics, which means that for him mathematical objects do exist. They are not human constructions. But this existence is related to hyperouranion, to the intelligible world, and not to our immanent world.

But the Greek geometers are makers: this is demonstrated by the presence and importance of diagrams in the teaching and in the demonstrations. The analysis of Euclid’s Elements confirms this. Geometry must therefore be strongly anchored in sensible reality.

But then, how to justify its axioms? Blåsjö demonstrates the limits of a series of solutions: the idea that axioms are elementary and self-evident propositions for all (how then to explain Newton’s physics?), the idea of a logical foundation (but logic is not reliable enough, because of its paradoxes)…

A schematic digression #

In order to make some order in all these possibilities and because the first goal of this post is for me to clarify to myself this problem, I will propose a schema to classify the different possible approaches. We can identify two polarities: realism vs. constructivism and rationalism vs empiricism (Blåsjö dedicates two episodes to the latter).

A brief explanations of these terms:

1. Realism is the vision according to which the world exists independently of us seeing, perceiving or thinking it. A realist is convinced that “there is a world out there”. A realist position with respect to geometry and mathematics consists in saying that geometry and mathematics exist independently of us. We can therefore discover them, we do not invent them.
2. Constructivism is the vision according to which the world only exist because we are in relationship with it - Quentin Meillassoux speaks about “correlationism”. We cannot say that there is a world out there because we only know that we perceive a world. A radical constructivism affirms that the world is completely our construction. As for geometry and mathematics: we construct them, they are our invention.
3. Rationalism is the vision according to which knowledge comes from pure thought and not from the relationship with the world. Rationalism explains very well why there are laws in nature and why geometry and mathematics can explain nature, but it presupposes a universal order governing the universe (so it ultimately presupposes a form of God). Rationalism imply a dogmatic metaphysics.
4. Empiricism is the vision according to which knowledge comes only and always from experience. Empiricism does not need any idea of God, it can get rid of metaphysics, but it cannot explain the apparent universality of some natural laws.

You can combine this 4 terms to have different interpretations of the world and, in this more precise case, of the foundations of geometry. So you have 4 possibilities:

1. realist and rationalist
2. realist and empiricist
3. constructivist and rationalist
4. constructivist and empiricist

Let us have a look at the four of them.

1. A realist and rationalist is typically Plato: he believes that the world exists but that the true world is not the sensible one but the intelligible one. A realist and rationalist believes that “ideas exist”. As for geometry: mathematics and geometry exist as ideas in the intelligible world, we can see them and know them. The sensible world is just an imitation of the intelligible world, this is why geometry and mathematics can be used to understand the sensible world. But mathematical and geometrical knowledge comes always only from reason and universal principles and never from things that one can observe.
2. A realist an empiricist position is the one of Hume, for example: the reality is out there and the only way to know it is to look at it. As for geometry and mathematics, they exist in the world and we can develop mathematical and geometrical knowledge only by studding the world, looking at all the particular cases that it offers and make synthesis and inductions: I see many triangles out there in the world, each of them has the same characteristics (say the sum of the angles is 180) so I induce that that must be a law for all triangles.
3. A constructivist and rationalist position is the one of Berkeley, for example: the world is just in our minds, in order to know it, we shall analyse our minds. There are no such things as triangles. Geometry and mathematics are the result of our thinking. We construct them, we make them up. In order to acquire knowledge on them, thus, we have just to study our own way of thinking. 4. A constructivist and empiricist. This is a very interesting case - showing at the same time the limits and the interest of this schema. One can imagine that it is impossible to reconcile a constructivist point of view with an empiricist point of view because the empiricist believes in a world that can be experienced while a constructivist thinks that the world is only our creation. A constructivist will always be a rationalist, therefore, and an empiricist will always be a realist. But the combination of constructivism and empiricism seems to me to indicate a series of attempts to solve all the problems of each of the other combinations. In this sense one can assume that Kant’s approach is a constructivist and empiricist one: the only way to talk about the world is to talk about our relationship with the world - so Kant is a constructivist - but this relation to the world is universal because there are “transcendental” structures like space and time. Blåsjö’s proposal (operationalism) also goes in this direction of combining constructivism and empiricism.

It is of course important to emphasize that this is only an outline and that most serious philosophical approaches try to find a balance within the two polarities. But this schema can help to better understand the issues that I will continue to discuss.

Back to geometry #

Several episodes of the podcast are dedicated to analyze the foundation problems of geometry. Let us try to summarize them in one unique problem: how to reconcile the universal validity of mathematical laws with the fact that these laws match with sensible experience?

The schema that I have just presented can help us to understand the problem. It is precisely an opposition between constructivism and empiricism.

If we have a constructivist approach (we invent mathematics), it is easy to explain its universality (we constructed it in order to be universal) but it is impossible to explain why we can use it in the real world. On the other hand, if we have an empiricist approach (we discover mathematics looking at the world), it is easy to explain why it fits our sensible experience, but more complicated to explain why its laws are so regular, universal and reliable.

In other words: we can say that geometry is only a formal system, that we invent axioms completely arbitrary and that the system works only as a system (like the casino example: coins have value only inside). We change the axioms, we have another geometry. This idea does not explain the fact that with the rules of geometry we can predict how much a wheat field will produce by calculating its surface. If geometry is just a formal game produced by our brains, why does it seem to work in the world? Why its coins have value outside the casino?

On the other hand we can say that axioms derive from our sensible experience, but never in the sensible experience do we have “axiomatic” certainties. The axioms are not always self-evident. And also, our experience is messy, never precise and we cannot actually derive laws from it.

Operationalism #

Blåsjö propose a deeply interesting paradigm in order to solve this problem: that of operationalism (episode Why construct? and following).

Citing the physicist Percy Bridgman, he defines the principle of operationalism as follows: “we mean by any concept nothing more than a set of operations; the concept is synonymous with the corresponding set of operations.”

Blåsjö continues:

For example, what does “triangle” mean? The operationalist answer is that “triangle” means: the figure obtained when drawing three intersecting lines with a ruler. This diagram is not a drawing of a triangle, or a physical instantiation of the formal concept of a triangle, or in some other way subordinated to or derived from some purer concept of triangle. No, a diagram resulting from these operations simply is what a triangle is. The act of drawing itself is the root meaning of the concept of “triangle.” The act of drawing is the foundational bedrock on which any claim about triangles ultimately rests. When Euclid says “let ABC be a triangle,” he strictly speaking simply means: draw one line, then another, then another (making three points of intersection).

Operationalism thus offers a very nice way to resolve the problem of geometry: geometry is the fact of giving well formed and unambiguous instructions to do things in the actual real world. So yes, we invent axioms, but we invent them by defining the operations that we need to realize in the world. Instead of saying: if you assume this principle then it follows that… we say: if you do this thing, if you realize this particular operation in the real world, then you will have this consequence.

I find this idea very appealing from a philosophical point of view. It is very close to what I am working on with the idea of “metaontology”. The problem is the one discussed for example by Quentin Meillassoux in his “After finitude”: it is the problem of how can we be realist today without being naive. After many poststructuralist excesses, we want to be realist today (i.e. to be able to grasp the world and talk about it believing that it is there), but we don’t want to be naive: we know that the world is just what we have access to and that it is not possible to deduce empirically any truths. Like Meilassoux, Blåsjö emphasizes that the dominant vision of today is to be radically constructivist: mathematics and geometry are only a formal system, closed, completely invented. This view allows mathematicians not to be bothered by ontological questions. But in mathematics, as in the rest of the disciplines, this approach is deeply problematic because it deprives us of the possibility to grasp the world. The political consequences of such a problem are well shown by Meillassoux.

Operationalism, in this respect, is close to Karen Barad agential realism and to my idea of metaontology: the world is, but its being is not given, it is the result of some actions. This idea is a way of reconcile realism and constructivism and possibly also rationalism and empiricism.

Blåsjö does not want to go that far and he limits the scope of operationalism to a strategy for opposing skeptics, saying that this approach allows one to say that anything that has been constructed following the indicated procedures will necessarily have certain properties. Then, says Blåsjö, the mathematician cannot demonstrate the universal validity of his proofs, but he can only say that they are valid until someone comes up with a counterexample.

So for example: if you draw a form in this particular way you will have a triangle and you will see that the sum of its angles will be 180. The mathematician can believe in his inner self that actually this proof is universal, but he can only say: ok, try it yourself and see if you are able to draw a triangle which does not have this property. Try to prove me wrong.

In other words, Blåsjö’s operationalism is a constructivist empiricism. The law is a law because we construct the thing in a particular way, but we can check its validity then only from an empirical point of view: one by one. It is already a good compromise because the constructivist part of it gives credibility to the laws and the empiricist part makes this approach realist - we know why it works out there in the real world.

Operationalism, as Blåsjö conceives it, already brings together several advantages of the different poles of polarities that I have schematized. Being constructivist, operationalism also ensures the rationality and universality of procedures and rules and being empiricist it ensures the relation to the world and a certain form of realism.

But the operationalism of Blasjo renounces to a stronger form of realism: the one that is based on an ontology. To renounce to ontology is a guarantee of scientific rigor, certainly. It means abandoning metaphysical assertions, following the good empiricist tradition à la Hume. To renounce ontology is a kind of scientific duty. The realism of operationalism is based exclusively on empiricism.
But this rigor comes at a high price: without an ontological foundation it is impossible to justify the universality of mathematics.

Can we go further? #

But in my opinion we can go further. We can actually avoid to renounce to the realist and rationalist advantages without falling into a metaphysical approach. In my opinion, it is possible to put together an operationalist approach with an - anti-metaphysical and anti-platonic - ontological foundation.

This requires a profound change in our idea of ontology. We must revise our way of thinking about Being. Being must no longer be thought of as something stable and predefined, but as a set of dynamic forces in action. Ontology must therefore no longer be centered on the notion of essence, but on the dynamics that afterwards determine the emergence of essences. Being is the result of some actions - or to say that in the terms of metaontology, Being is always originally mediated.

This is exactly what Karen Barad proposes in her Meeting the universe halfway with the idea of “agential realism”. Barad starts from Bohr’s texts on the double slit experience. From these texts Barad concludes that the concepts of particle and wave are (ontologically) physical arrangements. The material configuration of the observing device is the particle concept and a different and incompatible configuration is the wave concept. There is no wave or particle outside the material device of observation. Things are the result of intra-actions.

It is important to underline that in this way Barad is no longer a constructivist: this theory does not consist in thinking that the concepts of wave or particle are constructed by observers, but that Being is always the result of actions, of which the observational devices are part. Barad’s approach is thus properly realist because it affirms that reality is made in this way as such and not because “we” construct it this way.

The main scientific field that Barad has in mind is physics - actually Barad is a physicist. Let us try to apply this view to geometry and mathematics. Let us come back to the example of triangles. As an operationalist approach would state, the material conditions of the construction of the triangle are the very concept of a triangle. A triangle is not something which exists before these operations because, like all other essences, it is always the result of actions. This, however, is its true ontological nature. The triangle is this set of operations, not because we are not able to grasp its essence beyond its construction, but because it is its construction, or better, the set of material conditions of its construction. These conditions are objective, in the sense that they are not the fruit of an invention, but the result of an actual, real, process. I can build a triangle, but I cannot build a unicorn: the triangle exists because the set of material conditions of its construction are realized in a precise dynamic of drawing. In this sense the approach in question is not constructivist, because from a constructivist point of view one could as well build a unicorn. The set of operations necessary for the construction of a triangle are not just the result of human actions (I can also imagine a unicorn) but of a set of interactions - or to use Barad’s expression - of intra-actions. Intra-actions, not interactions, because the “things” between which these actions take place do not exist before the intra-actions. They are not interactions between, for example, human beings, paper, pen, compass… etc. They are intra-actions because the human beings, the pen, the compass and the triangle are only the results of these intra-actions: the essence comes after the relation.

This idea easily allows us to define a triangle essence and talk about all triangles. It’s not only “all the triangles you can draw”, but really all the triangles, because the triangle is only the set of conditions for its construction.

Now can this idea actually resolve the problem of universality? Can we in this way say that “all the triangles have this and this property” and found that kind of statement in a rigorous and certain way?

In a private email exchange, Blåsjö makes me the following objection:

I certainly see the appeal of saying that the set of all triangles simply is the set of all ways of drawing three line segments whose endpoints intersect. Still, the generality problem is not merely to be able to define this set meaningfully, but also to explain why for instance Euclid’s proof of that the angle sum of a triangle is two right angles proves the theorem for that entire set of entities (despite using merely a particular example in its diagram). This problems does not seem to have a straightforward counterpart in physics, since physics does not pretend to establish universal laws with absolute deductive certainty.

I think that answering this objection is the very interest of Barad approach: this objection stands only if we keep separating the essence from the construction. The construction would be particular and never universal: this particular diagram. But construction becomes universal if define it in an unambiguous and general way. We are not talking about the particular diagram, but about the set of rules that are to be applied in order to build it. A triangle is not a particular diagram. It is neither the sum of all the possible diagrams you can draw. It is the result of a particular set of actions which can be identified in an unambiguous way and thus replicated always in the same way. In this sense, geometry is not that far from physics: you have to define the set of conditions which determines the fact that an experiment will always give the same results.

That is not an answer #

I realize that in this way I do not completely answer Blåsjö’s objection. Because as I describe it, the universality of the results is still not an “absolute deductive certainty”. Mathematics is different from physics because it does not require experiments. It is not necessary to verify the result of a theorem by redoing the experiment with different diagrams. Why this?

I obviously do not have an answer to such a problem. But I have a lead, a first intuition. It seems to me that here we must appeal to the profoundly realistic element of agential realism. Reality is not something you can construct as you want. It is the objective result of a series of intra-actions. This means that the intra-actions are very real and defined. When we talk about mathematics, these intra-actions are of a particular type because they include some things and exclude others. As in the case of the double slit experiment: if we build a which path apparatus we have the concept of a particle and we cannot have the concept of a wave and vice versa.

As Barad puts it:

According to Bohr, either we can find out which slit an electron goes through by using the which-path apparatus, in which case the resulting pattern will be that which characterizes particles, or we can forgo knowledge about which path the electron goes through (using the original unmodified two-slit apparatus) and obtain a wave pattern - we can’t have it both ways at once. Barad, p. 107

Now we can imagine something similar with mathematics: the very nature of the operations needed to make mathematical objects - the very nature of the mathematical apparatus - includes the concept of universality and excludes the concept of particularity. By construction mathematics is universal. You cannot have at the same time mathematics and particularity, because the mathematical apparatus does not allow that.

Yet another much too long blog post #

A blog post should be shorter, I know. But I find myself writing a lot probably because my ideas aren’t clear enough. I apologize to those who read me: this writing is precisely a form of experience: putting together the material conditions of thought. Like defining the rules for drawing a triangle…

As far as I am concerned, this post allowed me to clarify my ideas a little. And it will be the starting point for an upcoming series on the role of humanities today. Stay connected!