thoughts on intuitionism

The following is a polished and expanded version of an email that I sent to Umut Eldem. You can read more about intutionism here.

The faculties we consider as "apriori" have developed out of a long string of interactions between a developing organism and its environment. In other words, there can be no apriori concept without environmental inputs prior to the full development of the faculty in question.

Space and time help us to survive and reproduce. They are certainly based on some aspects of the reality, but the determination of these aspects is beyond our reach. Why? Because the only way we can access reality is through these lenses.

Sometimes things go wrong in the developmental process. People end up having "distorted" senses of space and time. We call them mentally sick.

On the other hand, even cells have a sense of time. You do not need a huge, over-complicated brain to host a biological time keeping mechanism. There are many metabolic processes in our body that operate on pulsations which have such cellular origins.

Bacteria also have a sense of their surroundings. Through signalling mechanisms they can keep track of the local population of their kins. They can also sense food, light etc. Moreover they can skilfully navigate themselves within the space relevant to their existence. Nevertheless we do not consider bacteria to possess a sense of space and time. Why? Because our notions have an implicit prejudice to "awareness" whatever the hell that means. Space/time seems to be based on very basic and fundamental features of the reality. So basic that even the most simple life forms possess a mechanism that exploits it. But these life forms are not "aware" that they are exploiting something. That capability belongs only to certain vertebrates like us.

Make no mistake that we do not really know what we are exploiting neither. Our intuitive notions of space and time are based on the local environment that is relevant to our existence. These stuff stop making any sense in environments that have been irrelevant to our evolutionary history. Don't you get a funny feeling when a physicist utters sentences such as these:

"Expansion of the universe is stretching of the space-time fabric."

"The age of our universe is blablabla."

"Extremely massive objects tear the space-time fabric."

Bear in mind that there can be no non-classical mathematics. Yes, there is non-classical physics. (e.g. quantum phenomena) But the tools that scientists use to explain these phenomena are grounded in classical mathematics. This should not be surprising. Explanation means that a phenomenon is narrated in a language that is accessible to us. And what is accessible to us is mathematics that is born out of notions of space and time that were in turn born out of the part of physics that was relevant to our evolutionary history.

From these experiments it is seen that both matter and radiation possess a remarkable duality of character, as they sometimes exhibit the properties of waves, at other times those of particles. Now it is obvious that a thing cannot be a form of wave motion and composed of particles at the same time - the two concepts are too different... As a matter of fact, it is experimentally certain only that light sometimes behaves as if it possessed some of the attributes of a particle, but there is no experiment which shows that it possesses all the properties of a particle; similar statements hold for matter and wave motion. The solution of this difficulty is that the two mental pictures which experiments lead us to form - the one of particles, the other of waves - are both incomplete and have only the value of analogies which are accurate only in limiting cases... yet they may be justifiably used to describe things for which our language has no words. Light and matter are both single entities, and the apparent duality arises in the limitations of our language.

- Heisenberg as quoted on page 80 of Heisenberg by K. Camilleri

Evolution endowed us with intuition only for those aspects of physics that had survival value for our distant ancestors, such as the parabolic trajectories of flying rocks. Darwin's theory thus makes the testable prediction that whenever we look beyond the human scale, our evolved intuition should break down. We have repeatedly tested this prediction, and the results overwhelmingly support it: our intuition breaks down at high speeds, where time slows down; on small scales, where particles can be in two places at once; and at high temperatures, where colliding particles change identity. To me, an electron colliding with a positron and turning into a Z-boson feels about as intuitive as two colliding cars turning into a cruise ship. The point is that if we dismiss seemingly weird theories out of hand, we risk dismissing the correct theory of everything, whatever it may be. 

Tegmark - Mathematical Cosmos

What I am claiming is more radical then what Tegmark is saying. I am claiming that even the weird theorems, such as those put forth by Tegmark, will never be sufficient, because that weirdness is weirdness of classical descent. What physicists call non-classical is phenomenon that a) is observable by a mental faculty whose development was moulded within classical physics, b) takes place at scales irrelevant to the survival of these faculties. The weirdness stems from b), not from a).

Similar morphological structures, such as the eye, have emerged out of different evolutionary lineages, because we are all part of the same reality. (See biological structuralism.) For the purpose of its survival, an organism needs to make sense of its environment. That process involves certain simplifications and amplifications. Nevertheless what is filtered through can not be a hugely distorted version of the reality, because an exceptionally delusional organism sooner or later dies.

In some sense, it is not surprising that physics was so successful at explaining classical phenomena. After all, this is what our faculties, from which our mathematical constructs and analogies stemmed, were evolved for. (Of course, the fact that we could explain so much classical stuff using so few classical principles is still startling.)

Mathematics is effective in characterizing and making predictions about certain aspect of the real world as we experience it. We have evolved so that everyday cognition can, by and large, fit the world as we experience it. Mathematics is a systematic extension of the mechanisms of everyday cognition. Any fit between mathematics and the world is mediated by, and made possible by, human cognitive capacities. Any such "fit" occurs in the human mind, where we cognize both the world and mathematics.

- Lakoff & Nunez, Where Mathematics Comes From (Page 352)

I do not think that you need to have an in-built sense of time to be able to construct and make sense of mathematics. Whenever a time parameter is introduced in physics, its conceptual origin lies in geometric structures. Line is pictured as a continuum of points, and each point is pictured as an instant. In the discrete case, you don't even need the continuum construction.

On the other hand, the idea of space transformations have origins in time perception. This origin is not immediately apparent since the idea itself is stated in a "timeless" language.

I think the fact that all of mathematics looks "timeless" have misled a lot of adventurous minds who now believe that the next big discovery in physics will involve the removal of time from our theories. That will not be possible. Again this is simply because we can explain only what we experience.

The great irony in our discussion is that we forget that we are expressing ourselves in a language which is far more vague and ambiguous than what is being examined. As one can not measure physical distances with apparatuses whose resolution is cruder than the distances, one can not hope to understand a language by examining it with an inferior language. Mathematics precedes English, and ideas precedes all. (The fact that I can successfully convey the meaning of this sentence to you, by using English, is just tantalizing.)

Kant's notion of "apriori" has conceptual origins in time perception. (i.e. something that precedes) Hence the reason why I do not think that "apriori" is an enlightening concept. You can not explain something using that thing.

Let's now touch on the most fundamental and deepest of all of our ideas, namely the idea of contradiction/non-contradiction. Well... This, too, is a fiction. Although quite a useful one. (See this blogpost.)

Several specific points regarding Brouwer's Intuitionism and Formalism:

1) Brouwer's conception of science is out-of-date. The following extract sounds too simplistic after the discovery of quantum physics:

"And that man always and everywhere creates order in nature is due to the fact that he not only isolates the causal sequences of phenomena (i. e., he strives to keep them free from disturbing secondary phenomena) but also supplements them with phenomena caused by his own activity, thus making them of wider applicability. Among the latter phenomena the results of counting and measuring take so important a place, that a large number of the natural laws introduced by science treat only of the mutual relations between the results of counting and measuring."

2) Within ZFC (currently the most widely accepted set theoretical axioms), Burali-Forti "paradox" is no longer a paradox. This is for the same reason why Russell "paradox" is no longer a paradox. (See this Wikipedia article.)

3) Brouwer says:

"In this way the apriority of time does not only qualify the properties of arithmetic as synthetic a priori judgments, but it does the same for those of geometry, and not only for elementary two- and three-dimensional geometry, but for non-euclidean and n-dimensional geometries as well. For since Descartes we have learned to reduce all these geometries to arithmetic by means of the calculus of coordinates."

True. But the passage from geometry to arithmetic involves the introduction of real numbers, namely the continuum. Depicting a line by an equation is possible only if you view the line as a set of points, and if you let the function admit real numbers as inputs. Since the continuum is a highly controversial object from the point of view of constructivism, I do not understand how Brouwer can be in favor of the apriority of time for the above mentioned reason.

4) Brouwer says:

"But the most serious blow for the Kantian theory was the discovery of noneuclidean geometry, a consistent theory developed from a set of axioms differing from that of elementary geometry only in this respect that the parallel axiom was replaced by its negative."

No, it is not a blow to Kant. The so-called non-Euclidean geometries are built on hyperbolic or elliptical surfaces. In other words, they are embedded in the usual Euclidean space of 3 dimensions. Moreover, in these new geometries, the words "line" and "parallel" possess different meanings. People find it amazing when you tell them that there are geometries in which two parallel lines always meet. But if they bother to look up the precise mathematical definitions, they will discover that there is nothing special going on. The actual definition of two lines being parallel is very local in nature: Two lines are called parallel if they share a common perpendicular.(Look at the first picture in this Wikipedia article.)

Does Brouwer really understand what Kant means by apriori? Even the most contrived geometry that we can think of is a product of our mental facilities. It is just the meaning of the word "geometry" that is changing from theory to theory. There is still only "one" intuitive notion of space.

5) Brouwer says that he can not embrace the first limit ordinal as an object. Infinity is a process, and only processes can embrace other processes. He views number 5 as an object, but omega as an unfolding process.

The notion of a procedure is temporal in its nature, and Brouwer openly favours apriority of time perception over the apriority of space perception. I am wondering whether he would have been better off manipulating Turing machines rather than playing with numbers. He could have treated the Turing machine that generates five consecutive 1s as the number five, and the Turing machine that prints 1s without halting as omega. Unlike the process that it is capable of unfolding, a Turing machine is a perfectly tangible object that can be embraced as a whole.

I find it weird to debate whether certain ordinals can be granted existence or whether numbers are fundamental for human perception, while there is still an Amazonian tribe that does not have even the most basic number system. (See this blogpost.) Since this tribe is doing perfectly fine, I am wondering whether something is wrong with us instead... I think we are brainwashed into believing that the notion of "absolute" existence has an actual meaning. There is a big difference between the following:

a) When the red light in my right hand blinks, tell me whether there
exists an X on the wall in front of you.

b) Does X exist?

The difference is that the second does not make any sense.

Update (November 2010) : Apparently Konrad Lorenz was the first person to bring an evolutionary perspective to epistemology.