Now, Basti says, "when the final state of the dynamic process derives univocally from the initial moving cause [...], the physical explanation will assume a deductive character" and therefore in this case "the formal-final cause can be reduced, at least conceptually, to the moving one" : in this sense "modern Laplacian physics has developed in an operational (mathematical and experimental) form essentially the first part, i. e. the deductive one, of Aristotelian physics" (see Fig. 2a).
But when it doesn't happen, then "the moving cause is really and conceptually different from the final-formal cause. [...] In this case, indeed, it is necessary for a physicist [...] to individuate the other conditions, clearly distinct from the initial moving cause, which have made possible the achievement of that certain outcome" . Then, "the natural form simply denotes the specificity of a dynamics of a given (not linear and unstable) system which is going to stabilize. A specificity which can be reduced neither to the initial external movement (= the moving cause or the received
You would agree with me that convergences are really striking.
But, again, that's not all. If we accept to identify (even if only in principle, because in practice we are very far to be able to do it) a natural kind with a strange attractor and its members with its trajectories, each one describing, according with Basti, the natural form of the corresponding individual, then we can also understand better what Aristotle and Thomas really meant with this expression: not an unchangeable essence always identical to itself, but something very different and very more realistic, too. As Basti says very forcefully, indeed, individual essences must be "thought as thresholds of quantitative oscillations (alterations) admitted for the individual subject during its existence and depending on a concourse of contingent causes" . Obviously, not all these admitted (that is "present in potentia passiva") oscillations will really happen, without the subject stopping for this reason to be itself. Only when the oscillations get over threshold value (that is are such as to push the trajectory off the attractor) we have the destruction of the subject (namely, in the case of living beings, death). But if these oscillations happen during the initial phase of morphogenesis, instead of the destruction of the subject (which in this case it is not yet born) we'll have a phenomenon sometimes called "iperchaos" , that is a noise-induced jump from one attraction basin to another: in other words, a mutation (see Fig. 2b). And here we have the way open towards a really Darwinian and at the same time really Aristotelian evolution theory, too!
This way of reasoning is for me really fascinating. So, I'm very sorry to have to say you that not even it can solve the problem we have been confronted with in the former paragraph, that is ontological reductionism.
The reason is that, to put it with Doyne Farmer, "it's not sufficient to say emergence: cosmos is full of emergent structures as galaxies, clouds and snow crystals which are only physical objects and have not any autonomous life. It is necessary something more" .
But what can this "something more" be?
4. Intentionality and ontological anti-reductionism.
To firmly establish ontological anti-reductionism (at least for a particular class of complex systems, that is intelligent beings) we have to go on along a more indirect way, recurring to the notion of intellectus agens, better known today with the name of intentionality.
In the tradition of classical philosophy, indeed, from Plato to St. Thomas, passing from Aristotle, Plotino and all Christian philosophers, the most reliable proof of the fact that in man there is a non-material component has always been identified with the typically human capability of abstract thinking, and in particular with the capability of knowing the universal. In fact, since all that exists in material world is individual, it seemed evident that something non-individual could exist only by virtue of the action of something non-material. But in our century this argument has been called in question because of the claim made by the supporters of the so-called "strong" Artificial Intelligence program to be able to simulate human intelligence under every point of view with a completely mechanical system. Many objections has been advanced in time, but always at the philosophical level. Chaos, one time more, has deeply changed all the terms of the question.
On one hand, indeed, the use of non-linear dynamics in AI has made possible an entire set of applications which were simply unthinkable before: Arecchi's adaptive system for recognition and control of chaotic dynamics, which I'll speak about shortly, is an example of it, which I think well-known to you. But, at the same time, precisely these successes have pointed out more and more clearly the in-principle limits of this method.
Let me try to explain you what I mean. At the beginnings of AI all scientists' efforts were focused on the problem of simulating logical and mathematical reasoning. The adduced reason was that they are the paradigmatic forms of human intelligence: but the real reason was that they are not, and just for this reason they are the simplest to be simulated with a mechanical system. At a certain moment, also for the increasing use of robot in industry, it became clear that it could no more be sufficient, at least for the "strong" program, which wanted to reproduce not only this one or that one function of human intelligence, but human intelligence itself. For this purpose it would have been necessary that computers were able to perceive their environment and to communicate with it. Then the so-called problem of pattern recognition (which is the first step of intentionality) became central for AI. And it became soon clear, too, that this was a very more difficult problem than the reproduction of formal reasoning. In a first time the entered way was the traditional one, in which computer programs are seen, roughly speaking, as an implementation of Kantian philosophy: in fact, they analyse data in comparison with a set of fixed rules, and programmers' ability wholly consists in reducing as much as possible their number and in optimizing their efficiency.
But this approach is not successful when we have to face non-linear systems (that is, in practice, all real natural systems), because it leads very soon to intractability, requiring an exponentially increasing number of instructions as their complexity increases to enable the program to forecast every possible variation. As we have seen before, indeed, members of natural kinds are not identical, but only similar, and, even if they really share a common nature, they can be sometimes very different, just as the trajectories on a strange attractor.
For this reason it has been becoming more and more frequent the use of adaptive methodologies for this task, whose model is (explicitly or, more frequently, implicitly) St. Thomas Aquinas's theory of knowledge "by kind and specific difference". The idea is not to furnishing the system of a complete set of instructions about the pattern it must be able to recognize (which is impossible, as we saw), but, rather, to implement in it some instructions enabling it to continuously readapt its inner parameters depending on the inputs received from environment itself. A system like Arecchi's one, for instance , works by stroboscopically observing the wanted dynamics at not fixed time intervals (t), but at variable ones, depending on the difference between the first variations, which must be zero. In practice, the system makes its observation, then calculates the difference with respect to the previous one (dx, first variation) and then the difference between this variation and the former one (d(2)x = dxn+1 - dxn, second variation). If d(2)x = 0, then the system will make its next observation after the same time; but if d(2)x ¹ 0, then it will modify the time interval, and precisely it will increase it if d(2)x > 0, while it will diminish it if d(2)x < 0.>
The trick is that by operating in this way every dynamics generates a characteristic distribution of the intervals t between successive observations, namely: a point for a periodic dynamics, a straight line for a chaotic one and a dispersion all over the plane for a random one.
The discrimination among the different kinds of dynamics, therefore, is obtained here not in force of their direct comparison with a set of pre-programmed models, but basing on a set of data which are produced by the interaction of the particular shape of the dynamics itself with the computer-controlled measure apparatus. And this is the reason why I said previously that the implicit (but in Arecchi also explicitly declared) philosophical archetype of these systems is the Thomasian theory of knowledge instead of the Kantian one: because here every set of data represents a characteristic feature of the corresponding dynamics (its "specific difference") which doesn't pre-exist, in a Kantian way ("a priori"), in the "subject" (that is, in the computer program), but raises in it in the very moment of its interaction with reality.
So, my friend Arecchi always says that his system is a "not-with-fixed-rules" one. And every time I discussed with him I always objected that it is not true, because also in it there are fixed rules, namely: those individuating the parameter to be observed, those controlling the way in which its observation must be modified and, finally, those leading the evaluation of the so-obtained data. As you can see -you'll agree with me, I hope- they are not so few, after all! And furthermore each of them had to be chosen and set up very carefully -during a study many year long- to reach the proposed goal. Therefore I was very happy when I read, in the paper devoted to this subject in his recent Lexicon of Complexity , that "the main difference of an adaptive M apparatus with respect to a fixed one is that not all parameters are prepared by preliminary learning sessions" : that is just I have always been maintaining. Not all rules are fixed: this is correct. But many are. And -this is the point!- if they were not, then the system could not function, because it would be, so to speak, a pure syntax without any semantic.
So, we have the following paradox, which we could name "The machine-not-like-a-machine paradox":
Is it an effective demonstration that human intelligence is a non-material function requiring a non-material component in human beings? The correct answer, in my opinion, cannot be sharp.
Certainly I think that now we can be sure that human mind is not like a computer software. In fact, since also the "not-with-fixed-rules" systems require, in reality, fixed rules and models, it follows that every computer program can work only at a limited extent. But human mind has not such a limitation. Then, human mind, at least at some extent, does not work like a computer.
This is not still a demonstration that it is really emergent in a different and stronger way than Farmer's crystals and clouds, namely, as John Searle says, that it is "able to cause effects we cannot explain only by analysing neurones' behaviour" , but our analysis provides us strong and convergent circumstantial evidence in this sense, even if not a positive and conclusive proof.
Firstly, this hypothesis seems to be the best explanation of the former paradox. Indeed, if intentionality is not completely analysable in physical terms, then it is only logical, and by no means strange (as it would be in the opposite case), that a mechanical system not only cannot entirely simulate it, but, on the other hand, can do it the better, the less "mechanical" turns out to be.
In the second place, it is at least questionable that the problem of models regards only the computational approach and not the materialistic approach tout court. We can, indeed, admit that neurones have other causal powers besides the computational ones, but how could the evaluation of data happen otherwise than through some kind of comparison with some kind of physical model, if all is to be explained on a merely material plane? But in this case we have again the same problem we saw in the computational approach, that is: are these models preformed in us or not? If they were, they should be infinite in number, that is impossible. But if they are not, then it means that we construct them. But it is very difficult even to hypothesise how such a process of construction could happen without recurring again to some sort of models, since the most efficient material process we have been able to reproduce update needs them.
Finally, the various attempts to simulate intentionality have shown a scale of increasing difficulty going from "classical" objects to the "complex" ones to reach its maximum with the problem to construct a system able to recognize abstract concepts in the strict sense of the term. Here the situation is even tragic, because simply there are no ideas at all of how it could be done. The adaptive strategy, indeed, is not only limited, but completely useless for such a task, because it needs physical patterns to work: and abstract concepts have not (obviously I'm referring, as I said, to abstract concepts in the strict sense of the word, as, for instance, "good", "law", "philosopher", "science", and so on, and not to the ones corresponding to natural kinds of objects or living beings, which can be associated to physical patterns -even if without coinciding with them, but this would be too a long question to be treated here). Maybe you could object that abstract concepts are communicated through the language, which has recognizable patterns. But so you forget that concepts are only communicated linguistically: but the first time a concept is thought there is not a word to express it, and therefore this operation cannot have a pattern to base on. However, every analysis of the structure of a symbolic sequence can ever tell us only that it has (presumably) a meaning, and not -by no means- what it is. This is a conclusion that holds generally, and not only in the case of human language, but a complete discussion would be to long now. However, in the case of language the fact is so evident that every demonstration is useless: nobody but a fool could ever think to understand a stranger language only basing on a careful analysis of the sequence of its letters. But if meaning cannot be deduced by the structure of the symbolic sequence, and, on the other hand, you wanted to maintain a materialistic explanation of the phenomenon, how can you ever do it? Maybe trying to feel what taste it has?
The fact of the matter is that computational strategy was not chosen at random for this task, but for the very reason that it seemed (and despite all it seems still) the most promising, or, more exactly, the only. Therefore its failure in simulating intentionality is to be valued in the light of its successes (which are many and great) and, then, cannot be considered as a simple accident among others. In other words: maybe (better, certainly) human mind is not a computer software; but in this case what physical object could it ever be?
In any case it is, as I said before, only circumstantial evidence and not a conclusive proof. But it is very noteworthy that all today's scientific considerations converge with one another and also with the traditional philosophical arguments to indicate that the solution of the problem of intentionality -and therefore of human mind (or soul, if you prefer)- is to be searched out of the purely material dimension.
5. On the nature of mathematics.
Before ending I would like spend only few words about the perhaps only classical problem of philosophy that also scientists have always considered very carefully and with respect, that is: what is mathematics?
To answer this eternal question in an acceptable way would take at least an entire book (which I'm trying to write, indeed) and not a little paragraph in a paper like this. So, here I'll simply tell you what is my theory and what was the discovery from which it arose. In a nutshell, the core of my idea is that mathematics is a natural science just like all the other ones, only having the most possible generality of all, in which the experimental part is constituted by calculation.
And the crucial discovery was exactly the last one, namely the material nature of calculation.
This isn't a complete novelty, indeed. Maybe you know that during last years some mathematicians (among which Gregory Chaitin) have begun to speak about "experimental mathematics" , just for the reason that when we have to do with chaos and fractal we can no more solve the equations once and for all, but only study them numerically, case by case, in a pretty experimental way. But in their mind this should be only a branch of mathematics. On the contrary, I say this is a general and fundamental feature of the whole mathematics, which becomes in these cases only more evident, but is ever present.
Indeed, this was yet implicit in Chaitin's views themselves. In fact, if you consider, for instance, the Mandelbrot set, you find that the equation generating it (z ® z2 + c), if applied to the real numbers instead of the complex ones, generates only a well known, very trivial segment, going from the point corresponding to -2 to that corresponding to zero (see Fig. 3). But it would be absurd that something could be change its proper nature only by changing the objects to which it applies. So, if solving Mandelbrot equation when it applies to complex numbers requests an experimental procedure, then it means that this is a feature of Mandelbrot equation as such. But Mandelbrot equation is a very common one, formally indistinguishable from the others. Then, it means that solving equations is in general an experimental task.
This idea struck me hard. Then, I remembered of Percy William Bridgman and all became clear in my mind. Bridgman was, as you know, the main exponent of operationalism, that is a stream of thought maintaining that all in science can be reduced to operations. Since his opponents replied that he didn't take account of mathematics, Bridgman said that also in mathematics there are "operations with paper and pencil". Well, I'm not certainly an operationalist, but for other reasons: in fact, about this point Bridgman was really right. Maybe even more than he himself was thinking.
Indeed, Bridgman's reply was not considered seriously, because with "operations" he had always meant, until that moment, material operations, and those "with paper and pencil" looked to be all but this. But, on the contrary, they are.
It is a tragic, well known to each schoolchild truth that it is impossible to study mathematics without knowing multiplication table. And it is another tragic truth that it is necessary to learn it by heart: it is impossible to can grasp the right results only by reasoning on it.
The matter stands thus. But why?
The fact is that to know the right result of a mathematical operation we have to count. It's the material operation of counting which is the base of the whole mathematics. And all mathematical operations are nothing but methods to make it faster: in fact, using, for instance, decimal system, we count the number of the tens or of the hundreds and so on instead of directly the units' one. Certainly, we don't always count in the same way: when we have to make a sum we count in the most natural way, always going on, while to make a subtraction we must firstly go on and then go back. To perform even more complex operations (a square root, for instance) we have to obey to even more complex rules. But the nature of what we do is still the same, that is: counting a set of symbols according to a certain set of rules. Therefore, without any material items to be counted we'd be able simply to do nothing in mathematics. We forget it because when we have to do with not too big numbers we can make calculations only in our mind. But we do it by imagining the symbols representing the numbers, just like we do also, for instance, in the so-called Gedanken Experiment, in which we understand the outcome of a material operation without really performing it. But the fact that something is imagined does not mean it is not material as for itself. And it becomes clear immediately that this is the case if we try to do the same with numbers only a little bigger, that is impossible without a material external support. Big numbers, indeed, are big not only in our mind: they take space.
This has become particularly conspicuous with computers, in which every operation is, without any doubt, a material one, because a computer is nothing but a physical system evolving from a state to another. It's just for this reason that "strong" AI supporters claimed to have demonstrate that human mind is a completely physical phenomenon, since they were able to reproduce with such a system its (supposed) higher functions, i.e. deductive mathematical and logical operations. I said former that, on the contrary, they are the simplest ones to be simulated in this way. And, if I'm right, now we know also why: they are the simplest to be simulated by a mechanical system because they are mechanical in their deeper essence.
And thus, the ring is closed.
6. Conclusions.
Finally, I want to clarify a point. I'm very far to be completely satisfied of my analysis, because it is, so to speak, a minimalist one. I'm not a Cartesian and I'm personally sure there are not only two different ontological levels in reality, and also in man itself. A great problem, for instance, is whether perception is completely explainable in physical terms: personally I have many doubts. While I am not so sure, on the other hand, that, in the light of the most recent discoveries, it is still possible to consider intelligence as a peculiar and exclusive faculty of mankind (even if it is clear that mankind possesses it in an eminent way, to speak with medieval philosophers). More generally, it seems to me that traditional demarcation lines are no more completely adequate to the present situation of our knowledge and perhaps the whole question needs to be put in a partially new way. But I also think it cannot be solved once and for all with some kind of general demonstration, as in the case of the levels of knowledge. When we have to match with reality itself the problems are always more complex than when we have only to consider our way of representing it, because it's not us who made it: therefore I think it is highly improbable that a global solution can be reached basing on a single methodology as chaos theory, how general and powerful it can be. I'm actually working on this, and I'll be very happy for every suggestion you can give me. I hope to can tell you something of more complete and more satisfactory, soon or later. But today's paper was only devoted to the philosophical implications of chaos. And, at least for the moment, I assure you that it has been enough for me. And, I think, also for you.
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