Task of the Philosopher. Thought of the Day 75.0

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Poincaré in Science and Method discusses how “reasonable” axioms (theories) are chosen. In a section which is intended to cool down the expectations put in the “logistic” project, he points out the problem as follows:

Even admitting that it has been established that all theorems can be deduced by purely analytical processes, by simple logical combinations of a finite number of axioms, and that these axioms are nothing but conventions, the philosopher would still retain the right to seek the origin of these conventions, and to ask why they were judged preferable to the contrary conventions.

[ …] A selection must be made out of all the constructions that can be combined with the materials furnished by logic. the true geometrician makes this decision judiciously, because he is guided by a sure instinct, or by some vague consciousness of I know not what profounder and more hidden geometry, which alone gives a value to the constructed edifice.

Hence, Poincaré sees the task of the philosophers to be the explanation of how conventions came to be. At the end of the quotation, Poincaré tries to give such an explanation, namely in referring to an “instinct” (in the sequel, he mentions briefly that one can obviously ask where such an instinct comes from, but he gives no answer to this question). The pragmatist position to be developed will lead to an essentially similar, but more complete and clear point of view.

According to Poincaré’s definition, the task of the philosopher starts where that of the mathematician ends: for a mathematician, a result is right if he or she has a proof, that means, if the result can be logically deduced from the axioms; that one has to adopt some axioms is seen as a necessary evil, and one perhaps puts some energy in the project to minimize the number of axioms (this might have been how set theory become thought of as a foundation of mathematics). A philosopher, however, wants to understand why exactly these axioms and no other were chosen. In particular, the philosopher is concerned with the question whether the chosen axioms actually grasp the intended model. This question is justified since formal definitions are not automatically sufficient to grasp the intention of a concept; at the same time, the question is methodologically very hard, since ultimately a concept is available in mathematical proof only by a formal explication. At any rate, it becomes clear that the task of the philosopher is related to a criterion problem.

Georg Kreisel thinks that we do indeed have the capacity to decide whether a given model was intended or not:

many formal independence proofs consist in the construction of models which we recognize to be different from the intended notion. It is a fact of experience that one can be honest about such matters! When we are shown a ‘non-standard’ model we can honestly say that it was not intended. [ . . . ] If it so happens that the intended notion is not formally definable this may be a useful thing to know about the notion, but it does not cast doubt on its objectivity.

Poincaré could not yet know (but he was experienced enough a mathematician to “feel”) that axiom systems quite often fail to grasp the intended model. It was seldom the work of professional philosophers and often the byproduct of the actual mathematical work to point out such discrepancies.

Following Kant, one defines the task of epistemology thus: to determine the conditions of the possibility of the cognition of objects. Now, what is meant by “cognition of objects”? It is meant that we have an insight into (the truth of) propositions about the objects (we can then speak about the propositions as facts); and epistemology asks what are the conditions for the possibility of such an insight. Hence, epistemology is not concerned with what objects are (ontology), but with what (and how) we can know about them (ways of access). This notwithstanding, both things are intimately related, especially, in the Peircean stream of pragmatist philosophy. The 19th century (in particular Helmholtz) stressed against Kant the importance of physiological conditions for this access to objects. Nevertheless, epistemology is concerned with logic and not with the brain. Pragmatism puts the accent on the means of cognition – to which also the brain belongs.

Kant in his epistemology stressed that the object depends on the subject, or, more precisely, that the cognition of an object depends on the means of cognition used by the subject. For him, the decisive means of cognition was reason; thus, his epistemology was to a large degree critique of reason. Other philosophers disagreed about this special role of reason but shared the view that the task of philosophy is to criticise the means of cognition. For all of them, philosophy has to point out about what we can speak “legitimately”. Such a critical approach is implicitly contained in Poincaré’s description of the task of the philosopher.

Reichenbach decomposes the task of epistemology into different parts: guiding, justification and limitation of cognition. While justification is usually considered as the most important of the three aspects, the “task of the philosopher” as specified above following Poincaré is not limited to it. Indeed, the question why just certain axioms and no others were chosen is obviously a question concerning the guiding principles of cognition: which criteria are at work? Mathematics presents itself at its various historical stages as the result of a series of decisions on questions of the kind “Which objects should we consider? Which definitions should we make? Which theorems should we try to prove?” etc. – for short: instances of the “criterion problem”. Epistemology, has all the task to evoke these criteria – used but not evoked by the researchers themselves. For after all, these criteria cannot be without effect on the conditions for the possibility of cognition of the objects which one has decided to consider. (In turn, the conditions for this possibility in general determine the range of objects from which one has to choose.) However, such an epistemology has not the task to resolve the criterion problem normatively (that means to prescribe for the scientist which choices he has to make).

Comment on Purely Random Correlations of the Matrix, or Studying Noise in Neural Networks

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In the presence of two-body interactions the many-body Hamiltonian matrix elements vJα,α′ of good total angular momentum J in the shell-model basis |α⟩ generated by the mean field, can be expressed as follows:

vJα,α′ = ∑J’ii’ cJαα’J’ii’ gJ’ii’ —– (4)

The summation runs over all combinations of the two-particle states |i⟩ coupled to the angular momentum J′ and connected by the two-body interaction g. The analogy of this structure to the one schematically captured by the eq. (2) is evident. gJ’ii’ denote here the radial parts of the corresponding two-body matrix elements while cJαα’J’ii’ globally represent elements of the angular momentum recoupling geometry. gJ’ii’ are drawn from a Gaussian distribution while the geometry expressed by cJαα’J’ii’ enters explicitly. This originates from the fact that a quasi-random coupling of individual spins results in the so-called geometric chaoticity and thus cJαα’ coefficients are also Gaussian distributed. In this case, these two (gJ’ii’ and c) essentially random ingredients lead however to an order of magnitude larger separation of the ground state from the remaining states as compared to a pure Random Matrix Theory (RMT) limit. Due to more severe selection rules the effect of geometric chaoticity does not apply for J = 0. Consistently, the ground state energy gaps measured relative to the average level spacing characteristic for a given J is larger for J > 0 than for J = 0, and also J > 0 ground states are more orderly than those for J = 0, as it can be quantified in terms of the information entropy.

Interestingly, such reductions of dimensionality of the Hamiltonian matrix can also be seen locally in explicit calculations with realistic (non-random) nuclear interactions. A collective state, the one which turns out coherent with some operator representing physical external field, is always surrounded by a reduced density of states, i.e., it repells the other states. In all those cases, the global fluctuation characteristics remain however largely consistent with the corresponding version of the random matrix ensemble.

Recently, a broad arena of applicability of the random matrix theory opens in connection with the most complex systems known to exist in the universe. With no doubt, the most complex is the human’s brain and those phenomena that result from its activity. From the physics point of view the financial world, reflecting such an activity, is of particular interest because its characteristics are quantified directly in terms of numbers and a huge amount of electronically stored financial data is readily available. An access to a single brain activity is also possible by detecting the electric or magnetic fields generated by the neuronal currents. With the present day techniques of electro- or magnetoencephalography, in this way it is possible to generate the time series which resolve neuronal activity down to the scale of 1 ms.

One may debate over what is more complex, the human brain or the financial world, and there is no unique answer. It seems however to us that it is the financial world that is even more complex. After all, it involves the activity of many human brains and it seems even less predictable due to more frequent changes between different modes of action. Noise is of course owerwhelming in either of these systems, as it can be inferred from the structure of eigen-spectra of the correlation matrices taken across different space areas at the same time, or across different time intervals. There however always exist several well identifiable deviations, which, with help of reference to the universal characteristics of the random matrix theory, and with the methodology briefly reviewed above, can be classified as real correlations or collectivity. An easily identifiable gap between the corresponding eigenvalues of the correlation matrix and the bulk of its eigenspectrum plays the central role in this connection. The brain when responding to the sensory stimulations develops larger gaps than the brain at rest. The correlation matrix formalism in its most general asymmetric form allows to study also the time-delayed correlations, like the ones between the oposite hemispheres. The time-delay reflecting the maximum of correlation (time needed for an information to be transmitted between the different sensory areas in the brain is also associated with appearance of one significantly larger eigenvalue. Similar effects appear to govern formation of the heteropolymeric biomolecules. The ones that nature makes use of are separated by an energy gap from the purely random sequences.

 

Distributed Representation Revisited

Figure-132-The-distributed-representation-of-language-meaning-in-neural-networks

If the conventional symbolic model mandates a creation of theory that is sought to address the issues pertaining to the problem, this mandatory theory construction is bypassed in case of distributed representational systems, since the latter is characterized by a large number of interactions occurring in a nonlinear fashion. No such attempts at theoretical construction are to be made in distributed representational systems for fear of high end abstraction, thereby sucking off the nutrient that is the hallmark of the model. Distributed representation is likely to encounter onerous issues if the size of the network inflates, but the issue is addressed through what is commonly known as redundancy technique, whereby, a simultaneous encoding of information generated by numerous interactions take place, thus ameliorating the adequacy of presenting the information to the network. In the words of Paul Cilliers, this is an important point, for,

the network used for the model of a complex system will have to have the same level of complexity as the system itself….However, if the system is truly complex, a network of equal complexity may be the simplest adequate model of such a system, which means that it would be just as difficult to analyze as the system itself.

Following, he also presents a caveat,

This has serious methodological implications for the scientists working with complex systems. A model which reduces the complexity may be easier to implement, and may even provide a number of economical descriptions of the system, but the price paid for this should be considered carefully.

One of the outstanding qualities of distributed representational systems is their adaptability. Adaptability, in the sense of reusing the network to be applicable to other problems to offer solutions. Exactly, what this connotes is, the learning process the network has undergone for a problem ‘A’, could be shared for problem ‘B’, since many of the input neurons are bounded by information learned through ‘A’ that could be applicable to ‘B’. In other words, the weights are the dictators for solving or resolving issues, no matter, when and for which problem the learning took place. There is a slight hitch here, and that being this quality of generalizing solutions could suffer, if the level of abstraction starts to shoot up. This itself could be arrested, if in the initial stages, the right kind of framework is decided upon, thus obscuring the hitch to almost non-affective and non-existence impacting factor. The very notion of weights is considered here by Sterelny as a problematic, and he takes it to attack distributed representation in general and connectionsim as a whole in particular. In an analogically witty paragraph, Sterelny says,

There is no distinction drawable, even in principle, between functional and non- functional connections. A positive linkage between two nodes in a distributed network might mean a constitutive link (eg. Catlike, in a network for tiger); a nomic one (carnivore, in the same network), or a merely associative one (in my case, a particular football team that play in black and orange.

It should be noted that this criticism on weights is derived, since for Sterelny, relationship between distributed representations and the micro-features that compose them is deeply problematic. If such is the criticism, then no doubt, Sterelny still seems to be ensconced within the conventional semantic/symbolic model. And since, all weights can take part in information processing, there is some sort of a democratic liberty that is accorded to the weights within a distributed representation, and hence any talk of constitutive, nomic, or even for that matter associative is mere humbug. Even if there is a disagreement prevailing that a large pattern of weights are not convincing enough for an explanation, as they tend to complicate matters, the distributed representational systems work consistently enough as compared to an alternative system that offers explanation through reasoning, and thereby, it is quite foolhardy to jettison the distributed representation by the sheer force of criticism. If the neural network can be adapted to produce the correct answer for a number of training cases that is large compared with the size of the network, it can be trusted to respond correctly to the previously unseen cases provided they are drawn from the same population using the same distribution as the training cases, thus undermining the commonly held idea that explanations are the necessary feature of the trustworthy systems (Baum and Haussler). Another objection that distributed representation faces is that, if representations are distributed, then the probability of two representations of the same thing as different from one another cannot be ruled out. So, one of them is the true representation, while the other is only an approximation of the representation.(1) This is a criticism of merit and is attributed to Fodor, in his influential book titled Psychosemantics.(2) For, if there is only one representation, Fodor would not shy from saying that this is the yucky solution, folks project believe in. But, since connectionism believes in the plausibility of indeterminate representations, the question of flexibility scores well and high over the conventional semantic/symbolic models, and is it not common sense to encounter flexibility in daily lives? The other response to this objection comes from post-structuralist theories (Baudrillard is quite important here. See the first footnote below). The objection of true representation, and which is a copy of the true representation meets its pharmacy in post-structuralism, where meaning is constituted by synchronic as well as diachronic contextualities, and thereby supplementing the distributed representation with a no-need-for concept and context, as they are inherent in the idea of such a representation itself. Sterelny, still seems to ride on his obstinacy, and in a vitriolic tone poses his demand to know as to why distributed representation should be regarded as states of the system at all. Moreover, he says,

It is not clear that a distributed representation is a representation for the connectionist system at all…given that the influence of node on node is local, given that there is no processor that looks at groups of nodes as a whole, it seems that seeing a distributed representation in a network is just an outsider’s perspective on the system.

This is moving around in circles, if nothing more. Or maybe, he was anticipating what G. F. Marcus would write and echo to some extent in his book The Algebraic Mind. In the words of Marcus,

…I agree with Stemberger(3) that connectionism can make a valuable contribution to cognitive science. The only place, we differ is that, first, he thinks that the contribution will be made by providing a way of eliminating symbols, whereas I think that connectionism will make its greatest contribution by accepting the importance of symbols, seeking ways of supplementing symbolic theories and seeking ways of explaining how symbols could be implemented in the brain. Second, Stemberger feels that symbols may play no role in cognition; I think that they do.

Whatever Sterelny claims, after most of the claims and counter-claims that have been taken into account, the only conclusion for the time being is that distributive representation has been undermined, his adamant position to be notwithstanding.

(1) This notion finds its parallel in Baudrillard’s Simulation. And subsequently, the notion would be invoked in studying the parallel nature. Of special interest is the order of simulacra in the period of post-modernity, where the simulacrum precedes the original, and the distinction between reality and representation vanishes. There is only the simulacrum and the originality becomes a totally meaningless concept.

(2) This book is known for putting folk psychology firmly on the theoretical ground by rejecting any external, holist and existential threat to its position.

(3) Joseph Paul Stemberger is a professor in the Department of Linguistics at The University of British Columbia in Vancouver, British Columbia, Canada, with primary interests in phonology, morphology, and their interactions. My theoretical orientations are towards Optimality Theory, employing our own version of the theory, and towards connectionist models.

 

Simulations of Representations: Rational Calculus versus Empirical Weights

While modeling a complex system, it should never be taken for granted that these models somehow simplify the systems, for that would only strip the models of the capability to account for encoding, decoding, and retaining information that are sine qua non for the environment they plan to model, and the environment that these models find themselves embedded in. Now, that the traditional problems of representation are fraught with loopholes, there needs to be a way to jump out of this quandary, if modeling complex systems are not to be impacted by the traces of these very traditional notions of representation. The employment of post-structuralist theories are sure indicative of getting rid of the symptoms, since they score over the analytical tradition, where, representation is only an analogue of the thing represented, whereas, simulation with its affinity to French theory is conducive to a distributed and a holistic analogy. Any argument against representation is not to be taken as meaning anti-scientific, since it is merely an argument against a particular scientific methodology and/or strategy that assumes complexity to be reducible, and therefore implementable or representable in a machine. The argument takes force only as an appreciation for the nature of complexity, something that could perhaps be repeated in a machine, should the machine itself be complex enough to cope with the distributed character of complexity. Representation is a state that stands-in for some other state, and hence is nothing short of “essentially” about meaning. The language, thought that is incorporated in understanding the world we are embedded in is efficacious only if representation relates to the world, and therefore “relationship” is another pillar of representation. Unless a relationship relates the two, one gets only an abstracted version of the so-called identities in themselves with no explanatory discourse. In the world of complexity, such identity based abstractions lose their essence, for modeling takes over the onus of explanations, and therefore, it is without doubt, the establishment of these relations that bring together states of representations as taking high priority. Representation holds a central value in both formal systems and in neural networks or connectionism, where the former is characterized by a rational calculus, and the latter by patterns that operate over the network lending it a more empirical weight.

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Let logic programming be the starting point for deliberations here. The idea behind this is using mathematical logic to successfully apply to computer programming. When logic is used as such, it is used as a declarative representational language; declarative because, logic of computation is expressed without accounting for the flow of control. In other words, within this language, the question is centered around what-ness, rather than how-ness. Declarative representation has a counterpart in procedural representation, where the onus is on procedures, functions, routines and methods. Procedural representation is more algorithmic in nature, as it depends upon following steps to carry out computation. In other words, the question is centered around how-ness. But logic programming as it is commonly understood cannot do without both of them becoming a part of programming language at the same time. Since both of them are required, propositional logic that deals primarily with declarative representational languages would not suffice all alone, and hence, what is required is a logic that would touch upon predicates as well. This is made possible by first-order predicate logic that distinguishes itself from propositional logic by its use of quantifiers(1). The predicate logic thus finds its applications suited for deductive apparatus of formal systems, where axioms and rules of inferences are instrumental in deriving theorems that guide these systems. This setup is too formal in character and thus calls for a connectionist approach, since the latter is simply not keen to have predicate logic operate over deductive apparatus of a formal system at its party.

If brain and language (natural language and not computer languages, which are more rule-based and hence strict) as complex systems could be shown to have circumvented representationism via modeling techniques, the classical issues inherent in representation would be gotten rid of in the sense of a problematic. Functionalism as the prevalent theory in philosophy of mind that parallels computational model is the target here. In the words of Putnam,

I may have been the first philosopher to advance the thesis that the computer is the right model for mind. I gave my form of this doctrine the name ‘functionalism’, and under this name, it has become the dominant view – some say the orthodoxy – in contemporary philosophy of mind.

The computer metaphor with mind is clearly visible here, with the former having an hardware apparatus that is operated upon by the software programs, while the latter shares the same relation with brain (hardware) and mind (software). So far, so good, but there is a hitch. Like the computer side of metaphor, which can have a software loaded on to different hardwares, provided there is enough computational capability possessed by the hardware, the mind-brain relationship should meet the same criteria as well. If one goes by what Sterelny has hinted for functionalism as a certain physical state of the machine realizing a certain functional state, then a couple of descriptions, mutually exclusive of one another result, viz, a description on the physical level, and a description on the mental level. The consequences of such descriptions are bizarre to the extent that mind as a software can also find its implementation on any other hardware, provided the conditions for hardware’s capability to run the software are met successfully. One could hardly argue against these consequences that follow logically enough from the premisses, but a couple of blocks are not to be ignored at the same time, viz, the adequacy of the physical systems to implement the functional states, and what defines the relationships between these two mutually exclusive descriptions under the context of the same physical system. Sterelny comes up with a couple of criteria for adequate physical systems, designed, and teleological. Rather than provide any support for what he means by the systems as designed, he comes up with evolutionary tendencies, thus vouching for an external designer. The second one gets disturbing, if there is no description made, and this is precisely what Sterelny never offers. His citation of a bucket of water not having a telos in the sense of brain having one, only makes matters slide into metaphysics. Even otherwise, functionalism as a nature of mental states is metaphysical and ontological in import. This claim gets all the more highlighted, if one believes following Brentano that intentionality is the mark of the mental, then any theory of intentionality can be converted into a theory of of the ontological nature of psychological states. Getting back to the second description of Sterelny, functional states attain meaning, if they stand for something else, hence functionalism gets representational. And as Paul Cilliers says it cogently, grammatical structure of the language represents semantical content, and the neurological states of the brain represent certain mental states, thus proving without doubt, the responsibility on representation on establishing a link between the states of the system and conceptual meaning. This is again echoed in Sterelny,

There can be no informational sensitivity without representation. There can be no flexible and adaptive response to the world without representation. To learn about the world, and to use what we learn to act in new ways, we must be able to represent the world, our goals and options. Furthermore we must make appropriate inferences from these representations.

As representation is essentially about meaning, two levels are to be related with one another for any meaning to be possible. In the formal systems, or the rule-based approach, these relations are provided by creating a nexus between “symbol” and what it “symbolizes”. This fundamental linkage is offered by Fodor in his 1975 book, The Language of Thought. The main thesis of the book is about cognition and cognitive processes as remotely plausible, when computationally expressed in terms of representational systems. The language in possession of its own syntactic and semantic structures, and also independent of any medium, exhibits a causal effect on mental representations. Such a language is termed by him “mentalese”, which is implemented in the neural structure (a case in point for internal representation(2)), and following permutations allows for complex thoughts getting built up through simpler versions. The underlying hypothesis states that such a language applies to thoughts having propositional content, implying thoughts as having syntaxes. In order for complex thoughts to be generated, simple concepts are attached with the most basic linguistic token that combine following rules of logic (combinatorial rules). The language thus enriched is not only productive, with regard to length of the sentence getting longer (potentially so) without altering the meaning (concatenation), but also structured, in that rules of grammar that allow us to make inferences about linguistic elements previously unrelated. Once this task is accomplished, the representational theory of thought steps in to explicate on the essence of tokens and how they behave and relate. The representational theory of thought validates mental representations, that stand in uniquely for a subject of representation having a specific content to itself, to allow for causally generated complex thought. Sterelny echoes this when he says,

Internal representation helps us visualize our movements in the world and our embeddedness in the world. Internal representation takes it for granted that organisms inherently have such an attribute to have any cognition whatsoever. The plus point as in the work of Fodor is the absence of any other theory that successfully negotiates or challenges the very inherent-ness of internal representation.

For this model, and based on it, require an agent to represent the world as it is and as it might be, and to draw appropriate inferences from that representation. Fodor argues that the agent must have a language-like symbol system, for she can represent indefinitely many and indefinitely complex actual and possible states of her environment. She could not have this capacity without an appropriate means of representation, a language of thought. Mentalese thus is too rationalist in its approach, and hence in opposition to neural networks or connectionism. As there can be no possible cognitive processes without mental representations, the theory has many takers(3). One line of thought that supports this approach is the plausibility of psychological models that represent cognitive processes as representational thereby inviting computational thought to compute.

(1) Quantifier is an operator that binds a variable over a domain of discourse. The domain of discourse in turn specifies the range of these relevant variables.

(2) Internal representation helps us visualize our movements in the world and our embeddedness in the world. Internal representation takes it for granted that organisms inherently have such an attribute to have any cognition whatsoever. The plus point as in the work of Fodor is the absence of any other theory that successfully negotiates or challenges the very inherent-ness of internal representation.

(3) Tim Crane is a notable figure here. Crane explains Fodor’s Mentalese Hypothesis as desiring one thing and something else. Crane returns to the question of why we should believe the vehicle of mental representation is a language. Crane states that while he agrees with Fodor, his method of reaching it is very different. Crane goes on to say that reason: our ability as humans to decide a rational decision from the information giving is his argument for this question. Association of ideas lead to other ideas which only have a connection for the thinker. Fodor agrees that free association goes on but he says that is in a systemic, rational way that can be shown to work with the Language of Thought theory. Fodor states you must look at in a computational manner and that this allows it to be seen in a different light than normally and that free association follows a certain manner that can be broken down and explained with Language of Thought. Language of Thought.

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