Friday, October 20, 2017

Thoughts on how to find alternate algebra-like systems

It seems like there probably exists a category of systems which exhibit some of the most interesting and useful aspects of algebra, and yet are very dissimilar in other aspects. It’s probably a good starting point for seeking them out to consider what is arbitrary and what is significant in the operation of extant algebraic systems—in other words what is their ‘essence’, and what are their incidental aspects.

My understanding is that their main utility lies in their providing a systematic means of exploring alternate, equivalent representations of values. They let us take two representations which each evaluate to the same thing, and gives us a set of rules for transforming only the representations while the value stays the same (I call these transformations ‘ortho-semantic’ operations since they do not affect the meaning/evaluation of representations—they are orthogonal to semantics). By shifting representations around in this way, while maintaining consistent relations between them, we can discover new patterns and relationships which we would never otherwise have expected to exist.

One especially common/important goal of carrying out these transformations is to 'solve for' unknown elements by taking a pair of equivalent representations and transforming them until one consists only of a single atomic representational element whose value is unknown, while the other representation may be more complex, but readily evaluated. This may generalize to something like: representations may include ‘placeholder’ elements which cannot be evaluated in isolation; however, when equivalent representations are put into some kind of correspondence (e.g. like being placed on either side of an equals sign in traditional algebra), applying sequences of ortho-semantic operations to the representations may put the system into a state in which the correct evaluation for a ‘placeholder’ element is unambiguously revealed.

I think viewing the situation in that way frees the imagination a little to explore new forms of systems exhibiting those properties, while implementing them in potentially radically different ways.

My suspicion is that some of the specific features of algebra as we know it are accidental outgrowths of the historical fact that we had to do these representation transformations by writing symbols on paper. Especially given our familiarity with written language, this would bias us to a sequential, symbolic representation.  However, (what I claim are) the essential characteristics of algebra—i.e. the capability of systematically transforming equivalent representations without changing their value/meaning—do not depend intrinsically on symbol sequences.

And the kicker is: the ortho-semantic operations in the various algebras we use all depend on the notions of inverse and identity, since the two together provide a simple means of moving a symbol (or group of symbols) from one representation to its equivalent; and transforming the representations in that manner is of the essence if your machinery for carrying out and recording the transformations consists entirely in symbols drawn on paper via pencil by a human. That’s my central thesis here: we have assumed identity/inverses are part of the essence of systems that behave like algebras—but maybe they aren’t as necessary as they seem.

In the way mathematics has presently generalized specific algebraic systems into an abstract theory, inverse and identities play a very central role. I wonder two main things: 1) Could specific, alternate algebra-like systems be developed which have similar or greater power than traditional algebras, yet do not depend on inverses/identity? 2) Could a generalized mathematical theory be developed which deals more directly with what I claim is at the heart of algebra’s operation (systematically transforming equivalent representations without changing meaning/value), rather than focusing on a particular implementation of that behavior which happens to require inverses and identities? I know many will say that the rich theory around algebra is a sort of proof that it’s the ‘correct’ track—and I would counter that there’s plenty of space in the realm of pure mathematics for more such rich theories. One could also point to uses of abstract algebra in the physical sciences as a kind of proof of the same thing, but if I’m not mistaken it’s often the case already that the ground is covered by alternate theories as well, e.g. Category-theoretic formulations, so I don’t think of success of pre-existing systems as proof of any kind of ultimate correctness.

I think any algebra-like system must have these parts (and only these parts?):
  • A syntax (i.e. a definition of allowed symbols and how they may be arranged into statements; doesn’t actually have to be symbol sequences though, any consistent representation whose rules may be stated is fine).
  • A semantics: a mapping of syntactically valid statements into some other domain of ‘values’. For the system to work well it should be the case that the semantics frequently maps multiple unique syntactic statements to the same value.
  • A set of ortho-semantic operations, describing the ways in which one may convert sets of statements into equivalent sets of statements (equivalent in that the semantics would map the statements to the same values).

—maybe the general theory of these systems would use those ‘parts’ as its central terms?

One thing that occurs to me is that it’s not necessarily necessary to isolate single variables (i.e. ‘solve for’ single variables) in order to discover unknown values in other algebra-like systems. It’s necessary in algebra because if you had something like this: x + y + z = a + b + c, where x, y, and z are unknown and a, b, and c are known, it’s ambiguous which variable maps to which since addition is commutative. It’s possible that in alternate, algebra-like systems, you could perform some ortho-semantic operation which causes a number of representational parts to align unambiguously and have their meaning revealed through correspondence to the ‘partner representation’ (I would call each side of an equation one of the ‘partner representations’ in traditional algebra systems).

My best guess is that any alternate, algebra-like systems which are constructed will exist only in software, and would be very inconvenient to try drawing on paper (or at least the set of these is much larger and less explored, so we’re more likely to find something there). It could be that the ortho-semantic operations are much more complex, also, so that it’s not simple to state exactly what it does in any way than reading the algorithm that does it. So, a user of one of these alternate algebra-like systems would probably press buttons corresponding to the ortho-semantic operations in order to shift the representations around and investigate relationships.


Another mostly unrelated idea:

Why is it that we mathematically represent physical laws with equations? Generally speaking, what we’re attempting to document are how states of physical systems evolve in time after some operation occurs; so wouldn’t it make more sense to use a representation like

[state 1] {static collision} [state 2]

—where two states are related by an operation? If we were to develop an algebra-like system around this, every operation would have its own set of ortho-semantic operations (kind of like the different rules that exist if you relate two algebraic expressions by ‘<‘ or ‘>’ instead of ‘=’). Sounds like a lot of work, but it may be that there’s a more general system which can be used to automatically give sets of ortho-operations for particular physical operations (like ‘static collision’ in this example). 

Actually though, I don’t think this would work… —looking at a more concrete example:

[mass: 5, elasticity: 0.1, position: {0,0,0}, velocity: {0,0,0}], 
[mass: 8, elasticity: 0.25, position: {10,0,0}, velocity: {-2,0,0}]

{static collision} 

[mass: 5, elasticity: 0.1, position: {0,0,0}, velocity: {-3,0,0}], 
[mass: 8, elasticity: 0.25, position: {0.234,0.43,0}, velocity: {0.5,0,0}]

It is interesting though that scientific laws aren’t generally in the imperative form, like if I have a physical system in state X and do BLAH to it, Y will be the resulting state; my guess is that we use equivalence relations instead because it’s our only means of ‘doing theory’, by encoding results in equations and then looking for new relations by applying ortho-semantic operations to algebraic representations. However, that probably shapes our view of science quite a bit, in a very Sapir-Whorf manner.

Saturday, May 27, 2017

Reflections on "Toward a Noncomputational Cognitive Neuroscience"

Note on where this came from: I was bored one afternoon and one of my friends posted this paper ("Toward a Noncomputational Cognitive Neuroscience") on Facebook. I have nothing against it, I just wanted to do some writing and analysis since it had been a while, and this happened to show up. I do feel like my tone ended up being overly harsh, and I guess it did annoy me a little, but it also had some interesting ideas and who knows how off I am—so please do check it out on your own if you're curious.

Since the 'computational' is such an exceedingly far-reaching category, I keep an eye out for things which fundamentally can't be included in it. I only know of one subject totally outside its bounds: immediate conscious experience, all the qualia currently present—the subject matter of phenomenology.

Unfortunately, "Toward a Noncomputational Cognitive Neuroscience," doesn't seem to supply any new instances of the non-computational. Its central argument appears to be a straw man (but mine may in part be too, so make sure to check out [0]): it takes overly restrictive definitions of computation and information processing and then demonstrates how its alternate approach does not fall under said restrictive definitions. Yes, contemporary artificial neural nets don’t adjust their connection weights, transfer functions etc., while operating (i.e. after training), and a dynamical systems analysis of a system which does this is qualitatively different from one which does not—but computation has no intrinsic limits that would prevent someone from architecting such a neural net, even using the paper's narrow definition of computation which requires rules to be operating on representations. 

The paper's view on the difference between simplified and realistic nets is most concisely stated here:

"It is the processing of representations that qualifies simplified nets as computational. In realistic nets, however, it is not the representations that are changed; it is the self-organizing process that changes via chemical modulation. Indeed, it no longer makes sense to talk of 'representations.'"

Why would a computation which changes its own rules no longer be a computation? Such computations are at the very heart of computation theory! Additionally, the sense in which the system is no longer operating on representations can only be superficial, since at some level of interpretation, representations are still obviously a component of cognition. The difference is just that they emerge at a higher level, rather than being explicitly defined things which the base system explicitly operates on.

The separate argument about digital computers being classical, non-chaotic dynamical systems also falls flat, in my opinion. The sense in which computers are classical systems is superficial: if I can write a program for it which, at the appropriate level of interpretation, is nonlinear and chaotic—what does it matter if the substrate is classical? If you insist on only modeling the state space of the substrate, rather than something higher level, sure, it’s always classical—but what do you gain by doing that?

The bit on connecting Freud back to the super abstract dynamical systems stuff was a pretty neat idea I thought. Would be interesting to see if there’s a good fit with any ideas of, e.g., William James or Jung.

On the other hand, I suspect the paper's attempt to incorporate the ideas of Derrida et al is part of a flawed justification for considering its notion of the noncomputational to be more significant than it really is.

"Out of that intersection of ‘self’ and ‘other’ the dynamic whole evolves in its spontaneous, unexpectedly bifurcating manner. So the brain does not compute; it permits and supports ‘participation’ between self and other in the evolving whole."

I read that, and many other arguments from the paper, as being circumlocutions avoiding saying, “the system is bottom-up, not top-down.”

"The outside is not represented inside but participates on the inside as a constraint on a self-organizing process."

As far as I can tell, this usage of ‘to participate’ is just refers to something which: 'can’t be modeled top-down', and 'is one among other things involved.' Seems like it's largely, indirectly saying that cognition is emergent—and, erroneously, claiming that you can't create emergent, chaotic systems in classical computers:

"Computation as understood by the tradition is not performed by chaotic systems. Computer computation is not sensitively dependent on initial conditions."

And yet it is capable of executing programs which are.


[0] To be fair though the sense in which it is a straw man is only this: it doesn't say anything significant about computation in general, only about a special restricted Computation which it is interested in (and which I'm sure many other academics are interested in as well). The reason I take issue with it (it addition to the clickbaitiness of 'Noncomputational') is because the paper also comes with the suggestion of a paradigm shift for Cognitive Neuroscience—but this special Computation it's found a negation of isn't sufficient to constitute a paradigm shift. Anyway, considering that, much of my review may itself be a straw man. Oh well.