Do Machine-Learning Machines Learn?

The need for the qualifications (i.e. determinate, non-question-begging) should be obvious. The answer to the present paper’s title that a machine which machine-learns by definition learns, since ‘learn’ appears in ‘machine-learn,’ assumes at the outset that what is called ‘machine learning’ today is real learning—but that’s precisely what’s under question; hence the petitio.

All mathematical models of learning relevant to the present discussion that we are aware of take learning to consist fundamentally in the learning of number-theoretic functions from \(\mathbb {N} \times \mathbb {N} \times \cdots \times \mathbb {N}\) to \(\mathbb {N}\). Even when computational learning was firmly and exclusively rooted in classical recursion theory, and dedicated statistical formalisms were nowhere to be found, the target of learning was a function of this kind; see e.g. (Gold 1965; Putnam 1965), a modern, comprehensive version of which is given in (Jain et al. 1999). We have been surprised to hear that some in our audience aren’t aware of the basic, uncontroversial fact, readily appreciated by consulting the standard textbooks we cite here and below, that machine learning in its many guises takes the target of learning to be number-theoretic functions. A “shortcut” to grasping a priori that all systematic, rigorously described forms of learning in matters and activities computational and mechanistic must be rooted in number-theoretic functions, is to simply note that computer science itself consists in the study and embodiment of number-theoretic functions, defined and ordered in hierarchies (e.g. see Davis and Weyuker 1983). We by the way focus herein on unary functions \(f : \mathbb {N} \mapsto \mathbb {N}\) only for ease of exposition.

Not to be be confused with RL, reinforcement learning, in which real learning, as revealed herein, doesn’t happen.

As many readers will know, Searle’s (1980) Chinese Room Argument (CRA) is intended to show that computing machines can’t understand anything. It’s true that Bringsjord has refined, expanded, and defended CRA (e.g. see Bringsjord 1992, Bringsjord and Noel 2002; Bringsjord 2015), but bringing to bear here this argumentation in support of the present paper’s main claim would instantly demand an enormous amount of additional space. And besides, as we now explain, calling upon this argumentation is unnecessary.

Since at bottom, as noted (see note 2), the target of learning should be taken for generality and rigor to be a number-theoretic function, it’s natural to consider learning in the realm of the formal sciences.

Just as (computer) programs can be correct or incorrect, so too proofs can be correct or incorrect. For more on this, see e.g. (Arkoudas and Bringsjord 2007).

If we regard Turing to have been speaking of modern AI in his famous (Turing, 1950), note then too that his orientation is test-based: he gave here of course the famous ‘Turing Test.’.

In fact, this is why real learning for humans in mathematics is challenging; see e.g. (Moore 1994).

Our assumption here thus specifically invokes connectionist ML. But this causes no loss of generality, as we explain by way the “tour” of ML taken in Sect. 6.1, and the fact that the proof in the Appendix, as explained there, is a general method that will work form any contemporary form of ML.

This is a rough-and-ready extraction from (Jain et al. 1999), and must be sufficient given the space limitations of the present short paper, at least for now. Of course, there are many forms of ML/machine learning in play in AI of today. In Sect. 6.1 we consider different forms of ML in contemporary AI. In Sect. 6.2 we consider different types of “learning” in psychology and allied disciplines.

Lathrop (1996) shows, it might be asserted, that uncomputable functions can be machine-learned. But in his scheme, there is only a probabilistic approximation of real learning, and—in clear tension with (c1\('\))–(c3)—no proof in support of the notion that anything has been learned. The absence of such proofs is specifically called out in the formal deduction given in the Appendix.

A pair of additional works help to further seal our case: (Kearns and Vazirani 1994; Shalev- Shwartz and Ben-David 2014). Study of these texts will reveal that \(\mathcal {RL}\) as per (c1\('\))–(c3) is nowhere to be found.

We of course join epistemological cognoscenti in being aware of Gettier-style cases, but they can be safely left aside here. For the record, Bringsjord claims to have a solution anyway—one that is at least generally in the spirit of Chisholm’s (1966) proposed solution, which involves requiring that the justification in justified-true-belief accounts of knowledge be of a certain sort. For Gettier’s landmark paper, see (Gettier 1963).

Shakespeare himself, or better yet even Ibsen, or better better yet Bellow, couldn’t have invented a story dripping with this much irony—a story in which the machine-learning people persecuted the logicians for building “brittle” systems, and then the persecutors promptly proceeded to blithely build comically brittle systems as their trophies (given to themselves).

In which is by the way cited hypercomputational artificial neural networks.

E.g. even beginning textbooks introducing single-variable differential/integral calculus ask for verification of human learning by asking for proofs. The cornerstone and early-on-introduced concept of a limit is accordingly accompanied by requests to students that they supply proofs in order to confirm that they understand this concept. Thus we e.g. have on p. 67 of (Stewart, 2016) a request that our reader prove that. What machine-learning machine that has learned the function g here can do that?

This is essentially the Short Short Story Game of (Bringsjord 1998), much harder than such Turing-computable games as Checkers, Chess, and Go, which are all at the same easy level of difficulty (EXPTIME).

Outside of the present paper, we have carried out a second analysis that confirms this, by examining learning in AI as characterized in (Russell and Norvig 2009), and invite skeptical readers to carry out their own analysis for this textbook, and indeed for any comprehensive, mainstream textbook. The upshot will be the stark fact that \(\mathcal {RL}\), firmly in place since Euclid as what learning in the formal sciences is, will be utterly absent.

Instead of looking to published attempts to systematically present AI (such as the textbooks upon which we rely herein), one could survey practitioners in AI, and see if their views harmonize with the publications explicitly designed to present all of AI (from a high-altitude perspective). E.g., one could turn to such reports as (Müller and Bostrom 2016), in which the authors report on a specific question, given at a conference that celebrated AI’s “turning 50” (AI@50), which asked for an opinion as to the earliest date (computing) machines would be able to simulate human-level learning. It’s rather interesting that 41% of respondents said this would never happen. It would be interesting to know if, in the context of the attention ML receives these days, the number of these pessimists would be markedly smaller. If so, that may well be because, intuitively, plenty of people harbor suspicions that ML in point of fact hasn’t achieved any human-level real learning.

Luger’s book revolves around a fundamental distinction between what he calls weak problem-solving versus strong problem-solving.

There are a few exceptions. Hummel (2010) has explained that sophisticated and powerful forms of symbolic learning, ones aligned with second-order logic, are superior to associative forms of learning. Additionally, there’s one clear historical exception, but it’s now merely a sliver in psychology (specifically, in psychology of reasoning), and hence presently has insufficient adherents to merit inclusion in the ontology we now proceed to canvass. We refer here to the type of learning over the years of human development and formal education posited by Piaget; e.g. see (Inhelder and Piaget 1958). Piaget’s view, in a barbaric nutshell, is that, given solid academic education, nutrition, and parenting, humans develop the capacity to reason with and even eventually over first-order and modal logic—which means that such humans would develop the capacity to learn in \(\mathcal {RL}\) fashion, in school. Since attacks on Piaget’s view, starting originally with those of Wason and Johnson-Laird (e.g. see Wason and Johnson-Laird 1972), many psychologists have rejected Piaget’s position. For what it’s worth, Bringsjord has defended Piaget; see e.g. (Bringsjord et al. 1998).

We are happy to concede that years of laborious (and tedious?) study of conditioning using appetitive and aversive reinforcement (and such phenomena as inhibitory conditioning, conditioned suppression, higher-order conditioning, conditioned reinforcement, and blocking) has revealed that conditioning can’t be literally reduced to new reflexes, but there is no denying that in conditioning, any new knowledge and representation that takes form falls light years short of \(\mathcal {RL}\).

Note that all occurrences of ‘understanding’ in the itemized list that follows, in keeping with the psychometric operationalization introduced at the outset in order not to rely on the murky concept of understanding, could be invoked here; but doing so would take much space and time, and be quite inelegant.

This conception matches that of an agent in orthodox AI: see the textbooks, e.g. (Luger 2008; Russell and Norvig 2009).