The n-Category Café

JA: “Catch 1 would be that Bayes’ rule is a mathematical identity, an unimpeachable king of the hill, but only so long as it reigns in bounds of its own turf, in the chestnut shells of deductive truth.”
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SH: The earlier question in this thread was whether Category Theory (CT) would be helpful to Machine Learning. The learning involved is described as inductive rather than deductive. So is CT a better tool to map the inferences/inductions that children do (perhaps meta-meta reasoning) “routinely and effortlessly”? The next Toby Bartels quote seems to agree with the Gopnik remark that there seems to be an “analogy between learning in children and learning in science”. Can CT augment the creation of an algorithmic learning approach (CT -> human learning categories)?

TB: “In fact, I use inductive reasoning (and abductive reasoning) often when studying deductive reasoning; whenever I draw a conclusion (however tentative) about deductive logic that I haven’t proved but which I nevertheless believe due to trying out some possibilities and seeing analogies with facts that I know.”
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web.mit.edu/cocosci/Papers/devsci07_gopnik_tenenbaum.pdf
“The papers focus on two classes of learning problems: learning causal relations, from observing co-occurrences among events and active interventions; and learning how to organize the world into categories and map word labels onto categories, from observing examples of objects in those categories. Causal learning, category learning and word learning are all problems of induction, in which children form representations of the world’s abstract structure that extend qualitatively beyond the data they observe and that support generalization to new tasks and contexts. While philosophers have long seen inductive inference as a source of great puzzles and paradoxes, children solve these natural problems of induction routinely and effortlessly.

‘Theory theorists’ in cognitive development point to an analogy between learning in children and learning in science. Causal Bayesian networks provide a computational account of a kind of inductive inference that should be familiar from everyday scientific thinking: testing hypotheses about the causal structure underlying a set of variables by observing patterns of correlation and partial correlation among these variables, and by examining the consequences of interventions (or experiments) on these variables.”

TB: But reason is not three separate domains, each of which can say nothing about the others.

Of course not. Already in Aristotle we find him analys/zing reasoning by analogy, example, or paradigm as a mixture of inductive and deductive ingredients, and Peirce — long time passing — would analys/ze analogy and the whole cycle of inquiry into a cable-istic co-operation of inferential strands that involved a con-tri-bution coming from each and every one among the Big 3.

JA wrote: “Catch 2 would be that cause-effect is usually taken to be a 2-adic relation.”
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SH: I think you are comparing apples and oranges. Cause and effect is a general belief about the nature of the physical universe. Suppose you had a lamp with 3 switches for intensity. This analogy will use each setting of the switch for illumination, to either deductive, inductive or abductive reasoning. It wouldn’t matter which setting of the switch was being used, that would be irrelevant, to conclude that in each mode of illumination/thinking, that the underlying physical process was governed by the flow or non-flow of electricity, that’s the cause, which is independent of the mode of illumination/thinking.

I tend to quote authorities because I don’t trust the understanding of various people who have opinions about sundry topics. Thus you have to dispute the authority’s understanding, not mine.
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1.7.5.1 Abductive inference (Charles Sanders Peirce) CONTEXT AS OTHER MINDS by Talmy Givon
“Ever since Plato and Aristotle, empiricists have tended to emphasize induction as the prime mode of obtaining knowledge, while rationalists have tended to emphasize deduction. Echoing Kant’s rejection of both reductionist extremes, Peirce resuscitated Aristotle’s third mode, apagoge (‘reduction’), which he rechristened abduction. This is the mode by arguing by hypothesis, often by analogy. It is thus a pragmatic mode par excellence, dependent as it is on notions of similarity and/or relevance (Hanson 1958; see ch.8).

1.7.5.2 The interpretant as a third term in semiotics As noted earlier above (1.6.9), to the extent of Peirce’s notion of interpretant as the third term in the semiotic equation is distinct from Aristotle’s perfect-mirror mind, it may be interpreted as a stand in for the construed context.”
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Section 7: Causal and Counterfactual Reasoning Chapter 29: Causal Thinking by Lance J. Rips
“Pearl is explict about the fact that an important benefit of causal models over causal networks is that the models deal correctly with counterfactual conditionals - statements of the form “If X had happened, then Y would have happened, “like if Fred had taken the trouble to fix his brakes, he wouldn’t have had an accident. It’s been recognized at least since Goodman (1955) that there’s a close connection between counterfactuals and causation. The truth of many counterfactual conditionals seems to depend on causal laws that dictate the behavior of events. These laws hold not just in our current state of affairs, but also in alternative states that differ from ours but still obey the laws in question.”
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SH: Counterfactuals are often describes as ideas which that the What/If form. So there is a mathematical formalism for QM that agrees with 99.999999999~% accuracy with experimental observations. Yet, there are several Interpretations as to what these observations mean regarding the description of how the physical universe works. Each one of the Interpretations trys to fit an explanation with observed facts. This is done by hypothesizing which facts support the most consistent theory. Now whether the thinking involved at arriving at this hypothesis is deductive, inductive or abductive or a combination of all three, this provides no constraint to the goal of arriving at a causally consistent or reasonable explanation of how the universe works. Your original argument attempted to refute causality as a basic universal view with abductive reasoning, which is a mode of thinking to achieve a hypothesis often involving analogy. It wouldn’t matter if there were 17 different modes of thinking which described how or interpreted which causal relations were universally consistent -> that is the “construed context”.
QM is not yet a universal framework because it hasn’t been reconciled with SR and the human perception of time, which is called the problem of quantum gravity.
QM is still a microcosmic rather than a macrocosmic description although in the more recent era more people have started claiming that QM is universal.

Urk! Am I becoming even more cognitively impaired?! I thought 1.2 x 10^6 was
1,200,000 or 1.2 million? Does your computation use some other number base.

Toby, I see what you mean, 1.2 million is the result of doing a google search without quotes marks for “thinking reed”, from the original Pascal quote since Jon noted that he was a Pascal hacker. It didn’t occur to me that someone would do an unquoted search of _thinking reed_ to support the use of what I will charitably describe as an extended metaphor.

JA: …” and I felt that this was a clue to the trade-offs that create so much tension between diverse cognitive styles in human thinking reeds, whether between reeds or within the same reed.”

A phrase like Pascal’s “a thinking reed” or “a broken reed” are metaphors for humans, but how does that stretch to include a human as “within the same reed.”? That construction is quite illiterate so I originally corrected it very politely, but I realize now that was a mistake.