Archive for category Philosophy

Countable Infinity – Math or Metaphysics?

Are we too willing to accept things on authority – even in math? Proofs of the irrationality of the square root of two and of the Pythagorean theorem can be confirmed by pure deductive logic. Georg Cantor’s (d. 1918) claims on set size and countable infinity seem to me a much less secure sort of knowledge. High school algebra books (e.g., the classic Dolciani) teach 1-to-1 correspondence between the set of natural numbers and the set of even numbers as if it is a demonstrated truth. This does the student a disservice.

Following Cantor’s line of reasoning is simple enough, but it seems to treat infinity as a number, thereby passing from mathematics into philosophy. More accurately, it treats an abstract metaphysical construct as if it were math. Using Cantor’s own style of reasoning, one can just as easily show the natural and even number sets to be no non-corresponding.

Cantor demonstrated a one-to-one correspondence between natural and even numbers by showing their elements can be paired as shown below:

1 <—> 2
2 <—> 4
3 <—> 6

n <—> 2n

This seems a valid demonstration of one-to-one correspondence. It looks like math, but is it? I can with equal validity show the two sets (natural numbers and even numbers) to have a 2-to-1 correspondence. Consider the following pairing. Set 1 on the left is the natural numbers. Set 2 on the right is the even numbers:

1      unpaired
2 <—> 2
3      unpaired
4 <—> 4
5      unpaired

2n -1      unpaired
2n <—> 2n

By removing all the unpaired (odd) elements from the set 1, you can then pair each remaining member of set 1 with each element of set 2. It seems arguable that if a one to one correspondence exists between part of set 1 and all of set 2, the two whole sets cannot support a 1-to-1 correspondence. By inspection, the set of even numbers is included within the set of natural numbers and obviously not coextensive with it. Therefore Cantor’s argument, based solely on correspondence, works only by promoting one concept – the pairing of terms – while suppressing an equally obvious concept, that of inclusion. Cantor indirectly dismisses this argument against set correspondence by allowing that a set and a proper subset of it can be the same size. That allowance is not math; it is metaphysics.

Digging a bit deeper, Cantor’s use of the 1-to-1 concept (often called bijection) is heavy handed. It requires that such correspondence be established by starting with sets having their members placed in increasing order. Then it requires the first members of each set to be paired with one another, and so on. There is nothing particularly natural about this way of doing things. It got Cantor into enough of a logical corner that he had to revise the concepts of cardinality and ordinality with special, problematic definitions.

Gottlob Frege and Bertrand Russell later patched up Cantor’s definitions. The notion of equipollent sets fell out of this work, along with complications still later addressed by von Neumann and Tarski. Finally, it seems to me that Cantor implies – but fails to state outright – that the existence of a simultaneous 2-to-1 correspondence (i.e., group each n and n+1 in set 1 with each 2n in set 2 to get a 1-to-1correspondence between the two sets) does no damage to the claims that 1-to-1correspondence between the two sets makes them equal in size. In other words, Cantor helped himself to an unnaturally restrictive interpretation (i.e., a matter of language, not of math) of 1-to-1 correspondence that favored his agenda. Finally, Cantor slips a broader meaning of equality on us than the strict numerical equality the rest of math. This is a sleight of hand. Further, his usage of the term – and concept of – size requires a special definition.

Cantor’s rule set for the pairing of terms and his special definitions are perfectly valid axioms for mathematical system, but there is nothing within mathematics that justifies these axioms. Believing that the consequences of a system or theory justify its postulates is exactly the same as believing that the usefulness of Euclidean geometry justifies Euclid’s fifth postulate. Euclid knew this wasn’t so, and Proclus tells us Euclid wasn’t alone in that view.

Galileo seems to have had a more grounded sense of the infinite than did Cantor. For Galileo, the concrete concept of mathematical equality does not reconcile with the abstract concept of infinity. Galileo thought concepts like similarity, countability, size, and equality just don’t apply to the infinite. Did the development of calculus create an unwarranted acceptance of infinity as a mathematical entity? Does our understanding that things can approach infinity justify allowing infinities to be measured and compared?

Cantor’s model of infinity is interesting and useful, but it is a shame that’s it’s taught as being a matter of fact, e.g., “infinity comes in infinitely many different sizes – a fact discovered by Georg Cantor” (Science News, Jan 8, 2008).

On countable infinity we might consider WVO Quine’s position that the line between analytic (a priori) and synthetic (about the world) statements is blurry, and that no claim is immune to empirical falsification. In that light I’d argue that the above demonstration of inequality of the sets of natural and even numbers (inclusion of one within the other) trumps the demonstration of equal size by correspondence.

Mathematicians who state the equal-size concept as a fact discovered by Cantor have overstepped the boundaries of their discipline. Galileo regarded the natural-even set problem as a true paradox. I agree. Does Cantor really resolve this paradox or is he merely manipulating language?

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Let’s just fix the trolley

The classic formulation of the trolley-problem thought experiment goes something like this:

A runaway trolley hurtles toward five tied-up people on the main track. You see a lever that controls the switch. Pull it and the trolley switches to a side track, saving the five people, but will kill one person tied up on the side track. Your choices:

  1. Do nothing and let the trolley kill the five on the main track.
  2. Pull the lever, diverting the trolley onto the side track causing it to kill one person.

At this point the Ethics 101 class debates the issue and dives down the rabbit hole of deontology, virtue ethics, and consequentialism. That’s probably what Philippa Foot, who created the problem, expected. At this point engineers probably figure that the ethicists mean cable-cars (below right), not trolleys (streetcars, left), since the cable cars run on steep hills and rely on a single, crude mechanical brake while trolleys tend to stick to flatlands. But I digress.

Many trolley problem variants exist. The first twist usually thrust upon trolley-problem rookies was called “the fat man variant” back in the mid 1970s when it first appeared. I’m not sure what it’s called now.

The same trolley and five people, but you’re on a bridge over the tracks, and you can block it with a very heavy object. You see a very fat man next to you. Your only timely option is to push him over the bridge and onto the track, which will certainly kill him and will certainly save the five. To push or not to push.

Ethicists debate the moral distinction between the two versions, focusing on intentionality, double-effect reasoning etc. Here I leave the trolley problems in the competent hands of said ethicists.

But psychologists and behavioral economists do not. They appropriate the trolley problems as an apparatus for contrasting emotion-based and reason-based cognitive subsystems. At other times it becomes all about the framing effect, one of the countless cognitive biases afflicting the subset of souls having no psych education. This bias is cited as the reason most people fail to see the two trolley problems as morally equivalent.

The degree of epistemological presumptuousness displayed by the behavioral economist here is mind-boggling. (Baby, you don’t know my mind…, as an old Doc Watson song goes.) Just because it’s a thought experiment doesn’t mean it’s immune to the rules of good design of experiments. The fat-man variant is radically different from the original trolley formulation. It is radically different in what the cognizing subject imagines upon hearing/reading the problem statement. The first scenario is at least plausible in the real world, the second isn’t remotely.

First off, pulling the lever is about as binary as it gets: it’s either in position A or position B and any middle choice is excluded outright. One can perhaps imagine a real-world switch sticking in the middle, causing an electrical short, but that possibility is remote from the minds of all but reliability engineers, who, without cracking open MIL-HDBK-217, know the likelihood of that failure mode to be around one per 10 million operations.

Pushing someone, a very heavy someone, over the railing of the bridge is a complex action, introducing all sorts of uncertainty. Of course the bridge has a railing; you’ve never seen one that didn’t. There’s a good chance the fat man’s center of gravity is lower than the top of the railing because it was designed to keep people from toppling over it. That means you can’t merely push him over; you more have to lift him up to the point where his CG is higher than the top of railing. But he’s heavy, not particularly passive, and stronger than you are. You can’t just push him into the railing expecting it to break either. Bridge railings are robust. Experience has told you this for your entire life. You know it even if you know nothing of civil engineering and pedestrian bridge safety codes. And if the term center of gravity (CG) is foreign to you, by age six you have grounded intuitions on the concept, along with moment of inertia and fulcrums.

Assume you believe you can somehow overcome the railing obstacle. Trolleys weigh about 100,000 pounds. The problem statement said the trolley is hurtling toward five people. That sounds like 10 miles per hour at minimum. Your intuitive sense of momentum (mass times velocity) and your intuitive sense of what it takes to decelerate the hurtling mass (Newton’s 2nd law, f = ma) simply don’t line up with the devious psychologist’s claim that the heavy person’s death will save five lives. The experimenter’s saying it – even in a thought experiment – doesn’t make it so, or even make it plausible. Your rational subsystem, whether thinking fast or slow, screams out that the chance of success with this plan is tiny. So you’re very likely to needlessly kill your bridge mate, and then watch five victims get squashed all by yourself.

The test subjects’ failure to see moral equivalence between the two trolley problems speaks to their rationality, not their cognitive bias. They know an absurd hypothetical when they see one. What looks like humanity’s logical ineptitude to so many behavioral economists appears to the engineers as humanity’s cultivated pragmatism and an intuitive grasp of physics, factor-relevance evaluation, and probability.

There’s book smart, and then there’s street smart, or trolley-tracks smart, as it were.

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