The Beginning of Infinity: Explanations That Transform the World (55 page)

In my view this is a scientific experiment in form only. The substance of scientific theories is explanation, and explanation
of errors
constitutes most of the content of the design of any non-trivial scientific experiment.

As the above example illustrates, a generic feature of experimentation is that the bigger the errors you make, either in the numbers or in your naming and interpretation of the measured quantities, the more exciting the results are,
if true
. So, without powerful techniques of error-detection and -correction – which depend on explanatory theories – this gives rise to an instability where false results drown out the true. In the ‘hard sciences’ – which usually do good science – false results due to all sorts of errors are nevertheless common. But they are corrected when their explanations are criticized and tested. That cannot happen in explanationless science.

Consequently, as soon as scientists allow themselves to stop demanding
good explanations and consider only whether a prediction is accurate or inaccurate, they are liable to make fools of themselves. This is the means by which a succession of eminent physicists over the decades have been fooled by conjurers into believing that various conjuring tricks have been done by ‘paranormal’ means.

Bad philosophy cannot easily be countered by good philosophy – argument and explanation – because it holds itself immune. But it can be countered by
progress
. People want to understand the world, no matter how loudly they may deny that. And progress makes bad philosophy harder to believe. That is not a matter of refutation by logic or experience, but of explanation. If Mach were alive today I expect he would have accepted the existence of atoms once he saw them through a microscope, behaving according to atomic theory. As a matter of logic, it would still be open to him to say, ‘I’m not seeing atoms, I’m only seeing a video monitor. And I’m only seeing that theory’s predictions
about me
, not about atoms, come true.’ But the fact that that is a general-purpose bad explanation would be borne in upon him. It would also be open to him to say, ‘Very well, atoms do exist, but electrons do not.’ But he might well tire of that game if a better one seems to be available – that is to say, if rapid progress is made. And then he would soon realize that it is not a game.

Bad philosophy is philosophy that denies the possibility, desirability or existence of progress. And progress is the only effective way of opposing bad philosophy. If progress cannot continue indefinitely, bad philosophy will inevitably come again into the ascendancy – for it will be true.

Bad philosophy
   Philosophy that actively prevents the growth of knowledge.

Interpretation
   The explanatory part of a scientific theory, supposedly distinct from its predictive or instrumental part.

Copenhagen interpretation
   Niels Bohr’s combination of instrumentalism, anthropocentrism and studied ambiguity, used to avoid understanding quantum theory as being about reality.

Positivism
   The bad philosophy that everything not ‘derived from observation’ should be eliminated from science.

Logical positivism
   The bad philosophy that statements not verifiable by observation are meaningless.

– The rejection of bad philosophy.

Before the Enlightenment, bad philosophy was the rule and good philosophy the rare exception. With the Enlightenment came much more good philosophy, but bad philosophy became much worse, with the descent from empiricism (merely false) to positivism, logical positivism, instrumentalism, Wittgenstein, linguistic philosophy, and the ‘postmodernist’ and related movements.

In science, the main impact of bad philosophy has been through the idea of separating a scientific theory into (explanationless) predictions and (arbitrary) interpretation. This has helped to legitimize dehumanizing explanations of human thought and behaviour. In quantum theory, bad philosophy manifested itself mainly as the Copenhagen interpretation and its many variants, and as the ‘shut-up-and-calculate’ interpretation. These appealed to doctrines such as logical positivism to justify systematic equivocation and to immunize themselves from criticism.

In March 1792 George Washington exercised the first presidential veto in the history of the United States of America. Unless you already know what he and Congress were quarrelling about, I doubt that you will be able to guess, yet the issue remains controversial to this day. With hindsight, one may even perceive a certain inevitability in it, for, as I shall explain, it is rooted in a far-reaching misconception about the nature of human choice, which is still prevalent.

On the face of it, the issue seems no more than a technicality:
in the US House of Representatives, how many seats should each state be allotted?
This is known as the
apportionment problem
, because the US Constitution requires seats to be ‘apportioned among the several States . . . according to their respective Numbers [i.e. their populations]’. So, if your state contained 1 per cent of the US population, it would be entitled to 1 per cent of the seats in the House. This was intended to implement the principle of
representative government
– that the legislature should represent the people. It was, after all, about the House of Representatives. (The US Senate, in contrast, represents the
states
of the Union, and hence each state, regardless of population, has two senators.)

At present there are 435 seats in the House of Representatives; so, if 1 per cent of the US population did live in your state, then by strict proportionality the number of representatives to which it would be entitled – known as its
quota
– would be 4.35. When the quotas are not whole numbers, which of course they hardly ever are, they have to be rounded somehow. The method of rounding is known as an
apportionment rule
. The Constitution did not specify an apportionment rule; it left such details to Congress, and that is where the centuries of controversy began.

An apportionment rule is said to ‘stay within the quota’ if the number of seats that it allocates to each state never differs from the state’s quota by as much as a whole seat. For instance, if a state’s quota is 4.35 seats, then to ‘stay within the quota’ a rule must assign that state either four seats or five. It may take all sorts of information into account in choosing between four and five, but if it is capable of assigning any other number it is said to ‘violate quota’.

When one first hears of the apportionment problem, compromises that seem to solve it at a stroke spring easily to mind. Everyone asks, ‘Why couldn’t they just . . . ?’ Here is what I asked: Why couldn’t they just round each state’s quota to the nearest whole number? Under that rule, a quota of 4.35 seats would be rounded down to four; 4.6 seats would be rounded up to five. It seemed to me that, since this sort of rounding can never add or subtract more than
half
a seat, it would keep each state within half a seat of its quota, thus ‘staying within the quota’ with room to spare.

I was wrong: my rule violates quota. This is easy to demonstrate by applying it to an imaginary House of Representatives with ten seats, in a nation of four states. Suppose that one of the states has just under 85 per cent of the total population, and the other three have just over 5 per cent each. The large state therefore has a quota of just under 8.5, which my rule rounds down to eight. Each of the three small states has a quota of just over half a seat, which my rule rounds up to one. But now we have allocated eleven seats, not ten. In itself that hardly matters: the nation merely has one more legislator to feed than planned. The real problem is that this apportionment is no longer representative: 85 per cent of eleven is not 8.5 but 9.35. So the large state, with only eight seats, is in fact short of its quota by well over one seat. My rule under-represents 85 per cent of the population. Because we
intended
to allocate ten seats, the exact quotas necessarily add up to ten; but the rounded ones add up to eleven. And if there are going to be eleven seats in the House, the principle of representative government – and the Constitution – requires each state to receive its fair share of those, not of the ten that we merely intended.

Again, many ‘why don’t they just . . . ?’ ideas spring to mind. Why don’t they just create three additional seats and give them to the large state, thus bringing the allocation within the quota? (Curious readers
may check that no fewer than three additional seats are needed to achieve this.) Alternatively, why don’t they just transfer a seat from one of the small states to the large state? Perhaps it should be from the state with the smallest population, so as to disadvantage as few people as possible. That would not only bring all the allocations within the quota, but also restore the number of seats to the originally intended ten.

Such strategies are known as
reallocation schemes
. They are indeed capable of staying within the quota. So, what is wrong with them? In the jargon of the subject, the answer is
apportionment paradoxes
– or, in ordinary language,
unfairness
and
irrationality
.

For example, the last reallocation scheme that I described is unfair by being biased against the inhabitants of the least populous state. They bear the whole cost of correcting the rounding errors. On this occasion their representation has been rounded down to zero. Yet, in the sense of minimizing the deviation from the quotas, the apportionment is almost perfectly fair: previously, 85 per cent of the population were well outside the quota, and now all are within it and 95 per cent are at the closest whole numbers to their quotas. It is true that 5 per cent now have no representatives – so they will not be able to vote in congressional elections at all – but that still leaves them within the quota, and indeed only slightly further from their exact quota than they were. (The numbers zero and one are almost equidistant from the quota of just over one half.) Nevertheless, because those 5 per cent have been completely disenfranchised, most advocates of representative government would regard this outcome as much less representative than it was before.

That must mean that the ‘minimum total deviation from quota’ is not the right measure of representativeness. But what is the right measure? What is the right trade-off between being slightly unfair to many people and very unfair to a few people? The Founding Fathers were aware that different conceptions of fairness, or representativeness, could conflict. For example, one of their justifications for democracy was that government was not legitimate unless everyone who was subject to the law had a representative, of equal power, among the lawmakers. This was expressed in their slogan ‘No taxation without representation’. Another of their aspirations was to abolish
privilege
:
they wanted the system of government to have no built-in bias. Hence the requirement of proportional allocation. Since these two aspirations can conflict, the Constitution contains a clause that explicitly adjudicates between them: ‘Each State shall have at least one Representative.’ This favours the principle of representative government in the no-taxation-without-representation sense over the same principle in the abolish-privilege sense.

Another concept that frequently appeared in the Founding Fathers’ arguments for representative government was ‘the will of the people’. Governments are supposed to enact it. But that is a source of further inconsistencies. For in elections, only the will of
voters
counts, and not all of ‘the people’ are voters. At the time, voters were a fairly small minority: only free male citizens over the age of twenty-one. To address this point, the ‘Numbers’ referred to in the Constitution constituted the whole population of a state, including non-voters such as women, children, immigrants and slaves. In this way the Constitution attempted to treat the
population
equally by treating
voters
unequally.

So voters in states with a higher proportion of non-voters were allocated more representatives per capita. This had the perverse effect that in the states where the voters were already the most privileged
within
the state (i.e. where they were an exceptionally small minority), they now received an additional privilege relative to voters in other states: they were allocated more representation in Congress. This became a hot political issue in regard to slave-owners. Why should slave-owning states be allocated more political clout in proportion to how many slaves they had? To reduce this effect, a compromise was reached whereby a slave counted as three-fifths of a person for the purpose of apportioning seats in the House. But, even so, three-fifths of an injustice was still considered an injustice by many.
*
The same controversy exists today in regard to illegal immigrants, who also count as part of the population for apportionment purposes. So
states with large numbers of illegal immigrants receive extra seats in Congress, while other states correspondingly lose out.

Following the first US census, in 1790, notwithstanding the new Constitution’s requirement of proportionality, seats in the House of Representatives were apportioned under a rule that violated quota. Proposed by the future president Thomas Jefferson, this rule also favoured states with higher populations, giving them more representatives per capita. So Congress voted to scrap it and substitute a rule proposed by Jefferson’s arch-rival Alexander Hamilton, which is guaranteed to give a result that stays within quota as well as having no obvious bias between states.

That was the change that President Washington vetoed. The reason he gave was simply that it involved reallocation: he considered all reallocation schemes unconstitutional, because he interpreted the term ‘apportioned’ as meaning
divided
by a suitable numerical divisor – and then rounded, but nothing else. Inevitably, some suspected that his real reason was that he, like Jefferson, came from the most populous state, Virginia, which would have lost out under Hamilton’s rule.