Paradoxes in Sports Science

[Guest guest]

For several years I ran a special column on several lists called " Puzzles &

Paradoxes " (P & Ps) in Sports Science in which I presented apparent paradoxes

for analysis, because I believe that teaching and learning can be enhanced by

the use of such a teaching methodology. Some of these P & Ps are still

archived on the following biomechanics website:

<http://www.sportsci.com/SPORTSCI/JANUARY/archives2.html>

The following website also addresses this subject and is worth reading

through in some detail:The Power of Paradox:

<http://philosophy.wisc.edu/lang/pd/logic1.htm>

<There are at least two kinds of paradoxes. Propositions that logically

contradict themselves (such as the proposition that " Pencils are not

pencils " ) may be called logical paradoxes. Propositions that logically

contradict being known by us (such as the proposition that " It is raining,

but we don't believe it " ) may be called pragmatic paradoxes. Note that

logical paradoxes are paradoxical to everyone, but (some) pragmatic paradoxes

are only paradoxical to certain people. For example, the proposition " It's

raining, but I don't believe it " is paradoxical to me, but not to you. It

would be paradoxical for me to know it, for to do so would require that I

both believe and disbelieve that " It's raining " . However, nothing about this

proposition would make it especially paradoxical for you to know it, if it

were true.

Does the relativism of pragmatic paradoxes mean that we should treat them any

differently from logical ones? Not much. We should, of course, label the

paradoxes differently, but any reasons for disbelieving logical paradoxes are

also be reasons for us to disbelieve any propositions that happen to be

pragmatically paradoxical for us. The avoidance of logical paradoxes is

built-in to most systems of logic. Kant deserved some credit for realizing

that avoidance of pragmatic paradoxes should figure into such systems as

well. In sections three and four, I will suggest how this might be done.

Propositions that are paradoxical all by themselves are oddities that most

people encounter only rarely. More often, we deal with paradoxes that arise

from combinations of beliefs. For example, suppose thought that his

daughter, Mandy, was in school, but then saw her playing in a neighbor's yard

(which is not in school). Neither the proposition " Mandy is in school " , nor

the proposition " Mandy is in the neighbor's yard (which is not in school) " is

paradoxical by itself, but they are logically paradoxical in combination.

Thus, when sees Mandy in the neighbor's yard, he may feel obliged to

reevaluate his beliefs - Are his eyes deceiving him? Or was he wrong to think

that Mandy was in school? In such situations, recognition of paradoxes can be

extremely useful. It is only because has a policy of disbelieving

paradoxes that he can utilize his observation to recognize Mandy's truancy.

Similarly, even if he held no previous belief regarding the whereabouts of

his dog, Sam, 's policy of avoiding paradoxes would allow him use an

observation of Sam in a neighbor's yard to decide that Sam also is not in

school (nor in the house). Here takes advantage of the potential for

paradox in the combination of his beliefs.

The paradoxes that arise in combinations of beliefs are not only most often

useful - they are also the most likely to go unnoticed. Propositions can

combine in complex ways to yield implications. It has taken mathematicians

hundreds of years to figure out how certain propositions are implied by

certain small sets of axioms. Compared to such sets of axioms, our personal

sets of beliefs are tremendously numerous. Imagine, then, how much more

difficult it would be to identify all of the paradoxes and potential

paradoxes in a typical person's full set of beliefs! The grand prospects for

unrecognized paradoxes and potential paradoxes make it plausible that a great

deal of belief modification could result from mere reflection on our current

beliefs, independent of gathering new observations. Let's call this activity

of seeking paradoxes and potential paradoxes in our beliefs " paradox testing " .

What is the status of " paradox testing " relative to empirical testing? In his

Republic, Plato used his famous allegory of the Sun, Line and Cave to

describe the path to knowledge as consisting of four consecutive stages. The

first stage is that of holding beliefs. The second is that of knowing what

those beliefs are. The third is that of knowing that they do not contradict

each other (i.e. the result of paradox testing). The fourth is complete

knowledge of the truth. By this account, paradox testing comes before the

test of truth. In other words, non-paradoxical false beliefs may pass to the

third stage, but paradoxical true beliefs (if there are any) can only pass to

the second. Thus, if the truth were paradoxical, then those who follow Plato

would sooner choose to believe a non-paradoxical fantasy than the actual

(paradoxical) truth.

Of course, Plato probably expected truth to be non-paradoxical (as most of us

do), in which case he did not see himself as making a choice between true vs.

non-paradoxical belief. Rather, I suspect he intended to advocate a path

that would lead to beliefs that were both true and non-paradoxical

simultaneously. Nevertheless, it is telling that Plato's method employs

paradox testing prior to any other, including empirical testing. Perhaps this

was because Plato trusted his sense of logic more than he trusted any of his

other senses.

Modern scientists have also been known to set paradox testing above empirical

testing. For example, when observing a magic show, a scientist may observe

what appears to be a direct violation of physical laws. If he/she took

empirical evidence as the highest authority in guiding his/her choice of what

to believe in such situations, then he/she would believe that physical laws

had changed, that the previous experiments confirming the old laws had now

become obsolete. But we do not expect many scientists to react this way. Even

when the subject matter is not a magic show, even when it is an experiment in

their own controlled laboratories, we expect scientists to treat unexpected

observations with suspicion until working out a comprehensive non-paradoxical

body of theory to explain them.

Although paradox testing may have limited applicability regarding selection

of beliefs about the external world, scientists treat it as the ultimate

authority so far as its scope extends. Each scientist may hold a broader

variety of beliefs privately, but, in their professional circles, paradoxical

claims are strictly taboo. Scientists will not entertain paradoxical

propositions at scientific symposiums, nor teach them in science courses, nor

publish them in scientific journals. Paradox testing is so sacred an

institution of science, that certain scientists in most every scientific

field have devoted themselves primarily to its application. These scientists

are called " theorists " , and some of the most highly respected scientists have

been counted among their ranks....

If the Truth were Paradoxical

If it is true, as I have claimed, that scientists give paradox testing the

highest authority so far as its scope

extends, then there are four possibilities:

1. The truth is not paradoxical,

2. Paradox avoidance, rather than truth, is the goal of the scientific quest

for knowledge,

3. The epistemic practices of scientists are not aimed at their goal (and

perhaps ought, therefore, to be revised), or

4. Their quest for knowledge has multiple goals, one of which is truth, but

another of which is to develop non-paradoxical systems of belief.

If we can rule out the third possibility, then paradox testing is a valid

tool for science. That would yield new hope for foundationalism, the attempt

(championed by Descartes) to develop justification for our beliefs by

building on a core of non-controversial beliefs. It is said that such

projects never get off the ground, since we infer predictions, retrodictions

and generalizations from present observations plus universal laws about the

external world, but to infer the laws we would first have to infer such laws.

Paradox testing offers the following alternative: instead of inferring all

universal laws from observations, we might justify belief in some of them on

the grounds that their denial would be paradoxical. If the first possibly

(above) is the case, then the resulting beliefs would be true. If not, they

would still be " necessary for paradox-avoidance " and this would be sufficient

to achieve knowledge so long as the third possibility is not the case (since,

in the cases of the second or fourth possibilities, knowledge would not

always entail true belief).

Therefore, let us briefly imagine what would happen if scientists were to

discover good reason to believe that some paradoxes are true. I expect some

scientists would stop using paradox testing to guide their selection of

beliefs. However, I would also expect other scientists to continue the search

for non-paradoxical beliefs. That is, I would expect the scientific community

to split into two related families: one family seeking the truth about the

external world and another seeking non-paradoxical sets of beliefs about the

external world. I would expect both endeavors to be recognized by both

families as having merit (although of different kinds).

The merit of seeking truth need not be explained here. Suffice it to say that

anyone who knows many scientists knows that at least some of them care about

truth for truth's sake. However, I do need to explain why scientists might

see merit in paradox-avoidance independent of truth-seeking (in the normal

sense). My answer is that I know some scientists who perceive merit in

empowering people to make intelligent decisions. One of their scientific

goals is to produce documents that will inspire technology or policymaking.

In that endeavor, paradox testing by itself can be sufficient.

For example, imagine that a choice between two prospective policies were to

arise hanging on whether a certain proposition is true or false. Further,

imagine that, in our hypothetical paradoxical world, the proposition happens

to be BOTH true AND false (!). Finally, imagine that a scientist of the

paradox-avoidance family discovers that the proposition is paradoxical, but

fails to discover that the proposition also happens to be true.

Even in this strange world, the decision-maker could feel secure in choosing

the policy advocated by the anti-paradox scientist (i.e. the one associated

with the proposition being false). Granted, a scientist of the truth family

might later point out that the proposition was also true, but this would in

no way diminish the quality of the decision - the choice would remain the most

rational of the two options (since they'd happen to be equally rational).

Furthermore, beyond sufficiency, there may be cases in which paradox testing

is the only kind of testing that's feasible. Then it would certainly be

better to implement paradox testing than no testing at all!

Thus I offer two separate arguments to accept paradox testing as a legitimate

tool of science (please pick the one most appropriate to you):

1. If you believe that the truth about the external world is not

paradoxical, then I argue on the grounds that paradox testing helps us

identify truth.

2. If you believe that the [real] truth about the external world is

paradoxical, then I argue on the grounds that paradox testing is nevertheless

useful.

Hopefully, regardless of your philosophy of science, you will find at least

one of these arguments compelling...

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Dr Mel C Siff

Denver, USA

Supertraining/