Are Paradigms Incommensurable}
57
probability of Bob's death given that Cathy pulls her switch is 1/2. This is so
since if Cathy pulls her switch and Allen does not, the probability of death is
1/2, and if Cathy pulls her switch and Allen does, then the probability of
death is 1/2. Hence P(Bt.\A,., Cr) = P(Bt\C,~). Clause (Hi) is also met. We
have shown that the probability of B, given A,- and C,» is 1/2, and thus equal
to the probability of B, and A,-, which is also 1/2. Hence P(B,.\AV, C,-)
^ (Bt I A,-). From this it can be seen by the definition of 'spurious cause' that
At. is not a genuine cause of Bt since there exists an event C,», earlier than At.,
which 'screens off At. from Bt. This result, however, runs contrary to our
assumption that At. preempts Ct~ in causing Bt.
DOUGLAS EHRING
Southern Methodist University, Dallas
ARE PARADIGMS INCOMMENSURABLE?*
Thomas Kuhn [1970] and others, including Paul Feyerabend [1975] and
Barry Barnes [1982], have argued that there can be no comparison of
competing paradigms based solely on experimental evidence. 'There is no
appropriate scale available with which to weigh the merits of alternative
paradigms: they are incommensurable' [Barnes [1982], p. 65]. Briefly stated,
the argument is as follows. There can be no neutral observation language.
All observation terms are theory laden and thus we cannot compare
experimental results because in different paradigms terms describing
experimental results have different meanings, even when the words used are
the same.1 An example would be the term 'mass' which in Newtonian
* This material is based upon work supported by the National Science Foundation under
Grant No. SES-8204074. Any opinions, findings, conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of
the National Science Foundation.
1
Kuhn offers the following statement on incommensurability. 'These examples point to the
third and most fundamental aspect of the incommensurability of competing paradigms. In a
sense that I am unable to explicate further, the proponents of competing paradigms practice
their trades in different worlds. . . . Equally it is why, before they can hope to communicate
fully, one group or the other must experience the conversion that we have been calling a
paradigm shift. Just because it is a transition between incommensurables, the transition
between competing paradigms cannot be made a step at a time, forced by logic and neutral
experience' ([1970], p. 150). 'This need to change the meaning of established and familiar
concepts is central to the revolutionary impact of Einstein's theory' ([1970], p. 102). Kuhn has
recently modified his view. In an address to the Philosophy of Science Association (Oct. 1982)
he spoke of 'local incommensurability', which I take to mean the difficulty of context
dependent translation Feyerabend is even harder to pin down. At times he states that theories
are incommensurable only if they are not interpreted in an 'independent observation
language' ([1975], p. 274). Elsewhere he seems to deny the possibility of such a language.
'Adopting the point of view of relativity we find that the experiments, which of course mill tuna
be described in relativistic terms, (italics in original) are relevant to the theory, and we also find
that they support the theory. Adopting classical mechanics we again find that the experiments
which are now described in the very different terms of classical physics are relevant, but we also
find that they undermine classical mechanics' ([1975], p- 282).
58 Allan Franklin
mechanics is a constant, while in Einstein's special theory of relativity it
depends on velocity. I believe this view is incorrect.
In this note I will demonstrate that an experiment, described in
procedural, theory-neutral (between the two competing paradigms or
theories) terms, gives different theory-neutral results when interpreted in
the two alternative paradigms. Thus a measurement of the quantity derived
will unambiguously distinguish between the two. I will take as an example
one of Kuhn's own exemplars, the difference between Newtonian and
Einsteinian mechanics. The case in point can be loosely described as the
scattering of equal 'mass' objects.
The experimental procedure is as follows. Consider a class of objects, let
us say billiard balls.1 The objects are examined pairwise by placing a
compressed spring between them. The spring is allowed to expand freely
and the velocities of the two objects measured. Because we restrict ourselves
to a single frame of reference in the laboratory the measurement is theoryneutral. We then select two balls whose velocities are equal. Of course, a
Newtonian would interpret this as giving two objects for which the
Newtonian masses MN are both equal and constant, while an Einsteinian
would say that the relativistic masses MR = MoR y/i — V2/C2 are equal and
thus that the rest masses MoR are equal. This is agreed, but the point is that
the procedure itself is theory-neutral. One of the objects is placed at rest in
the laboratory and the other given a velocity V± (again theory-neutral) and
the particles allowed to scatter off each other. Care is taken to make the
collision elastic, i.e. no energy of any sort is emitted. The final velocities of
the objects V2 and V3, respectively, are measured as is the angle between
these velocities. It is again true that the assignment of momenta and energies
to the particles will be different but the measurement of the angle does not
depend on these assignments. As is well-known, and demonstrated below,
the predicted value of this angle differs in Newtonian and relativistic
mechanics. For a Newtonian 6N = 90.0, while for an Einsteinian 6R < 900. In
the calculation below we shall use units such that C = 1 (the velocity of
light). We shall also assume for calculational convenience that the two
outgoing velocities are equal. This apparent loss of generality is unimportant, although the proof can be generalised.2 The fact that any single
experiment can distinguish between the two paradigms is enough to show
that they are commensurable.
1
The analysis also seems to apply to proton-proton or electron-electron scattering. An antirealist might object, however, that protons and electrons are theoretical entities whose
definition includes mass. Thus the two views of the experiments would be incommensurable
because they are talking about different entities. The advantage of billiard balls is that they
can be defined ostensively. You can hold one in your hand or point to it.
2
Dr Michael Redhead has graciously provided such a general proof. Let 0* and <j>* be the
scattering angles in the centre of mass system (Note: 8* + (j>* = 180°) and let 0, 0 be the angles
in the laboratory system. Then tan 0 = ^/2/y + 1 tan^0# where y = I / ^ / I —Kf/c2, and
similarly for $. Then it can be shown quite easily that tan ( 6 + 0 ) = (2^/y+i /y — 1 )/sin 0* so
that except for 0* = o" tan (0+(j>) < 00 or 0+tf> < 90*.
Are Paradigms Incommensurable? 59
Let us first consider the experiment from a Newtonian point of view. Let
Pifi, P2s> P3N be the initial and final momenta and E1N, E2N, E3N be the
initial and final energies. We further note that in this case EiN = PfNlzMN.
We now apply the Newtonian of conservation of momentum.
PIN
=
P
This gives cos 9N = (PiN-P22N-PlN)l2P2NP3N
(1)
Applying Newtonian conservation of energy we have
^ l \ = &2N + #3JV
or
P21NlzMN = P22Nl2MN +
PlNl2MN.
This gives P2N-P22N-PlN = o (2).
Combining (1) and (2) we get the result
cos 6N = o or 0N = 90*, because P2w ano< P3N ^ °Let us now consider the experiment interpreted in relativistic terms.
Using relativistic conservation of momentum we have Pm = P2R + P3R and
we obtain by a calculation identical to that above
(PJR-P22R-PlR)l2P2RP3R.
COS GR =
We may rewrite this using relativistic energy and momentum as
cos 0* = (Elu-Mln-Elx
+ MZn-Eln +
2
cos 0R = GE?« - E\R - E
MUlzP^P^
2
R
+M
0R)lzP2RP3R
(3)
Using relativistic conservation of energy we have
E1R+M0R = E2R + E3R.
(4)
Squaring both sides gives
E\R + zEiMon+MlR = E\R + zE2RE3R+E\R
(S)
Rewriting we have
E\R-E22R-E23R+M20R
= zE2RE3R-zExM0R
(6)
We can now show that the expressions given in Eq. (6), which are the
numerator in the expression for cos 9R (Eq. (3)) are greater than zero and this
cos 6R > o or 6R < 90°.
Consider zE2RE3R — zElRM0R. We now consider the case that E2R = E3R
= (E1R + M0R)lz from Eq. (4).
60 Allan Franklin
We now have
2(.(ElR
=
+M0R)l2)2-2E1RM0R
E\Rl2+ElRMOR+M2ORl2-2ElRMOR
= E21Rl2-E1RM0R+MlRl2
=
«ElR-M
By the condition of our experiment ElR > M and thus this expression is
greater than o.
Thus 2E2RE3R — 2ElRM0R = ((ElR — M0R)ly/2)2 in this case but 2E2RE3R
— 2E1RM0R = E\R — E\R—E\R + M%R and must be greater than o and hence
cos 0R > o and 0R < go", because P2N = P3R # o.
We have thus shown that in this procedurally defined, theory-neutral,
experiment the predicted results when interpreted within the two paradigms give unambiguous and different results i.e. 8N = 90°, 6R < 90°. Thus a
measurement of the angle between the velocities of the two outgoing objects
will clearly distinguish between the two paradigms and the paradigms are
commensurable. The fact that two paradigms are commensurable does not
require a scientist to accept the one consistent with the experimental results,
namely relativity. A clever classical physicist might very well accommodate
the results by postulating new mechanical effects depending on motion
through the ether.1 These two issues of commensurability and theory choice
are often conflated, as exemplified in the statement by Barnes cited earlier,
but it is useful to keep them separate and distinct.
ALLAN FRANKLIN
University of Colorado
REFERENCES
BARNES, B. [1982]: T. S. Kuhn and Social Science. Macmillan.
FEYERABEND, P. [1975]: Against Method. New Left Books.
KUHN, T. S. [1970]: The Structure of Scientific Revolutions. Chicago University Press, 2nd ed.
EINSTEIN AND SPECIAL RELATIVITY: WHO WROTE THE ADDED
FOOTNOTES?
A recent note in this Journal by Cullwick [1981] attributes a certain footnote
in the English translation of Einstein's original paper on special relativity
(published in Lorentz et al. [1923]) to Sommerfeld. A recent book by Miller
[1981] attributes to Sommerfeld all those footnotes that appear in the
English translation but not in the original paper.
1
I am grateful to the anonymous referee for making this point.