Do you agree that from scientific discoveries and theories we can infer the nature of empirical reality, at least insofar as determining if an entity constitutes (some part of) empirical reality, or concluding that the essential terms of scientific theories refer to something objectively existing? This is just the thesis of scientific realism. “Scientific realism is a realism about whatever is described by our best scientific theories”. http://plato.stanford.edu/entries/scientific-realism/
E=mc2, F=ma, F=k(q1q2)/d2 (Coulomb’s law), ∆S ≥ 0 (second law of thermodynamics), Schrodinger’s equation, the laws of conserved quantities, etc., etc. These are the sorts of entities or facts that physicists discover about empirical reality. Obviously these are mathematical relations, i.e., relations between quantities. Energy is a quantity whose value is the product of two other quantities--the mass of a system multiplied by the speed of light squared. Granted, all the particular mathematical relations or laws just noted may be inexact approximations, their applicability limited to a particular domain or specific conditions. Nevertheless, we can hardly imagine the discovery of more exacting or truer statements about the nature and operation of empirical reality than relations between quantities.
These quantities and relations seem to possess all the credentials of objective existence, first and foremost by the fact that they are discovered rather than invented. We can only assume that energy, momentum, angular momentum, charge (and mass insofar as it is considered to be energy) were conserved quantities before humans were on the scene to conceive these rules and make such calculations, and we can assume these will be conserved quantities in a closed system until the crack of doom. We can say without shame that what makes the written representations of these relations or laws true statements is the same as what makes any other true statement true: they are referential of objective reality--in this case, referential of objectively existing relations between quantities.
Perhaps one wishes to claim that physicists discover other kinds of stuff in addition to mathematical relations, such as, say, the discovery of the electron. Few people would quibble with that. I would note that, even so, everything we know about electrons pertains to their mathematical (measurable) nature and their relations with other quantities or mathematical relations. J. J. Thomson’s cathode ray experiments did not involve the discovery of anything of a non-mathematical nature about empirical reality: he deduced the particulate aspect of the rays (which had been speculated), that the particles have a negative charge, and he arrived at estimates of the size of charge and mass by way of calculations and deduction premised on the degree the rays were bent by electrical currents of differing strengths. The discovery of the electron was ultimately the discovery of a quantity or a set of quantities and mathematical relations.
Structuralism or structural realism is a species of scientific realism that recommends that the mathematical relations described in physics reveal or represent abstract structures as the objective constituents of reality. (I tried to express both the ontological and epistemological versions of the thesis in that sentence, perhaps not successfully.) Philosophers, mathematicians and physicists at least as far back as Henri Poincare, Arthur Eddington, Ernst Cassirer, Rudolf Carnap and Bertrand Russell have expressed various rudimentary notions of structuralism, but in a highly influential paper in 1989, philosopher John Worrall resurrected and developed the thesis as a solution to some of the objections to standard scientific realism, namely the perceived discontinuity of scientific theories and their posits from one theory to the next. According to Worrall (all emphasis his):
The rule in the history of physics seems to be that, whenever a theory replaces a predecessor, which has however itself enjoyed genuine predictive success, the ‘correspondence principle’ applies. This requires the mathematical equations of the old theory to reemerge as limiting cases of the mathematical equations of the new. [. . .] I can see no clear sense in which an action-at-a-distance force of gravity is a “limiting case” of, or “approximates” a space-time curvature. Or in which the ‘theoretical mechanisms’ of action-at-a-distance gravitational theory are “carried over” into general relativity theory. Yet Einstein’s equations undeniably go over to Newton’s in certain limiting special cases. In this sense, there is “approximate continuity” of structure in this case.
[. . .]
On the structural realist view what Newton really discovered are the relationships between phenomena expressed in the mathematical equations of his theory. http://joelvelasco.net/teaching/3330/Worrall 1989 Structural Realism .pdf
To the best of my reckoning, the “relationships between phenomena expressed in the mathematical equations” are just relations between quantities. But if anyone wishes to reify the term “phenomena” in that sentence in some other way, be my guest.
Thus, taking my cue from some of the characteristic claims of scientific realism--
“The entities described by the scientific theory exist objectively and mind-independently.” https://en.wikipedia.org/wiki/Scientific_realism “The central terms of the best current theories are genuinely referential.” http://www-personal.umd.umich.edu/~delittle/Encyclopedia entries/scientific realism.htm
--I wish to state an argument such as:
P1: All central terms of fundamental scientific laws are genuinely referential.
P2: All central terms of fundamental scientific laws are quantities (/mathematical relations).
C: Therefore, some quantities (/mathematical relations) are genuinely referential.
(AAI-3)
or
P1: All entities (/structures) discovered by physicists using the scientific method are objectively existing.
P2: Some mathematical relations are entities (/structures) discovered by physicists using the scientific method.
C: Therefore, some mathematical relations are objectively existing.
(AII-1)
Of course, the above arguments are just my own redneck way of going about the Quine-Putnam Indispensability Argument, which professor Colyvan renders thus:
(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
(P2) Mathematical entities are indispensable to our best scientific theories.
(C) We ought to have ontological commitment to mathematical entities
http://plato.stanford.edu/entries/mathphil-indis/
Colyvan contends that it is the best argument for mathematical realism (I agree), and notes that the argument throws down the gauntlet to those who profess anti-realism, as it puts nominalists in the position of needing to show where the argument goes wrong. I, too, would like to know why one shouldn’t deduce mathematical realism from the premise of scientific realism. What are the reasons or motivations from resisting such a conclusion?
Of course, some people apparently disagree with the thesis of scientific realism, and thus reject P1 in all the arguments above. In which case, I ask: Why shouldn’t we assume that the entities or relations described by the best theories of physics exist objectively? Why shouldn’t we assume that the essential terms of the scientific laws discovered by physicists are genuinely referential? What is a better way for determining the nature of empirical reality than by inference from the discoveries and theories of science?
E=mc2, F=ma, F=k(q1q2)/d2 (Coulomb’s law), ∆S ≥ 0 (second law of thermodynamics), Schrodinger’s equation, the laws of conserved quantities, etc., etc. These are the sorts of entities or facts that physicists discover about empirical reality. Obviously these are mathematical relations, i.e., relations between quantities. Energy is a quantity whose value is the product of two other quantities--the mass of a system multiplied by the speed of light squared. Granted, all the particular mathematical relations or laws just noted may be inexact approximations, their applicability limited to a particular domain or specific conditions. Nevertheless, we can hardly imagine the discovery of more exacting or truer statements about the nature and operation of empirical reality than relations between quantities.
These quantities and relations seem to possess all the credentials of objective existence, first and foremost by the fact that they are discovered rather than invented. We can only assume that energy, momentum, angular momentum, charge (and mass insofar as it is considered to be energy) were conserved quantities before humans were on the scene to conceive these rules and make such calculations, and we can assume these will be conserved quantities in a closed system until the crack of doom. We can say without shame that what makes the written representations of these relations or laws true statements is the same as what makes any other true statement true: they are referential of objective reality--in this case, referential of objectively existing relations between quantities.
Perhaps one wishes to claim that physicists discover other kinds of stuff in addition to mathematical relations, such as, say, the discovery of the electron. Few people would quibble with that. I would note that, even so, everything we know about electrons pertains to their mathematical (measurable) nature and their relations with other quantities or mathematical relations. J. J. Thomson’s cathode ray experiments did not involve the discovery of anything of a non-mathematical nature about empirical reality: he deduced the particulate aspect of the rays (which had been speculated), that the particles have a negative charge, and he arrived at estimates of the size of charge and mass by way of calculations and deduction premised on the degree the rays were bent by electrical currents of differing strengths. The discovery of the electron was ultimately the discovery of a quantity or a set of quantities and mathematical relations.
Structuralism or structural realism is a species of scientific realism that recommends that the mathematical relations described in physics reveal or represent abstract structures as the objective constituents of reality. (I tried to express both the ontological and epistemological versions of the thesis in that sentence, perhaps not successfully.) Philosophers, mathematicians and physicists at least as far back as Henri Poincare, Arthur Eddington, Ernst Cassirer, Rudolf Carnap and Bertrand Russell have expressed various rudimentary notions of structuralism, but in a highly influential paper in 1989, philosopher John Worrall resurrected and developed the thesis as a solution to some of the objections to standard scientific realism, namely the perceived discontinuity of scientific theories and their posits from one theory to the next. According to Worrall (all emphasis his):
The rule in the history of physics seems to be that, whenever a theory replaces a predecessor, which has however itself enjoyed genuine predictive success, the ‘correspondence principle’ applies. This requires the mathematical equations of the old theory to reemerge as limiting cases of the mathematical equations of the new. [. . .] I can see no clear sense in which an action-at-a-distance force of gravity is a “limiting case” of, or “approximates” a space-time curvature. Or in which the ‘theoretical mechanisms’ of action-at-a-distance gravitational theory are “carried over” into general relativity theory. Yet Einstein’s equations undeniably go over to Newton’s in certain limiting special cases. In this sense, there is “approximate continuity” of structure in this case.
[. . .]
On the structural realist view what Newton really discovered are the relationships between phenomena expressed in the mathematical equations of his theory. http://joelvelasco.net/teaching/3330/Worrall 1989 Structural Realism .pdf
To the best of my reckoning, the “relationships between phenomena expressed in the mathematical equations” are just relations between quantities. But if anyone wishes to reify the term “phenomena” in that sentence in some other way, be my guest.
Thus, taking my cue from some of the characteristic claims of scientific realism--
“The entities described by the scientific theory exist objectively and mind-independently.” https://en.wikipedia.org/wiki/Scientific_realism “The central terms of the best current theories are genuinely referential.” http://www-personal.umd.umich.edu/~delittle/Encyclopedia entries/scientific realism.htm
--I wish to state an argument such as:
P1: All central terms of fundamental scientific laws are genuinely referential.
P2: All central terms of fundamental scientific laws are quantities (/mathematical relations).
C: Therefore, some quantities (/mathematical relations) are genuinely referential.
(AAI-3)
or
P1: All entities (/structures) discovered by physicists using the scientific method are objectively existing.
P2: Some mathematical relations are entities (/structures) discovered by physicists using the scientific method.
C: Therefore, some mathematical relations are objectively existing.
(AII-1)
Of course, the above arguments are just my own redneck way of going about the Quine-Putnam Indispensability Argument, which professor Colyvan renders thus:
(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
(P2) Mathematical entities are indispensable to our best scientific theories.
(C) We ought to have ontological commitment to mathematical entities
http://plato.stanford.edu/entries/mathphil-indis/
Colyvan contends that it is the best argument for mathematical realism (I agree), and notes that the argument throws down the gauntlet to those who profess anti-realism, as it puts nominalists in the position of needing to show where the argument goes wrong. I, too, would like to know why one shouldn’t deduce mathematical realism from the premise of scientific realism. What are the reasons or motivations from resisting such a conclusion?
Of course, some people apparently disagree with the thesis of scientific realism, and thus reject P1 in all the arguments above. In which case, I ask: Why shouldn’t we assume that the entities or relations described by the best theories of physics exist objectively? Why shouldn’t we assume that the essential terms of the scientific laws discovered by physicists are genuinely referential? What is a better way for determining the nature of empirical reality than by inference from the discoveries and theories of science?
