Can the truth of this statement be established with the predictive precision of mathematics? Obviously not. Therefore, any claim that the above is true must be declared "illusory".
This is a typical self-refuting statement, like the statement that "there is no truth", or the contention that: "Whatever cannot be known by the method of empiriological physics is not knowable or even worth knowing." The conclusion that "whatever cannot be known by the method of empiriological physics is not knowable or even worth knowing" cannot itself be established by the method of empiriological physics. It should therefore follow that we cannot know that "whatever cannot be known by the method of empiriological physics is not knowable or even worth knowing"; or if we could, it isn't worth knowing.
The original statement above is not a mathematical statement, but a philosophical one. The mathematician in fact does not study math. It is the philosopher who studies math, for mathematics is a science, and it is the philosopher that studies the science of math, that is, what it is we do when we do math, the object of mathematical knowledge, the nature of mathematical entities, etc. The mathematician studies quantities, that is, numbers, magnitudes and their relations.
The term empiriological (as in "empiriological physics") is used here to express both the experimental character (empeiria) of the science as well as its theoretical (its logos) character. Now, the investigative sciences resolve their conclusions in the observable and measurable. But such resolution is not enough. It is in deduction that the most perfect type of scientific explanation is to be found, and that is why empiriology has sought to link itself to a deductive science. There are but two deductive sciences of a pure type, and these are mathematics and philosophy. Mathematical physics is precisely empiriological knowledge linked to the deductive science of mathematics, which plays a formal and directive role with regard to the experience of the physicist. In this case, empiriological science is subalternated to a deductive science.[1]
Subalternation
A science is subalternated to another when it derives its principles from this other science, which Maritain calls the subalternant.[2] Now the subalternate science (in the case of mathematical physics, the subalternate science is physics) does not by itself resolve its conclusions into the first principles of reason, but the subalternant science resolves its own conclusions into first principles and these conclusions of the subalternant becomes the principles for the subalternate science. Classic examples include geometry (subalternant science) with regard to optics (subalternate), which, with geometrical laws, explains the properties of light rays. Acoustics is a subalternate science to arithmetic, and astronomy is as well subalternate to mathematics. Maritain calls this type of empiriological analysis in which the sensible is interpreted mathematically, empiriometrical.
So the mathematical sciences, which are deductive and explanatory sciences, draw the sensible real into their own proper domain in order to "explain" the sensible real and to construct a system of explanatory reasons and causes. This system takes in all the sensible real and explains it not by real or ontological causes and principles (which are real entities or 'real being' of the intelligible order) but by mathematical or logical entities ('logical being', or beings of reason, or ideal entities). As Maritain puts it, "there is a constant coming and going from observed and measured real beings to mathematical beings of reason and vice versa".[3] The mathematical science will use mathematically constructed entities (logical being as opposed to real being) to explain the sensible real, and so the danger here is that of mistaking these mathematically constructed, ideal entities with their grounding in reality, for ontological causes ('real being', entia realia), which alone explain the essence of the physical real.
This confusion of logical being with real being is very commonplace in the world of science and is in fact the root of the self-refuting contention that without the predictive precision of mathematics, any claim to "truth" is illusory.
Is it true that "If you cannot express your knowledge in mathematical form, you may know something; you may have the beginnings of knowledge, but your knowledge is inevitably of a rudimentary and incomplete form"?
Another classic self-refuting statement. For no part of the above statement is expressed in mathematical form, nor can it be so expressed. If the author really believed what he said, you wonder why he didn't even attempt to express his thought or idea in mathematical form. And if the statement were true, it would only represent the beginnings of knowledge, an inevitably rudimentary and incomplete knowledge. But the truth of the matter is precisely the opposite. The only thing that can be expressed in mathematical form is that which is quantifiable. And, what is expressed mathematically is itself in an incomplete and rudimentary form with regard to knowledge proper.[4] Consider graphs, opinion polls, and anything else that are mathematicized. The mathematical expression leaves out all sorts of things that are knowable and considers only the quantifiable. Consider what an opinion poll tells us, for example, about a people, and/or about the political party that is ahead in the polls. Opinion polls provide us with knowledge that is very useful, but nonetheless incomplete. The mind needs more than a number, even though the numbers may prove very useful.[5]
Consider the Guttmacher Institute's statistic on teen pregnancy. The statistic tells us little about the cause of increased sexual activity among teenagers, if anything at all. In itself, the statistic tells us nothing about the character of the teenagers involved nor the culture in which they live.[6]
The mathematicization of time, moreover, does not tell us what time is. Scientists and philosophers still wonder about the nature of time, even though we've measured time for centuries.
So it is not true that if we
find
we are unable to express our knowledge in mathematical form, we only
have
the beginnings of knowledge, but fundamentally a knowledge of a
rudimentary
and incomplete nature. The truth is in the opposite of the statement.
The
mathematical is always rudimentary (for elementary school children are
taught mathematics, but they are not taught philosophy) and incomplete,
which is why it fails to satisfy the mathematical physicist.[7]
For he still seeks to know what the electron is. He wants to understand
what is really going on at the subatomic level despite probability
equations,
which is why they still ask the "quantum-reality" question: because
mathematical
entities do not reach ontological causes. If mathematical expression
really
did amount to complete knowledge, physicists would not seek to know
whether
in reality the electron is a wave or a particle, and whether it is
actually
something independent of our act of perceiving. For the complete
contains
the incomplete, not vice versa. But since the physicist does indeed ask
further questions about an area of reality for which he has more than
ample
mathematical knowledge, it follows that he feels his present knowledge
(for the most part mathematically realized) to be woefully inadequate.
Notes
1See Jacques Maritain. Philosophy of Nature (New York: Philosophical Library, 1951) 102.
2Ibid., 102-114.
3Ibid., 105.
4"Physico-mathematics works in the terms of the physical real, but in order to envisage them from the formal standpoint of mathematics, and of mathematical laws which connect together the measurements collected by our technical instruments from nature. All its concepts are resolved in the measurable. And what verifies the deductive synthesis which it erects is simply the coincidence of its numeric results with the measurements given by experiment; it does not follow that the mathematical beings which intervese in this synthesis represent determinatively real causes and entities which are like the ontological articulations of the world of the sensible nature. Physical theory is verified en bloc, by means of the correspondence established between the system of signs which it employs and the measurable data which have been recognised by experiment.... The system of mathematical relations which it seeks to establish between sensible phenomena, and which constitutes its highest formal object, does not in itself sufficiently satisfy or stimulate the mind of the scientist. His interest is directed towards the physically real." Jacques Maritain. The Degrees of Knowledge (London: Centenary Press, 1937) 168-69.
5"Considering things from the standpoint, not of the physicist, but of the philosopher, and to express ourselves in his language, quantity, i.e. the extension of substance and the metaphysical unity of its parts which are diverse with regard to position, is a real property of bodies. There are, in nature, dimension, numbers, real measurements, real space, real time, and it is under the conditions and modalities of this real quantity, quantitatively measured and regulated, that the interacting causes in nature develop their qualitative activities....Physical reality breeds a rich harvest of entitative riches irreducible to terms of quantity; but by reason of its materiality, and becuase it emanates from the substance of bodies mediatized by quantity, this world of qualities is intrinsically subject to quantitative determinations (that is why it is accessible by our extrinsic and artificial measurements)....But quantity can be considered in another way: when disengaged from its subject by abstractio formalis, set before the mind in itself, as constituting in itself a separate universe of knowledge (the universe of the preter-real), it is then treated no longer ontologically and from the point of view of being, but quantitatively or from the standpoint of those relations of order and measurement which sustain the objects of thought so discernible as the forms or essences which are proper to them." Ibid., 172-73.
6"Mathematics alone among the sciences deliberately 'leaves out' much that is knowable in the things with which it deals. It omits, in fact, everything knowable in them except quantity." Joseph Owens. The Doctrine of Being in the Aristotelian Metaphysics (Toronto: Pontifical Institute of Mediaeval Studies, 1978), 384.
7Cf.
supra, n. 60.
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Douglas
P. McManaman
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