“If all the arts aspire to the condition of music, all the sciences aspire to the condition of mathematics.”– George Santayana. ` `
Last time we looked at whether mathematics corresponds to absolute truth; today we investigate mathematics as a tool of science particularly as the substructure of scientific certainty. Nowadays we see math as so integral to science that we might think they developed in lockstep, but that is not the case. Mathematics likely started as counting possessions such as fingers, children, goats, or coins; followed by the geometry necessary for land measurement and construction. At that time, science was predominantly observational – e.g. identification of the constellations and planets and the four ancient elements of earth, water, air, and fire – or speculative as in the case of Democritus’ atomism.
Some ancient geniuses transposed common mathematics onto the mystery of nature most famously in an increasing understanding of the motion of the planets and sun (Thales predicted an eclipse in the sixth century B.C.E.) and on the harmonics of stringed instruments (Pythagoras; also the sixth century B.C.E.). 
But successors failed to follow up on their insights, so in fact the greatest scientist of the ancient world, Aristotle, studied zoology and botany only by observation and description; while its greatest mathematician, Archimedes, took mathematics much further, but mostly for technology rather than for the analysis of nature.
Successive cultures in Rome, Arabia, India, and even medieval Europe advanced in pure and applied mathematics, but failed to identify its utility in elucidating nature. That seems to appear suddenly in the works of Copernicus, Kepler, and particularly Galileo who rejected scholastic views of knowledge and subjected observational and experimental data to mathematical analysis in formulating theories. Arguably their insight that mathematics can explain data was their greatest contribution to science and one of the great feats of humanity. Newton, Pascal, Lavoisier, Faraday, Einstein and countless others followed, all using mathematics as the scapel by which to dissect out the hidden structure of reality.
Thereafter for centuries, mathematics and science grew in parallel without impediment until coming up against two fundamental challenges. First, mathematics itself showed defects as outlined in the last blog. Second, in the desire for solid foundations for scientific theories, mathematics was overstretched to fit some theories, and increasingly modified or invented merely to permit models of nature that transcend any experience of reality at all. It is this latter trend in the relationship of science to mathematics that is most disconcerting with respect to its certainty. Next time we will look at some important examples including (1) chaos theory and complex systems, (2) quantum mechanics, uncertainty, and the spontaneous appearance of matter, and (3) string theory.
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Alfred North Whitehead goes somewhat further in defining mathematics as “the science concerned with the logical deduction of consequences from the general premises of all reasoning,”3 though for our purposes we will use the use the stricter Webster definition.
Meanwhile Gregor Cantor demonstrated that rules of infinity broke basic rules of mathematics, effectively proving that two unequal numbers can be equal. By mapping infinite series such as all integers, against all even numbers, he showed that while there are clearly more integers than even numbers, there are in fact an infinite number in each series. .
As Morris Kline wrote in The Loss of Certainty in 1980, “It behooves us therefore to learn why, despite its uncertain foundations and despite the conflicting theories of mathematicians, mathematics has proved to be so incredibly effective.”4
“In our infinite ignorance, we are all equal.” – Karl Popper.
The crux of the positivist position is that meaningful statements in general can be demonstrated by empirical means, that is, they are verifiable. Thus the statement, “all swans are white” is meaningful since this can be verified by simply looking at swans, while the statement “reality is one” cannot be verified by experience making it meaningless. A powerful corpus of positivist philosophy followed by thinkers like Bertrand Russell, Ludwig Wittgenstein, and A. J. Ayers all directed at the thesis that metaphysical statements and theology are effectively non-sensical. Since there is no way to test the existence of God, the affirmation of the divine, by this thinking makes no sense.
He feels scientific theories should not be presented as certainty, but only as more or less likely. For instance he challenges the big bang theory of the origin of the universe (now accepted as fact by most cosmologists) based on the significant problems with the theory. However falsification has flaws as well since it assumes verifiability of the falsifying fact. That would not be possible if we modify the earlier statement to “most swans are white.”
Thus for example, paleontology relies on theories such as carbon dating or the increasing age of deeper sedimentary layers which in turn depend on the indefeasibility of the logic behind these methods. Alternatively, science insists on the corollary that no text or teaching is inviolate; all must stand up to ongoing scrutiny.
Science embraces Occam’s razor, the theory that the least complex explanation is most likely. As a result, additional inferences are made such as the prediction of a ‘theory of everything’ and the seemingly irrational theory that everything material ultimately came from nothing.
“What science and the quest for knowledge are after is irrefutable truth; that is, propositions that human beings are not free to reject – that are compelling. They are of two kinds, as we have known since Leibniz: truths of reasoning and truths of fact.” – Hannah Arendt.
Fourth is the presumption of a uniformity in nature. By this the scientist asserts that what applied in the past will apply in the future and what is true in one place is true everywhere else under the same conditions. Thus the boiling temperature of water is the same today as it was one thousand years ago and as it will be in one thousand years and stars in other galaxies act in the same way as stars in our galaxy. From this uniformity, a limited number of measurements can be projected on to nature itself, allowing the induction of scientific laws.
“If a man will begin with certainties, he shall end in doubts; but if he will be content to begin with doubts, he shall end in certainties.” – Francis Bacon.
Science uses a rigorous method consisting of observation, experimentation, and mathematical and statistical analysis. Procedures, data, and principles are typically public allowing others to confirm or dispute conclusions. But perhaps most significant is that science allows predictions and applications that strengthen its validity. No other system yet derived by humanity – definitely not theology or philosophy – is so internally consistent and externally useful. Inevitably we are led to the core of its power; science is singular in conforming to the three theories of truth, that is, it appears to correspond to reality, requires coherence of principles and facts, and demonstrates pragmatic instrumentality.
“I can live with doubt and uncertainty and not knowing. I think it’s much more interesting to live not knowing than to have answers which might be wrong. – Richard Feynman, Nobel Laureate in Physics, 1965.
In theory, coherentism does not rely on basic foundational truths constructing a brick wall, but instead on a ‘web’ of items of knowledge interconnected by strands of evidence. (This reminds me of the metaphysics of Buddhism.2) Julian Baggini sides with this more modest framework of truth, but notes the difference may be illusory since most foundationalist models also depend on interlocking beliefs. Other coherentists appear to agree; Baggini cites Susan Haack who argues for ‘foundherentism’ where experience is the foundation of a coherent system. Ludwig Wittgenstein and Bertrand Russell also seem to concede foundational aspects, most importantly that logic must be trusted to inform the web. In fact ‘critical nodes’ on the web are vital to keep it intact. For instance, the principle of noncontradiction is essential to any system of knowledge making it ultimately indispensable rather than indisputable.3
My last example comes from the plot line of The Bhagavad Gita where Arjuna, the protagonist, must decide between his duty to his side in a battle where the opposing side includes his friends, teachers, and even family (alternatively you may consider choosing the union side in the American Civil War). It appears impossible to determine whether duty to some of our friends and family is ethically correct compared to avoiding harm to others of them– here we have almost no level of certainty. The pragmatic solution is to do both. While we have a duty to provide service to our side, we can choose service that does not entail harming others – we can choose to be medics, or unarmed messengers, staff personnel, or other non-combatants – many of which involve opportunities for the epitome of heroism and sacrifice.
At the suggestion of a subscriber, I read this 17 page essay by Professor Robert Pasnau (University of Colorado at Boulder) published in 2015 in the Journal of the American Philosophical Association. It can be accessed at 
