“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 knife 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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“As far as the laws of mathematics refer to reality they are not certain; and as far as they are certain, they do not refer to reality.”– Albert Einstein.

In our investigation of science as certainty, we arrive now at the complementary discipline of mathematics. If we accept the definition of science as “a branch of knowledge or study dealing with a body of facts or truths systematically arranged and showing the operation of general laws,”1 then mathematics appears to be the first science methodically uncovered by the ancients. This seems confirmed by the Webster definition of mathematics as “the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities…”2 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.

In today’s blog we examine the certainty of mathematics while next time we will look at mathematics as a scientific tool. Grade school mathematics particularly addition and subtraction appear to be the pinnacle of unquestionable truth, but remain difficult to prove philosophically. The analytic philosophers in the early 19th century worked tirelessly at demonstrating that mathematics could be validated using only rigorous logic. However this  effort proved hopeless once Kurt Godel developed his ‘incompleteness theorem’ showing that any system that proves all true statements, also permits paradoxes that make no sense (such as “this sentence is false”), whereas any tinkering to the system to eliminate paradoxes results in some true statements no longer being demonstrable.

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. .

Other fields in mathematics also show areas of concern. Euclidean geometry looks ironclad, but  late 19th century mathematicians discovered non-Euclidean forms of geometry that were equally coherent, but gave different results (consider a triangle projected on a sphere versus a plane). Worse yet, some  scientists argue Euclidean geometry is not verifiable in the real world at all.

Chaos theory shows that unpredictable results or non-linearity occur in complex mathematical systems, thus undermining the presumption that all mathematic relations are absolute. In fact unpredictability is predictable in such systems, which appears inconsistent with our usual understanding of mathematics.

Also troublesome is meta-mathematics; which questions what numbers are (Platonic, self-existing, ideas vs. formalistic or logic-based entities) and the sublime question of whether mathematics is discovered or invented by the human mind (Einstein believed the latter). Such fundamental questions do not diminish the practical nature of mathematics, but do subvert our trust in its absolute certainty. 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


1Webster’s New Universal Unabridged Dictionary, Barnes & Noble, Inc. 2003. ISBN 0-7607-4975-2, p. 1716 – definition 1.

2Webster’s New Universal Unabridged Dictionary, Barnes & Noble, Inc. 2003. ISBN 0-7607-4975-2, p. 1186 – definition 1.

3Fadiman, Clifton, Editor, The Treasury of the Encyclopaedia Britannica.Viking Penguin, New York, 1992. ISBN 0-670-83568-4, page 659.

4Ferris, Timothy, Editor, The World Treasury of Physics, Astronomy, and Mathematics. Little, Brown, and Co., Boston, 1991. Page 525.

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“In our infinite ignorance, we are all equal.” – Karl Popper.



As natural philosophy separated from the rest of philosophy in the 17th and 18th century, it came to be known as science. This led to a continental drift in philosophy by the 19th century when logical positivism appeared most famously in the thinking of Auguste Comte. Positivism asserts that rationally justifiable statements can always be scientifically verified or are capable of logical or mathematic proof; hence rejecting metaphysics and most other branches of philosophy basically as meaningless.

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 is, by this thinking makes no sense.

The ensuing debate is complicated and impossible to summarize, but at a minimum, verification theory appears to be flawed as science is limited to a finite number of observations, measurements, or experiments “confirming” any hypothesis. Thus scientific theories can never in fact be verified and arguably become nonsense by this definition as well. Albert Einstein challenges verification theory on the basis that geometry itself cannot be verified in the real world, effectively leading to an a reductio counterproof.1

Karl Popper who claims to be a strong believer in science, but is skeptical of its certainty, argues the more rational approach is falsification. Basically no matter how many times observation, measurement, or experiment supports a theory, one instance of discordance disproves it. For example, the theory that all swans are white cannot be proven by looking at any number of white swans, but can be refuted by finding one black swan. 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.”

It is not my intent in this brief essay to argue that science is defective; in fact I do believe most accepted scientific theories are highly probable, even justify the presumption of truth. Rather I believe we must factor in the possibility of human error in our understanding of nature as integral to the procedures we use in science and mitigate for that potential error. In the synopsis of this topic we will come back to the limits of verification and the ceiling falsification places on scientific doctrines.

1Schilpp, Paul Arthur (editor), Albert Einstein: Philosopher-Scientist. Open Court, La Salle, Illinois, 1970. Page 676-677.

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Last time I noted there are ten fundamental assumptions underlying the seemingly ironclad truth of science, and I discussed the first five: empiricism, material monism, mathematics, the uniformity of nature, and causality.  The sixth is universal applicability, meaning science as a method is assumed to apply to all of nature and even to the human realm. Therefore science is considered the best means not only to study the earlier history of the universe but also the very existence of it. Likewise not only can human behavior be categorized scientifically, but subjective reasoning and consciousness also can (or will) be explained, even predicted. Some scientists argue that humans lack free will and that all human action in theory can be explained through material causation. Others contend that ethics, happiness, and human meaning can be elucidated using scientific techniques.

Next is the axiom that human reasoning in general and the experimental method specifically are valid in the interpretation of data and informing reality. Demonstration of the reliability of human reasoning is necessarily circular as it must be used to justify itself. The scientific method – hypothesis, experiment, measurement, confirmation – seems rational, but is itself confirmed only by repetition of the method since further demonstration is indirect, for example by validity of prediction or application. 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.

The eighth assumption is that each field of science interconnects with the others. For instance, the quarks of the physicist become the elements and molecules of the chemist which in turn make up the organic matter of living things. The implication is that instinct, love, and human creativity, for example, can be explained by mere chemical structure and reactions.

The ninth assumption is that most physical processes can be distilled down to a limited number of scientific laws. Laws of mechanics, wave and field theory, relativity, and string theory are the progressively inscrutable explanations for common experience like the growth of an oak from an acorn or human romanticism around the full moon. 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.

Last is utterly critical – the denial of the supernatural. If any events in the universe occur outside of natural law, then all natural laws become suspect as either wrong, unprovable, or local. In science, the explanation of any event must not require something outside of the sensible universe. Thus ESP, ghosts, souls, miracles, and God cannot be implicated or acknowledged as they cause the scientific matrix to unravel.

Of course there is significant overlap among these ten fundamental assumptions that underpin science. But it seems science has developed not as coherent truth, but as a modern form of foundationalism. We will return to this in the synopsis of science as certainty, but first we need to explore falsification and verification challenges to scientific certitude.

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“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.


Science, I observed last time, is the paradigm of a system of coherence, but also fits as well as any other system the two other concepts of truth – correspondence (to reality) and pragmatic (instrumentality). Today we delve further into the question of whether science can claim certainty by investigating its ten foundational principles:

1.   Empiricism

2.   Material monism.

3.   Mathematics/statistics

4.   The uniformity of nature/induction

5.   Causality

6.   Universal applicability

7.   Validity of human reasoning/scientific method

8.   Interoperability/transferability between fields

9.   Refinement into a limited (or singular) underlying laws.

10. The non-existence of the supernatural.

Again I would like critique these individually, starting with empiricism. While all of these basic assumptions are necessary for its coherence, science requires one to accept that the human experience of nature and of natural and experimental events is the only legitimate basis for evaluating reality. This is the sine qua non of all science and the greatest lesson of Galileo. If it turns out our senses in fact fail to examine the physical world, then science is at worst simple conjecture and at best only an explanation of human phenomenologic experience, but by no means correspondence with reality.

The second axiom of science is material monism, the belief that there is no separate nonmaterial existence. For instance consciousness is only a manifestation of organic brain activity. The great metaphysical question of the existence of the human soul is dispensed with by simple denial. Some scientists may be dualists, but even they tend to accept science alone describes physical reality.

Third is the validity of mathematics especially in its application to systematic observations yielding for example, Newton’s “force equals mass x acceleration” or Einstein’s “e=mc2.” The axiomatic truth of mathematics applies to all measurements in scientific research and underpins the statistics essential to most scientific conclusions.

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.

Fifth is causality – the belief that preceding events cause later events and no event is uncaused. If sequential measurements show the earth’s average temperature is rising, there must be a reason; we cannot say this rise is instantiated in the universe itself. While apparently self-evident, Hume argued this belief may be merely a trick of the human mind and that causality cannot be demonstrated beyond question.

(continued next post)

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“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.



Last time we considered coherentism as an alternative to the absolute certainty of foundationalism or the complete uncertainty of radical skepticism, and I ended with the proposition that science is humanity’s best model of coherentism. Western science dates back at least to the ancient Greeks with their surprisingly accurate astronomy (Thales predicted an eclipse in the 6th century B.C.E.), the mathematics and geometry of Pythagoras and Euclid, and the physics and biology of Aristotle, but most authorities date the origin of modern science to the 16th and 17th centuries with Copernicus, Kepler, Galileo, and especially Newton.

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.

Science with its amazing achievements appears to be the paradigm of a system of truth. However the ultimate question for us is whether science offers the absolute certainty denied by the radical skeptic. The following areas color the clarification on this question:

1.    Foundational assumptions of science

2.   Theories of falsification and verification

3.   The limits of mathematics    

4.   Issues of connection.

The next eight posts will address these systematically. Thereafter I hope to synthesize the best possible attitude for the philosopher with regards to science in considerations of metaphysics and ethics.

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“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 our investigation of certainty, so far we touched on the definition of knowledge, the place of subjectivity in the concept of truth, and the challenge of skepticism. If we concede that truth for humans does not refer to the actual correspondence of idea and reality and that foundationalism fails rigorous examination, but shudder at the thought of radical skepticism, perhaps we can find a safe harbor in the intermediate position of coherentism. The coherence theory of truth sees true statements as those that cohere with a system of other statements. Thus the coherentist concludes “our beliefs must fit together, and fit with the evidence in such a way as to add up to the most overall coherent picture of reality.”1

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

To some extent, I am a coherentist, but then I feel the difference between foundationalism and coherentism is  mostly semantic. The proponents of each must either consider their basic beliefs or their ‘critical nodes’ as certain (beyond a shadow of a doubt as Adler might say) or concede the viability of the radical skeptic’s position. I still believe the best approach of the practical philosopher is to utilize all three theories of truth – correspondence, coherence, and pragmatism – to guide understanding and conduct in an uncertain world, rather than default to any one of the three or abandon oneself to hopeless skepticism.

Next time we will look at humanity’s most successful effort to date at such a threefold reinforcing system of knowledge – science – and ponder how close it comes to certainty.


1Baggini, Julian, The Edge of Reason. Yale University Press, New Haven, 2016. ISBN 9780300208238, page 25.

2See Suffering – The First Noble Truth – Part II on this site – February 26, 2020.

3Baggini, Julian, The Edge of Reason. Yale University Press, New Haven, 2016. ISBN 9780300208238, page 26-28.

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Last time I reviewed Dr. Robert Pasnau’s essay Snatching Hope from the Jaws of Epistemic Defeat where he discusses radical skepticism and offers his response -the hopeful affirmation of evidence-based credence – arguing the goal of finding truth and avoiding error is trumped by the importance of being able to live a full life within the bounds of uncertainty. I enjoyed reading his article and appreciate his solution to the problem of skepticism, but today I wish to offer an alternative to mere hope.

It seems to me the practical philosopher should not choose wholeheartedly to embrace propositions based on credibility alone. Rather I think we must choose courses of action within our uncertainty that offer the best outcome should we be wrong, that is, pragmatic conduct. I offer three examples of increasing uncertainty.


First, absolute evil – take for instance whether the blinding of innocent animals is good or evil. While this seems self-evident, the radical skeptic may derive an argument I cannot that this action is justifiable or desirable. Pragmatically however, I choose not to blind innocent animals as there is no apparent value to me as a person; the choice not to commit this apparent evil has no untoward consequences.

Second is the near certain proposition that i should try to make a good life for myself within the limit of not interfering with this goal for others. I may be wrong; in fact, making a good life for myself may be impossible, but by attempting I have only the possibility of making a good life for myself in which case my life is good, or failing in which case I am no worse off than if I thought I should not make a good life for myself. Pragmatism succeeds again (although hope might here as well).

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.

The reason I took on the project to develop practical philosophy from the teaching of the great thinkers was to offer ethical balance in conduct within the framework of a life full of uncertainty. Hope is a valuable human emotion, but a meaningful life requires virtue, and that demands we factor in the uncertainties in reality. In that sense I would consider myself, following Pasnau’s lead, a pragmatic epistemic defeatist, although I dislike the term.

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 Snatching Hope from the Jaws of Epistemic Defeat1

“Hope is the only God common to all men; those who have nothing more, possess hope still.” – Thales of Miletus

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 https://spot.colorado.edu/~pasnau/inprint/. He defines ‘epistemic defeatism’ as “the view that we have no good evidence for the truth of any proposition.”1 His goal is to untangle it from related views and establish its independence from questions of knowledge. He traces strict skepticism back to Pyrrhonism with its cardinal doctrine that for every good argument there is an equal opposing argument forcing the lover of truth to suspend judgment in all affirmations of knowledge. The Pyrrhonists offer at least seven challenges to formulations of certainty: illusion, perpetual variation, disagreement, cultural relativity, infinite regress of premises, circularity of arguments, and groundless assumptions.2

He divides skepticism into two branches: ‘weak’- which concedes beliefs are based on evidence that is insufficient for certainty (e.g. Descartes), and ‘strong’- which denies any belief is based on ultimately reliable evidence (e.g. Hume).  Pasnau thinks this latter case of epistemic defeatism, is more interesting philosophically. Strong skepticism is usually ignored or refuted a reductio, that is, since we obviously have knowledge, epistemic defeatism must be absurd, but he thinks other responses – reliabilism, default to truth, language game theory, changing context theory, consequence-based belief, and coherence – are better.2

While he maintains that knowledge need not be identical with certainty, he is intrigued by Hume’s arguments that our firmest beliefs lack even probability, contending that “…what could possibly be of greater philosophical significance than the thesis that, in the final analysis, we have no good evidence for the truth of any proposition?”3 He considers various approaches such as rational expectation, probability, antirealism (e.g. Berkeley), divine knowledge, and evidentialism,  but thinks none is ideal .2

He eventually comes to Augustine and Al-Ghazali who rely on divine inspiration or faith, but he finds problems with this position such as what happens when faith clashes with evidence, questions of degrees of knowledge to accept on faith, and the risk of epistemic chaos.

Instead he sees the crux of the claim to knowledge as the tension between our fear of being wrong and our hope of being right. Based on credence – one’s attitude regarding the chance of a proposition’s obtaining – one may choose to be optimistic and confident while abandoning the concern about being certain. He admits this ‘hopeful defeatism’ may not count as knowledge, but it does recognize “we care about many things other than truth and falsity- inasmuch as we want to live rich, engaged lives.”4 I am reminded of William James’ essay The Will to Believe.

However, I believe there is a stance more reasonable to the practical philosopher which is the subject of my next post.


1Journal of the American Philosophical Association/ Volumes 1 / Issue 2 / June 2015. Page 257.  Note: I refer to this as ‘radical skepticism’ in my last two blogs.

3As there is not room in this essay even to summarize these, I leave it to the interested reader to go to the source.

3Ibid., page 262.

4Ibid., page 274.

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Last time we reviewed the origins and a limited history of skepticism and ended on the destructive nature of its radical form. Some philosophers go further, arguing complete skepticism is, in fact, unintelligible, because it assumes or requires no difference between certainty and uncertainty at all. Others see it as devolving into near irrationality – propositions are denied to be statements in any meaningful sense, as if one is saying something like “middle C is soluble in water.”7

Be that as it may, to me, universal skepticism fails the common sense sniff test. If one denies proof of one’s existence, then why not jump off a cliff ? If the answer is the skeptic does not wish to jump off a cliff, then a belief in personal wants is affirmed. If the skeptic says he doesn’t know why he doesn’t jump off a cliff; ask why he eats and drinks? If the answer is he doesn’t know, than suggest he find out the truth by stopping. Refusal is proof that he does exist and wishes to continue to exist or his lack of desire to find truth. Even if he says he refuses to stop eating and drinking but does not know why, then he has proven he is a being that refuses to stop eating and drinking. At the end of the day, the circularity of radical skepticism appears even more severe than that of foundationalism, and so likewise fails.

I suspect it is some form of this reasoning that leads Hume to ‘mitigated skepticism.’ While he rejects absolute skepticism, he points out that no one believes all things are knowable, therefore the difference among people is one of degree. The crux then is really the criteria for knowledge. Hume thinks the only certainties are quantity and number or mathematics and relationships of ideas8 such as “all bachelors are unmarried.” In an effort to avoid the parsing of epistemology, I would suggest the simplest basic criteria are disciplined reason and consistent empirical evidence. The reader will have to decide for himself or herself whether these are sufficient and exactly how to define and utilize them.

Our next schedule topic is the intermediate theory of coherentism, but before that, in the next two posts, I will explore an essay brought to my attention by a reader that dovetails nicely with the last two blogs on skepticism.


1 Other cultures have skeptics – for example the Charvakas in ancient India and the Taoists in China.

2Of course Socrates can be seen as the first Greek skeptic. When told that the Oracle of Delphi had pronounced him the wisest of the Greeks, he interpreted this to mean that compared to other Greeks, at least he knew that he knew nothing.

3Hallie, Philip P. (editor), Sextus Empiricus. Hackett Publishing Company, Indianapolis/Cambridge,1985. ISBN 0-87220-006-X, p 10-13.

4Ibid., page 7


6 Ibid., page 27.

7Edwards, Paul (editor), The Encyclopedia of Philosophy. Macmillan Publishing Co., Inc. & The Free Press, 1972.   Volume 2, page 68.

8Adler, Mortimer J, et. al., The Great Ideas – A Syntopicon of Great Books of the Western World, Volume II, Encyclopaedia Britannica, Inc., Chicago, 1952. Page 884.

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