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Is a Judge of Knowledge Shipwrecked by the Laughter of the Gods?

Roger McCann tries to identify some of the attributes of knowledge.

“Whoever undertakes to set himself up as a judge of Truth and Knowledge is shipwrecked by the laughter of the gods” – Albert Einstein

Knowledge is a strange concept. It is used as though its meaning were as well understood as ‘dog’. Even though dogs come in innumerable sizes, shapes, and colors, there is rarely disagreement as to whether something is, or is not, a dog. The same is not true for knowledge. In common usage, ‘knowledge’ can be both transient (It is no longer tenable that the Earth is flat) and eternal (In Euclidean geometry the sum of angles in a triangle is 180°); both irrational (‘Jesus is both God and human’) and rational (the contents of Euclid’s Elements); both confusing (the interdependency of life-forms in nature) and clarifying (Kepler’s laws of planetary motion). Perhaps a suitable definition of knowledge would remove these ambiguities.

In common usage, knowledge is often defined as a familiarity with something, such as facts, information, or skills acquired through thought or experience. However, there is no absolutely accepted philosophical definition of knowledge. In Plato’s dialogue Theaetetus (c.369 BCE), Socrates and Theaetetus discuss three definitions of knowledge: knowledge is perception, knowledge is true belief, and knowledge is justified true belief. Philosophers have never been entirely satisfied with any of them. Bertrand Russell (1872-1970) summarizes the problem with defining knowledge in his ‘Theory of Knowledge’ (1913): “at first sight it might be thought that knowledge might be defined as belief which is in agreement with the facts. The trouble is that no one knows what a belief is, no one knows what a fact is, and no one knows what sort of agreement between them would make a belief true.”

In any system of thought there are undefined concepts, called ‘primitive notions’, that are formulated by appealing to intuition or experience, and are thought to need no further justification. Since at least 2,500 years of trying has failed to produce a satisfactory definition of knowledge, let us assume that knowledge is a primitive notion. This does not mean that knowledge is necessarily an amorphous concept: a primitive notion can often be at least partially described by listing some of its attributes, such as Islam’s 99 names of Allah.

The main purpose of this article is to present examples that will help identify, or rule out, attributes of knowledge. These examples are drawn mainly from mathematics and physics. I do not claim that the attributes I propose are definitely attributes of knowledge, but I do claim that they are plausible attributes of knowledge. The reader is left to decide how plausible. If they are found to be implausible, then that would itself suggest other attributes of knowledge. Along the way we will see that ‘knowledge’ as proposed by certain seventeenth century philosophers was made untenable by a twentieth century mathematician-logician, and that the supposed ‘knowledge’ of Aristotle (384-322 BCE) was made untenable by scientists in the sixteenth century. In addition, we will consider two fundamental questions about knowledge: (1) Does knowledge exist? and, if knowledge does exist, (2) What are possible sources of knowledge?

We’ll begin by considering the latter question.

Sources of Knowledge

In the West, knowledge has often been identified as the result of rational thought – the truth of which idea has been disputed by philosophers for centuries. Some philosophers, rationalists such as Baruch Spinoza (1632-1677) and Gottfried Leibniz (1646-1716), argued that one can deduce all possible knowledge from a suitable set of axioms; other philosophers, empiricists such as John Locke (1632-1704) and David Hume (1711-1776), argued that knowledge comes solely from experience of the world. Most philosophers fall in the gap between these two extremes, by arguing that both reason and experience are necessary for knowledge. In short, a major portion of Western philosophy is based on the premise that knowledge follows either from rational thought, experience, or from a combination of the two.

But these are not the only possible sources of knowledge. Muslims believe that Muhammad (c.570-c.632) received the Koran by divine revelation. Thomas Aquinas (1226-1274) also considered divine revelation to be a source of knowledge, in his case for the Bible. In Zen Buddhism, the primary purpose of a koan (a paradoxical question, such as ‘What is the sound of one hand clapping?’) is to exhaust the rational intellect, thereby leaving the mind open for an intuitive experience, and with it a type of knowledge. The Indian mathematician Srinivasa Ramanajan (1887-1920) credited his mathematical accomplishments to his family goddess, while his colleague and collaborator G.H. Hardy (1877-1947) claimed they were achieved by other means, including intuition. Anyone who doubts that divine revelation or intuition can be a possible source of knowledge should consider the work of Ramanajan.

However, divine revelation and intuition suffer from many of the same problems, among which are: (1) What is divine revelation or intuitive knowledge to some may be utter nonsense to others; (2) Divine revelation and intuition frequently conflict with other divine revelations and intuitions; (3) Divine revelation and intuition are often influenced by tradition, conventional wisdom, or other biases; (4) Divine revelation and intuition can conflict with rational thought or experience; and (5) Divine revelation and intuition cannot be replicated or verified.

There are also innumerable other methods people have used to gain ‘knowledge’; for example, scapulimancy (predicting the future by reading cracks on heated shoulder bones). Before we disparage so-called ‘primitive’ societies for such methods, we need to remember that astrological forecasts still appear in many Western newspapers, that phrenology has had various levels of acceptance during its very recent history, and that both Ouija Boards and Tarot cards are readily available, even at Walmart and Amazon. Such methods are well documented to be even more unreliable than the other mentioned sources of knowledge, and will not be considered further.

There is yet another, seldom recognized, source of ‘knowledge’, namely legislation. Can knowledge be legislated? The question may seem absurd, but attempts have been made to legislate knowledge. For example, in 1897, the Indiana House of Representatives unanimously passed a Bill to legislate a value for pi. Correcting a possible transcript error in the records gives pi a value of 9.2376…. Fortunately, the Indiana Senate voted to postpone consideration of the Bill, and it never became law. (See Petr Beckmann’s book, A History of π.) What would have happened if the Bill had been enacted is anybody’s guess. But pi is what it is. No legislation, presidential decree, royal edict, fatwah, papal bull, or any other human proclamation, can make it otherwise. I propose that this is an attribute of knowledge in general. That is, knowledge cannot be established solely by human proclamation.

On this theme, since the early 1900s there have been numerous attempts in the United States to ban the teaching of evolution and require the teaching of creation science. A law advancing the latter was declared unconstitutional by the U.S. Supreme Court in 1987 because the court considered the law an attempt to advance a particular religion. During the hearing, numerous scientists and scientific organizations filed amicus briefs describing creation science as being composed of religious tenets. In short, the battle was over whether ‘knowledge’ from divine revelation supersedes ‘knowledge’ from experience and rational thought. Thus the age-old philosophical debate over the essence of knowledge has in some quarters degenerated into a legal issue. Shakespeare’s comment, “The first thing we do, let’s kill all the lawyers” from Henry VI, Part 2, was not made by a character seeking knowledge, but it could have been.

The above description of possible sources of knowledge confirms Einstein’s observation about gods’ laughter shipwrecking whoever undertakes to be a judge of knowledge. Nonetheless, we will now follow philosophical tradition, and focus on rational thought and experience as possible sources of knowledge.

Rational Thought

Rational thought has been used to justify concepts that some find ludicrous (e.g., the existence – or non-existence – of God) or abhorrent (e.g., principles of eugenics as interpreted by the Nazis). Nonetheless, many people believe that a statement such as ‘All knowledge comes from rational thought’, can be established as either true or false by logical means. Unfortunately, in 1931 Kurt Gödel (1906-1978) proved his Incompleteness Theorem, which says that within all but the most trivial logical systems capable of doing arithmetic, statements exist that are neither provable nor disprovable using the axioms that define the system. I can conceive of no reasonable attribute of knowledge that would keep the Incompleteness Theorem from being knowledge.

Rational thought was used to prove the Incompleteness Theorem, but the Theorem itself is independent of rational thought, just as an arch is independent of the centering used to construct it. Therefore, we will say that the Incompleteness Theorem is absolute knowledge. In addition, there is no apparent experience that would yield the Incompleteness Theorem. I propose therefore that the Incompleteness Theorem is absolute knowledge that cannot be attained by experience. Consequently, ‘Knowledge exists’, ‘Some knowledge is absolute’ and ‘Some knowledge cannot be attained from experience’ are also plausible attributes of knowledge.

Was the Incompleteness Theorem knowledge before Gödel proved it? This question can be formulated more generally as, Does knowledge exist independently and is merely discovered, or do humans create knowledge? I cannot give a definitive answer to this question, but I can give a partial answer.

Prime numbers are widely used to secure information, such as bank card transactions and email accounts. Thus, there are few whose lives are not affected daily by prime numbers. I can conceive of no reasonable attribute of knowledge that would exclude knowing whether a number is prime from being knowledge. Euclid (323-283 BCE) is credited with an elegant proof that there are infinitely many prime numbers. So suppose N denotes the largest known prime number – which as of early last year has over 17.4 million digits. The next prime number larger than N exists according to Euclid’s proof, and so awaits discovery. Thus, at least some knowledge is discovered and not created. This is another plausible attribute of knowledge.

Sherlock Holmes, Hercule Poirot, Jane Marple, and other fictional detectives deceptively lead us to believe that rational thought can resolve the most complex enigmas. In reality, many issues may be too complex for purely rational resolution, especially philosophical or religious issues. Rational arguments both for and against concepts such as free will or the existence of God abound – thereby creating a prima facie case against rational thought being a reliable source of knowledge. But this may be only an illusion due to improper application of rationality. A valid rational argument is based on a set of postulates. Unfortunately, many so-called ‘rational arguments’ are based upon ill-defined, if not undefined, postulates. This allows ‘rational arguments’ to validate both a concept and a conflicting concept, thereby creating an absurd situation in which arguments are simultaneously both rational and irrational! Without clearly knowing the postulates upon which it is based, a ‘rational argument’ can be little more than intimidation by strength of personality or erudition, or its conclusion simply a statement with superficial appeal. An example of the former is Aristotle’s influence suppressing for roughly seventeen hundred years the notion of a sun-centered solar system, proposed by Aristarchus of Samos in c.270 BCE. Political slogans illustrate the latter.

However, it also needs to be noted that knowledge attained by rational argument is relative to a set of postulates, and so not absolute. Geometry provides a good example of this.

illusion wheels
Can we say we know what our senses tell us?

Euclidean geometry is based on five postulates. The Fifth Postulate, called ‘the Parallel Postulate’, is that there is exactly one straight line that can be drawn through a point not on a given line which will be parallel to the given line. If the phrase ‘there is exactly one straight line’ in the Fifth Postulate is replaced by ‘there exists no straight line’, or by ‘there exist at least two straight lines’, Euclid’s first four postulates together with one of these new postulates describe an equally valid ‘non-Euclidean’ geometry. These replacements were done around 1830 by Janos Bolyai (1775-1856) and Nicolai Lobachevsky (1792-1856). Pythagoras’s Theorem is valid in Euclidean geometry, but is not valid in the non-Euclidean geometries. Thus, if Pythagoras’s Theorem is knowledge, it is knowledge relative to the postulates of Euclidean geometry and not relative to the postulates of non-Euclidean geometries. However, not accepting Pythagoras’s Theorem as knowledge would ignore the fundamental fact that the Theorem is utilized in everyday engineering and construction. Few would consider this acceptable. Consequently, ‘Some knowledge comes from rational thought’ and ‘Some knowledge is relative’ are both plausible attributes of knowledge.

A major problem with a rational argument is the appropriateness of the postulates on which the argument is based. The appropriateness of geometric postulates is sometimes easy to resolve. Euclidean geometry is appropriate for calculations on an everyday scale, while non-Euclidean geometries are appropriate for calculations on a large scale where general relativity applies, such as in cosmology and celestial mechanics. Unfortunately, it is sometimes impossible to resolve the appropriateness of postulates, as is the case in the debate over free will. People who believe the postulate that everything that happens is determined by physical laws believe that free will does not exist; while people who believe the postulate that mind can make choices independent of the influence of physical law believe that free will does exist. Each argument is valid only relative to its own postulates.

One might try to avoid the problem of appropriate postulates by saying that the conclusion of any rational argument is knowledge relative to the associated postulates. But if we do say this, what do we do with cases in which the postulates are nonsensical, contradictory, or paradoxical? We surely do not want to consider results based on such postulates as knowledge. Thus, we would need an attribute of knowledge that precludes inappropriate postulates. Unfortunately, I have no candidate.

Before we leave this discussion of rational thought, please note that there is no universal agreement as to what ‘rational thought’ is. The technique of reductio ad absurdum – a method of indirectly proving a proposition by assuming its negation to be true, and showing that this leads to a contradiction – has been used by philosophers from the fifth century BCE onward, with Heraclitus (c.535-c.475 BCE) and Hegel (1770-1831) being notable exceptions. The discovery of contradictions at the apparent foundations of mathematics around the beginning of the twentieth century, such as Russell’s Paradox, led a few philosophers, such as Graham Priest (1948-) to develop logic systems in which some statements can be both true and false. Consequently, what is knowledge in a traditional logical system may not be knowledge relative to one of these other logic systems, and vice versa. Even though these alternative logic systems are not widely accepted, they nevertheless complicate the concept of ‘rational thought’.


Another plausible attribute of knowledge is ‘Some knowledge comes from experience’.

Using the Incompleteness Theorem as an example, we have already concluded that some knowledge does not come from experience. Therefore, it does not seem possible to increase ‘Some’ to ‘All’ in this attribute. Nonetheless, it is interesting to consider the validity of ‘All knowledge comes solely from experience’ from two other points of view.

First, suppose the statement ‘All knowledge comes solely from experience’ is true, then this is surely a piece of knowledge. But since it is impossible for us to experience all possible knowledge, experience could not establish this particular piece of knowledge. This reductio ad absurdum seems to prove the opposite, that some knowledge does not come from experience. However, there are two other conclusions possible from this argument: (1) the validity of the statement ‘All knowledge comes solely from experience’ is unknowable; or (2) it is true but its truth is incompatible with standard rational thought. (Ironically, Locke, Hume, and others devoted hundreds of pages of rational argument trying to justify the validity of the proposition ‘All knowledge comes solely from experience’.)

Second, since antiquity, rational mathematical arguments have shown that the ratio of a circle’s circumference to its diameter (pi) does not depend on the particular circle chosen. Experience obtained by measuring the circumferences and diameters of numerous circles might also lead one to conjecture that the ratio of a circle’s circumference to its diameter does not depend on the particular circle chosen. However, it is not possible to establish this solely by experience, due to inesscapable measurement error (no measurement is truly exact), and also to the impossibility of measuring all circles. So if we accept ‘All knowledge comes solely from experience’, then an essential fact in mathematics, science and engineering that affects everyday life would not be knowledge. Few would consider this an acceptable conclusion.

Euclid taking a measured approach to knowledge

Detractors of experience as a source of knowledge cite deficiencies analogous to those I listed for divine revelation as a source of knowledge. For example, we have all been deceived by experiences. Optical illusions, such as water appearing to flow uphill, are common in amusement parks. Magicians and con-men make their livings by creating deceptive experiences. So not all experiences are reliable. This idea is supported in an extreme sense by the Buddhist concept that ‘all is illusion’, and by Western writers such as Edgar Allen Poe (1809-1849), who wrote: “All that we see or seem is but a dream within a dream.”

Another primary difficulty in basing knowledge on experience is that experiences vary from person to person, from culture to culture, and from time to time. For instance, until the beginning of the twentieth century, experience showed that something cannot be in two different states simultaneously. Quantum physics altered that dramatically. The 2012 Nobel Prize in Physics was awarded to Serge Haroche (1944-) and David J. Wineland (1944&-) for work that included experiments that put an ion in two different energy states simultaneously. Thus, what appeared to be knowledge for millennia is no longer knowledge.

The quantum physics example raises a fundamental question: How do we know that future experiences will not negate other current ‘knowledge’ attained from experience? Closely related to this question is: How does one know which experiences yield knowledge (if any)?

One might suppose that the answer is ‘experience that is confirmed by another source of knowledge’, such as rational thought. Unfortunately, this is not necessarily the case, as the geocentric model for the cosmos illustrates. The geocentric model – with the Earth at the center of the cosmos and all celestial bodies orbiting it – was the dominant cosmological system of many ancient civilizations. This model was supported by experiences including (1) the stars, the sun, and planets appearing to revolve around the Earth; and (2) the Earth not seeming to move for someone on the Earth, and also by rational thought, as well as divine revelation (e.g., Joshua 10:12, where the Sun and Moon are said to stop in the sky, and Psalms 93:1, where the world is described as never moving). Eventually, other experiences and rational arguments, by Nicolaus Copernicus (1473-1543), Tycho Brahe (1546-1601), Galileo Galilei (1564-1642), Johannes Kepler (1571-1630) and others, showed the geocentric model to be untenable for most people. However, this does demonstrate that experience supported by other sources of knowledge does not necessarily yield knowledge. Is there an attribute of knowledge that identifies which experiences do yield knowledge? If so, I do not have even an inkling of what it may be.


We have considered several possible sources of knowledge, none of which have proved entirely satisfactory. However, our examples, mostly based on rational thought and experience, indicate several plausible attributes of knowledge:

(1) Knowledge exists;

(2) Some, but not all, knowledge comes from rational thought;

(3) Some, but not all, knowledge comes from experience;

(4) Some knowledge is absolute, while other knowledge is relative to postulates;

(5) At least some knowledge is discovered and not created; and

(6) Knowledge cannot be established by human proclamation.

We apparently need to develop further attributes to prevent the use of inappropriate postulates in rational thought and to identify which experiences yield knowledge. If these attributes exist, then we can hardly maintain the assumption that knowledge is a primitive notion.

Due to the infinite variety of life and thought provided by different cultures, I doubt the latter attributes exist. However, I fear that as advancing communication technology homogenizes cultures, formulations of such attributes identifying knowledge will appear plausible, and may enable the creation of totalitarian societies, as depicted in the novels Fahrenheit 451, Brave New World, and 1984. In short, I fear that what passes for knowledge can and will be both created and destroyed. This certainly seems to be a major goal of some political campaigns.

Furthermore, if these latter attributes are found, they would greatly reduce the importance of faith in all its aspects, and generally make life much less interesting and amusing, by removing its diversity. I hope instead that a paraphrase of Mark Twain’s observation about religion applies to knowledge: Humankind is the only animal that has the True Knowledge – several of them. If so, a judge of Knowledge is indeed shipwrecked by the laughter of the gods; and hopefully by our laughter as well.

© Roger McCann 2014

Roger McCann was a professor of mathematics for seventeen years, then spent fifteen years at the research lab of the Exxon Mobil Corporation. He is retired in the mountains of western North Carolina.


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