Proof

While I will be talking about “proof,” it is important to note that the background concept of “justification” is what is generally of interest here. As we begin our discussion, we need to note the following:

-justification generally involves appeals to (and evaluations in terms of) rules, goals, standards, or criteria—that is, there are a variety of types of justifications—among them:

--moral justification (involving appeal to moral rules, laws, or considerations),

--pragmatic justification (involving appeals to goals), and of special interest to me

--epistemic justification (involving appeals to “first principles”—or fundamental principles of reason);

-finally, justification is a matter of degree—we must distinguish between justification as “absolute,” and as a “relative.” The latter type, but not the former, is a “threshold” one.

Talk of proof and justification arises when we are discussing our rational beliefs, actions, or selves. As Peter Klein notes,

…being able to produce reasons for beliefs is a distinctive characteristic of adult human knowledge. Apparently, nothing else knows in this way. Of course, many things have knowledge that is not rational belief. Dogs scratch at doors knowing, in some sense, that they will be opened. But dogs do not have reasons. Even adult humans know (in that sense) when they do not have reasons. As Fred Dretske says, when adult humans are in Minnesota in mid-winter, they know that it is cold without having reasons.[1]

While my arriving at this room on time for the lecture may have required that I have “dog-like knowledge” of how to navigate round the campus, open doors, etc.; when I claim to know that triangles have 180°, I can be called upon to provide a proof or justification or proof for that claim. Now, of course, we all sometimes claim knowledge which we can not justify or prove:

sometimes we lie, sometimes we thought we knew, sometimes we took something on trust which we should not have….

But if these were the paradigmatic cases, the very enterprise of claiming to know would make no sense. It is like promise-making: if the standard cases of promising were akin to the politician’s promises, the very institution of promise-making would be pointless.

II. The Dominant Western Characterization of “Proof” and “Justification:”

This proof appeals to the fundamental principles of Euclidean Geometry, and for over two thousand years such deductive mathematical proofs have served as the paradigm of rigorous proof. The sorts of reasons they provide are taken to legitimate our certainty in the assertion and beliefs in question. This sort of proof has been taken as the mark of knowledge and rationality.

The Greek thinkers of Euclid’s era “began” the “development” of the now-common view of the universe as rationally ordered. That is the view that all events are “governed” by what we now call “laws of nature.”[2] As the commonly-told story goes, the gift of the Ancient Greek (and Roman) philosophers to Western Civilization was the idea of there being a fixed and rationally ordered reality.

This view was opposed to the then-common world-view (which held that any extensive inquiry into inherent plans would be inadequate, since there are unpredictable [or chance] acts [of the various deities] which do not [or at least, do not necessarily] follow any detailed or specific overall plan). The sort of reception which Socrates receives in the Athenian court, and some of the Ancient portrayals of the other philosophers of the period show that these thinkers were by no means simply relaying the common view of their culture! It would be a mistake to believe that everyone (or, perhaps, even the majority of people) in the Ancient and Roman adopted the view of the universe as a rationally-ordered thing—there was, and remains, a multitude of “takes” on this! Nonetheless, I will continue to speak of what I take to be the “dominant cultural views and ignore the underlying diversity.”

The Ancient Period was followed by the Medieval one. As was the case with the Ancient view, this world view offered a teleological conception of the universe where value and fact infused one another. Teleological explanations occur when past and present events are explained in terms of future events (they are “goal-oriented” explanations). They are often contrasted with mechanical explanations which hold that present and future events are to be explained in terms of past mechanical events and their consequences.[3]

The core modification to the Ancient teleological view is to be found in St. Augustine’s [352-430 C.E.] thought. He held that there was a transcendental, absolute, and personalistic standard against which all judgments were to be measured—one provided by the Catholic deity. In this teleology, everything had its purpose, and that purpose was supplied by this deity. While the Ancient philosophers had teleological systems, they were certainly not Catholic, and, in fact, were frequently adverse to the appeal to deities at all.

A discussion of a number of St. Augustine’s [353-403] views helps clarify this period. In his The Enlightenment: An Interpretation: The Rise of Modern Paganism, Peter Gay, maintains that at the end of the Roman Period Augustine

...recommended the gradual replacement of pagan by Christian classics, and the expurgation of all obnoxious passages from ancient literature, so his very commendation of the human understanding has a new and unclassical tone. Ipsum credere nihil aliud est quam cum assensione cogitare—“to believe is itself nothing but to cogitate with assent,” might be read (and has been read by [Christian] apologists)[4] as the demand that religious faith be tested by rational investigation. But the statement is antithetical to [the] antique [that is, Ancient]...conception of philosophy: it stresses, not the will to criticism, but the will to believe. Augustine sees man as unhappy; puzzled by himself, his world, and his destiny. All men want happiness, and all philosophers seek the way to it, but without divine aid all fail: “Thou has made us for Thyself, and our heart is restless until it rest in Thee”—this famous exclamation from [Augustine’s]...Confessions is the exclamation of a tormented soul weary of mere thought, weary of autonomy, yearning for the sheltering security found in dependence on higher powers. When Augustine speaks of understanding or reason, these words have a religious admixture: philosophy to him is touched by the divine.[5]

Augustine’s dictum stands the…method of classical philosophizing on its head: God, who to the ancients was the result of thought, now becomes its presupposition. Faith is not the reward of understanding; understanding is the reward of faith. Man may search for the explanation of his situation by his humble reason; he may even try to order his moral conduct through the understanding. But the explanation for the human condition is a myth—the Fall; the guide to his salvation is a supernatural being—Jesus Christ; the proof text for the primacy of faith over reason is a divinely inspired book—the Bible; and the interpreter of this Book is an infallible authority—the Church. All four testify to the collapse of [ancient philosophers’] confidence in man’s unaided intellect.

Hence, nisi credideritis, non intelligetis: “unless you believe, you will not understand.”[6] This injunction is the center of Augustine’s doctrine on the relation of philosophy to theology, and through its enormous authority, it became the center of medieval speculation on the same subject, although the Scholastics, as the philosophers knew, provided intellect with much room for play....But faith [for Anselm] imposed on the believer the obligation to strive within his limited means to understand what he believes. True faith is a kind of love, the highest kind of love, and a true lover does not love ignorantly: like all other medieval philosophers, Anselm accepted Aristotle’s dictum that man naturally strives for knowledge.[7]

While the Medieval concept of proof and justification remained one of deductive reasoning, it now was now placed in the Catholic context; and was premised upon a presumption that there was a complete concordance of faith and reason. The philosophically-minded theologians of this period held that the deity’s reasons were, were paradigmatically, rational. Just as this deity could not sin because it was perfectly moral, so it could not act irrationally because it was perfectly rational. As was the case with Ancient philosophical thought, so in Augustine (and for Medieval philosophers generally, proof was deductive and was to produce certainty. Thus, in distinguishing eternity from time, and in discussing the creation of the universe by the deity, Augustine says:

again, sacred and infallible Scripture tells us that in the beginning God created heaven and earth in order. Now, unless this means that nothing had been made before, it would have been states that whatever else God had made before was created in the beginning. Undoubtedly, then the world was made not in time but together with time. For, what is made in time is made after one period of time and before another, namely, after a past and before a future time. But, there could have been no past time, since there was nothing created by whose movements and change time could be measured.[8]

Augustine is wrestling here with the differences between time and eternity, and with the question of the relationship of time and creation—and all of these are central to the Catholic world-view of the Medieval [and later periods]. Note the nature of the reasoning. Here human deductive rationality is at work endeavoring to deductively comprehend the Scripture he believes!

The Renaissance led to a significantly modified world-view. An example of the “new” orientation is provided by Galileo in his “Letters on Sunspots.” In a Letter to the Grand Duchess Christina (1615) he says:

…I think that in discussion of physical problems we ought to begin not from the authority of scriptural passages, but from sense-experiences and necessary demonstrations; for the holy Bible and the phenomena of nature proceed alike from the divine Word, the former as the dictate of the Holy Ghost and the latter as the observant executrix of God’s commands. It is necessary for the Bible, in order to be accommodated to the understanding of every man, to speak many things which appear to differ from the absolute truth so far as the bare meaning of the words is concerned. But Nature, on the other hand, is inexorable and immutable; she never transgresses the laws imposed upon her, or cares a whit whether her abstruse reasons and methods of operation are understandable to men.[9]

While, for example, the theology professors at the University of Padua would have refused to look through Galileo’s telescope at the orbiting moons of Jupiter because they knew (on Medieval standards of proof) that such orbiting could not happen (it would conflict with theological faith), Galileo looked—the difference here is the presence or absence of the new orientation!

Theologians of the Medieval Period and of Galileo’s day had the authority of the Bible to rest upon—and geocentrism is deeply entrenched in the Biblical view: on the first day, according to it the deity created the Earth, and it was not until day four that the sun, moon, and other stars were created.[10] Clearly, defenders of geocentrism contend, the Earth can not be said to circle something whose existence is subsequent to its own existence. Today’s internet has many sites which continue to defend geocentrism, and they can assist us in understanding how “radical” Galileo’s views must have seemed. In one such defense of geocentrism, Gerardus D. Bouw maintains:

to hear tell, geocentrism, the ancient doctrine that the earth is fixed motionless at the center of the universe, died over four centuries ago. At that time Nicolaus Copernicus…a Polish canon who dabbled in astrology, claimed that the sun and not the earth was at the center of the universe. His idea is known as heliocentrism. It took a hundred years for heliocentrism to become the dominant opinion, and it did so with a complete lack of evidence in its favor.[11]

And the sun stood still, and the moon stayed, until the people had avenged themselves upon their enemies. Is not this written in the book of Jasher? So the sun stood still in the midst of heaven, and hastened not to go down about a whole day.

Here the Moderator of Scripture, the Holy Ghost Himself, endorses the daily movement of the sun and moon. After all, God could just as well have written: “and the earth stopped turning, so that the sun appeared to stand still, and the moon seemed to stay….” The wording would be no more “confusing” the reader than anything in Job chapters 38 through 41….

….The Copernican Revolution, as this change of view is called, was not just a revolution in astronomy, but it also spread into politics and theology. In particular, it set the stage for the development of Bible criticism. After all, if God cannot be taken literally when He writes of “the rising of the sun,” then how can He be taken literally in writing of “the rising of the Son?”[12]

This discussion helps us understand a difference between the Medieval and what is now called the “Modern” world-views: while the former “begins in faith,” the latter wants to “begin in human reason.” To get at this, ask yourself:

“What is being “rediscovered” (or re-born) in the Renaissance?” A core of the answer here will be that the Ancient thinkers’ emphasis upon unaided human reason is at the core of the “rebirth.”

This is not to say that the Modern thinkers were unreligious, most of them were clearly within the Christian tradition (though the advent of the Reformation gave them “options” unavailable to the Medievals). The contrast with the Medieval thinkers that is important for our purposes today is that the Modern starting point was not be to faith, but instead “eternal truths of human reason” which were conceived of as unchanging, universal, certain, self-evident, and accessible via unaided human reason.

While the Modern scientists wanted to recognize the importance of using our sensatory experiences to find knowledge (see the citation from Galileo earlier), however, they held that ultimately science was to be thought of along the models of Euclidean geometry. Ideally there would be rationally-guaranteed first principles which were the “eternal truths,” and scientific laws could be deduced from (or rendered consistent with) them.

Here our example will be Descartes’ cogito argument. He developed this argument specifically to address the rise in skeptical thought which accompanied the transition from the Medieval world-view to the Modern one.

Note that what he is offering isn’t, strictly speaking, an appeal to our experience. Instead he is offering a rational argument—one which contends that, upon reflection, any rational agent will rationally see that the effort to skeptically doubt her own existence must be self-refuting!

Famously, in his effort to achieve human certainty, Descartes posited the possibility that there is some evil genius who constantly works to deceive us wherever we think we have the best reasons for our beliefs. The idea is nicely captured in the movies in the Matrix series—we are left with the possibility that all we know and experience might well be illusory. Here we encounter a classic attempt to resolve one of the central philosophical problems which arises when we speak of proof or justification:

whatever one appeals to in providing such proofs and justifications, questions can arise as to what justifies these appeals.

The skeptic effectively challenges the important notions of proof and justification, maintaining that all we can legitimately have are beliefs—not knowledge. Since we can always ask for “reasons for our reasons,” we can never, truly, complete the justificatory task necessary for knowledge—we are either going to find an infinite regress, be led in a vicious justificatory circle, or stop the justificatory activity at an arbitrary stopping point.

The story I have told thus far of the dominant model of proof and justification throughout our culture, but by the time of Galileo, and especially as we move forward, appeal to our sensory experience became of increasing, then of predominant, importance. Throughout human history, of course, “seeing has been believing,” and appeals to our sensory experience has been an important form of justification and proof.

-the obvious answer to the question “How did you know the parking spot on campus you parked in was empty?” is “I saw it was so!” Like Galileo, many Ancient and Medieval thinkers practiced science! It was, however, the dominant model of proof or justification—that was clearly conceived to be deductive.

Indeed appeals to human sensory experience can not provide the level of certainty which was considered necessary for fully secure knowledge. The underlying reason for this is that our sensory experiences seem subject to far more legitimate skeptical questions than do our exercises of deductive rationality:

misperceptions, perceptual misidentifications, perceptual illusions, and hallucinations, etc. are all too common, and these serve to undermine proofs and justifications which appeal to these experiences.

Given that the dominant model of proof and justification required certainty, it seemed as if appeals to our sensory experience could not provide what was wanted. Given the difficulty of making significant progress in refuting the skeptical argument via appeals to human reason, we seem left in an awkward place as we seek rigorous proofs which will provide the knowledge we seek.[13]

-second, to argue that ultimately all attempts to provide proof or justification involve an appeal to “faith”—whether a “faith in a deity,” or a “faith in science.”

In the above discussion about “geocentricity,” we saw that Gerardus D. Bouw rejects “heliocentrism” contending that the view is “without evidence.” He doesn’t explicitly take either a skeptical stand, nor does he advance the second sort of use of skepticism. The two readings for this lecture from Michael Lynch, however, help characterize this second use of skepticism, and help us understand that the importance of discussions regarding the nature of proof and justification for our contemporary society.

This second use of the skeptical arguments puts them in the service of a view often called fideism—the view that ultimately our efforts at providing rational justification and proof must end with commitments which can not be rationally defended but, instead, are matters for faith. Ancient Philosophers worked with a presumption that the rational enterprise of proof and justification was completable, while Medieval philosophers believed that faith and reason were intimately but consistently intertwined, and Modern philosophers believed that human reason could uncover the eternal truths which were the basis of the rational universe (created by their deity). I highly recommend Colin Well’s “How Did God Get Started”[14] for a discussion of the development of the Greek perspective, and for a discussion of the development of the monotheistic idea of a deity—it provides an excellent characterization of the Ancient Greek polytheistic perspective and of the development of both the “rational” and “monotheistic” perspectives in the ancient period, as well as a compelling account of how the two perspectives (of faith and reason) developed together.

The readings I asked you to do for this class, however, turn our attention back to the beginning of this lecture and the point that proof and justification generally involves appeals to (and evaluations in terms of) rules, goals, standards, or criteria. The issue to be foregrounded finally here is with the “standards” (etc.) themselves—how are they “proved,” rationally supported,” or “justified?” This is, to mind one of the most difficult, and central questions in philosophy, and it is clearly the issue between the followers of reason, skepticism, and fideism!

In his “Thomas Kuhn, Rocks and the Laws of Physics,” Richard Rorty offers a discussion which may help set the stage for our discussions:

the trouble is that intersubjective agreement about who has succeeded and who has failed is easy to get if you lay down criteria of success in advance [and, he suggests, this is what scientists are able to do]. If all you want is fast relief, your choice of analgesic is clear (though the winning drug may have unfortunate belated side effects). If you know that all you want out of science is accurate prediction, you have a fast way to decide between competing theories (though this criterion by itself would, at one time, have led you to favor Ptolemaic over Copernican astronomy). If you know that all you want is rigorous demonstration, you can check out mathematicians’ proofs of theorems and award the prize to the one who has proved the most (although the award will then always go to a hack, whose theorems are of no interest). But intersubjective agreement is harder to get when the criteria of success begin to proliferate, and even harder when those criteria themselves are up for grabs [as they are, he suggests, in philosophy].[15]

Michael Lynch contends in his “Reasons For Reason” that many for many of the individuals who agree with Texas Governor Rick Perry’s dismissal of Darwinian evolution

…scientific evidence (or its lack) has nothing to do with it. Their belief in creationism is instead a reflection of a deeply held epistemic principle: that, at least on some topics, scripture is a more reliable source of information than science. For others, including myself, this is never the case.[16]

Lynch believes that we must deal with the differences which people have over their “fundamental epistemological standards” if we are to avoid intractable civic problems. Following David Hume, he contends that the ideal of civility requires that we conceive of a democracy as “a space of reasons,” and notes that we will never be able to agree on facts, actions, or evaluations if we can not come to some concurrence on such standards.

But many in Hume’s time rejected his commitment to reasons and the atheism he embraced. He could not secure a university job because of his views (except a brief appointment as a university librarian, which ended because of some of the purchases he approved).

In his “Science, Faith and First Principles: A Response,” Lynch maintains that although

…we can’t give epistemic reasons for epistemic first principles. We have to give reasons of a different sort.[17]

Lynch notes that many would appeal to pragmatic reasons—to the utility of scientific reason. He argues, however, for an appeal to the fundamental presuppositions of democracy, and I concur with his view. His “parallel earth” thought experiment is deeply indebted to the thought of John Rawls and Hillary Putnam, and it is an interesting argument, but I believe that he doesn’t pick up on his own core sufficiently.

Democracies rely upon having citizens who are willing to openly and freely debate issues without resting upon appeals to authorities which can not be questioned. The commitment to rational debate is so central to democracy that it must even allow for rational debate regarding the commitment to democracy. As Thomas Jefferson in his Inaugural Address in 1801: “if there be any among us who would wish to dissolve this Union or to change its republican form, let them stand undisturbed as monuments of the safety with which error of opinion may be tolerated where reason is left free to combat it.”[18] As Jefferson demonstrated, however, he had an alternative course of action ready for those who wished to depose democracy by force.

Notes: to return to the text related to the note, click on the note number.

[1] Peter Klein, “Human Knowledge and the Infinite Regress of Reasons,” in Epistemology: An Anthology (2^nd edition), eds. Ernest Sosa, Jaegwon Kim, et. al. (Malden: Blackwell, 2008), pp. 165-185, p. 168. The essay originally appeared in Philosophical Perspectives v. 13 (1999), pp. 297-325. Notes are to the reprint. Emphasis (bold) is added to the passage.

[2] I refer here to a broad period from Thales, who is widely regarded as the founder of the Ionian school of natural philosophy in the 580’s B.C.E. up to Socrates and Plato in the 400s-350s B.C.E.

[3] The contrast is well-stated by Wilber Long in his entry under “teleology” in Dictionary of Philosophy, ed. Dagobert Runes (N.Y.: Philosophical Library, 1960), p. 315.

[4] Christian “apologists” were theologians who endeavored to offer rational arguments and proofs for Christianity.

[5] Peter Gay, The Enlightenment: An Interpretation: The Rise of Modern Paganism (N.Y.: W.W. Norton, 1966), p. 230. Emphasis (bold) added to passage twice.

[6] Gay’s footnote here reads: “this much-quoted passage is from the Septuagint version of the Bible, from Isaiah, VII, 9. All other versions translate the Hebrew differently. The King James Version has, “if you will not believe, surely ye shall not be established.” The Septuagint is an ancient Greek version of the Old Testament Scriptures that was made by between seventy and seventy-two translators between 208 and 130 B.C.E. Emphasis is added to the cited passage.

[7] Ibid., pp. 230-231. Emphasis is added to the cited passage.

[8] Saint Augustine, The City of God [426], Book XI, Chapter 6, Gerald Walsh, Demetrius Zema, Grace Monahan, and Daniel Honan, trans. (Garden City: Image, 1958), p. 212.

[9] Galileo Galilei, “Letter to the Grand Duchess Christina” [1615], trans. Stillman Drake, in The Philosophy of the 16^th and 17^th Centuries, ed. Richard H. Popkin (New York: Free Press, 1966), p. 62.

[10] The Bible, Genesis I, 1-20.

[11] Gerardus Bouw, “Why Geocentricity?”, http://www.geocentricity.com/geocentricity/whygeo.html, last modified May 7, 2001, and accessed on May 5, 2011. Of course the current view can be correctly characterized as :heliocentric,” since our sun is but a tiny star revolving around the center of a small galaxy far, far away from the center of the universe.

[12] Ibid.

[13] There is much to say about the development of theories of empirical conceptions of proof and justification. This would require discussion of “British Empiricism,” “logical positivism,” the role of hypotheses, “inductive reasoning,” statistical and probabilistic reasoning and the influence of Darwinism on theories of proof and justification. This portion of the story is just too long to try and fit into this lecture. Moreover, it leads to an essential tension in the contemporary discussions of justification and proof—a tension between such accounts and the still largely dominant one of deductive reasoning. For this reason, I will leave this story untold.

[14] Colin Wells, “How Did God Get Started?”, Arion v. 18 (Fall 2010). Available online at:

http://www.bu.edu/arion/archive/volume-18/colin_wells_how_did_god_get-started/ .

[15] Richard Rorty, “Thomas Kuhn, Rocks and the Laws of Physics” [1997], in his Philosophy and Social Hope (London: Penguin, 1999), pp.175-189, p. 180. The essay originally appeared in Common Knowledge v. 6 (1997).

[16] Michael Lynch, “Reasons for Reason,” The New York Times Opinionator, October 2, 2011, available online at:

http://opinionator.blogs.nytimes.com/2011/10/02/reasons-for-reason/ .

[17] Michael Lynch, “Science, Faith and First Principles: A Response,” The New York Times Opinionator, November 3, 2011, available online at:

http://opinionator.blogs.nytimes.com/2011/11/03/science-faith-and-first-principles-a-response/ .

[18] Cited by Fritz Machlup in his “On Some Misconceptions Concerning Academic Freedom,” which is “Appendix B” of Academic Freedom and Tenure: A Handbook of the American Association of University Professors, ed. L. Joughin (Madison: Wisconsin, 1969), p. 197. Originally the essay appeared in: AAUP Bulletin v. 41 (1955), pp. 753-784. Machlup indicates that the citation is from Thomas Jefferson’s Inaugural Address March 4, 1801, and it is reprinted in The Writings of Thomas Jefferson (Thomas Jefferson Memorial Association, 1903) v. III, p. 319.

Revised on 08/27/2012

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