Mathematics, famously, starts from assumptions, like the definition of a circle or the parallel postulate. Of these, definitions and common notions seem to be no more than features of the terrain in which one works. If one has points, lines, and planes, one will have circles, so that, if it is useful to recognize and label them, one is allowed. Or, if one has numbers, one can designate a unit and designate the incremental operation of counting, since they are there within the concept of number, or perhaps it is better to say that they are necessary in generating the concept of number.
But among mathematical assumptions, those called axioms are of particular interest because they seem to be chosen or constructed, rather than simply recognized or acknowledged.
Recognizing that an axiom is constructed, chosen as one possibility from among a set of possibilities, is not always evident from the get-go. It took over 20 centuries to realize that the parallel postulate was a choice and not a necessary starting-point.
Let us now distinguish between primary mathematical assumptions, like the parallel postulate, called axiomsand secondary mathematical assumptions, called conjecturesbefore they are proven and theoremsafter.
Taking this a little further, mathematics and the natural sciences each depends on two kinds of hypotheses, foundational hypotheses, the rules of logic for mathematics and the scientific method for science (including the principles of causalityand measurability), and provisional hypotheses, the conjectures of mathematics and the empirically-testable hypotheses of science.
Now, in between these, mathematics also has axioms like the parallel postulate, assumptions that enable particular mathematical systems and can be peculiar to those particular systems, as is the parallel postulate to Euclidean geometry. The nearest to that in science would be fundamental concepts of a discipline, like matter and energy in physics, the atom in chemistry, or organisms and cells in biology. But those have to be discovered and proven empirically, while axioms need only be posited to see if they are productive of provable theorems. In other words, axioms are provisional hypotheses, not of particular theorems, but of whole mathematical systems of theorems.
The equivalent in science would be something like other universal laws of physics within other universes, or, say, ecosystems on alien planets based on silicon rather than carbon.
Sand, typically silicon-based
Oddly, a planet with too much carbon would likely be waterless, and lifeless, too, as water is “the solvent essential to life as we know it”, according to Jonathan Lunine of Cornell University. (I tell you this because I couldn’t find a cool picture of what silicon-based life forms might look like.)
Carbon Worlds May Be Waterless, jpl.nasa.gov
This reflection on assumptions (i.e., hypotheses that go beyond the evidence) was triggered by Angie Matute’s comparison of provisional hypotheses as used in science and mathematics, in answer to the Prescribed Title prompt: “The production of knowledge requires accepting conclusions that go beyond the evidence for them.” Thank you, Angie.
After having put in a long day at one’s job or one’s studies, most of us want to spend what time we have left with our family, our friends, our pastimes, doing the things that make life new and exciting (some of the time) and warm and comfortable (the rest of the time).
So no one’s going to kill themselves examining claims of knowledge that might force us to curb our pleasures (e.g., smoking and vaping are bad for you), our livelihoods (e.g., hydrocarbon fuel use contributes to global warming), or our sense of who we are, and what group we belong to.
And the group many of us see ourselves as belonging to is no longer Americans, but the Blue Man Group or the Red Man Group.
Blue Man Group: the strong should care for the weak, anything called green is good; and at its fringes: democratic socialism works, the police are unreliable, America is a global aggressor.
Red Man Group: democratic socialism doesn’t work; and at its fringes: we’re the ones who belong here, we made this country, our family values over everyone else’s, the police serve and protect (us), American is special and can do no wrong.
In polarized political atmospheres, mindsets double-down, and the spectrum of viewpoints is compressed into two opposed ideological mindsets.
In ideological mindsets, partial truths and half-truths become articles of faith, oversimplification becomes reality, and those who disagree are demonized.
Balance and pragmatism are whittled away between extremes hell-bent on forcing everyone else to choose sides.
There are comfortable beliefs matching those two mindsets (or any other). For what is a sense of identity but a claim about what kind of person one should be — and, by inference, is.
People mostly seek out comfortable beliefs, and only secondarily true ones, with a crucial exception. People do seek out true beliefs where the consequences of falsity recoil upon them painfully, quickly, and undeniably.
The biggest difference in this regard between age and youth, is that the older you are, the more likely you are to value lasting long-term consequences above fleeting short-term ones. This explains the gulf between generational mindsets about sex, drugs, and rock-and-roll.
We all like to think that whatever group we accept (and that accepts us) is the right group, whether we chose it or were born into it.
And everyone believes in that part of science, i.e., technology, which increases their powers. Yes, you’ll find red zone climate-change deniers and blue zone anti-business activists, but none of them are throwing away their smartphones. If you can use it, it’s gotta be true; that part of science everyone loves! Too bad science isn’t a buffet, where you pick and choose only the dishes you like!
But science is an all-or-nothing proposition. Accept the unpleasant truths along with the pleasant ones or you’re not entitled to any of it.
Like that will stop you! Too many people treat science the way a controlling boyfriend/girlfriend treats their partner — like a project, accepting their positive features, while reserving a limitless right to edit or redact their undesired features!
So, what makes for a comfortable belief?
Comfortable beliefs are those that tell you to go right on doing what you are already doing (in other words, keep doing what pleases, enriches, or otherwise benefits you, because it’s not just to your advantage, it’s true, by Golly!) You’ve hit the trifecta: it pleases you, it pleases the group you want to fit in with, and it’s true! You’ve found the ultimate triple-strength sugar rush: dopamine, peer pressure appeasement, and the Absolute!
But, as Nietzsche says, “truth has no survival-value”.
You can always find the Easy Way Out — it’s where the crowd is.
Blue Man Groups and Red Man Groups have different Easy Ways Out, but red and blue are equal in their devotion to their own Easy Way Out.
Me? I’m taking refuge behind this George Orwell quote: “If liberty means anything at all, it means the right to tell people what they do not want to hear.”
John Locke memorial stone, Christ Church Cathedral, Oxford: I know there is truth opposite to falsehood that it may be found if people will & is worth the seeking.
Next time you’re in the Auditorium, notice the flag of the great seal of Florida, to your right, when seated.
The seal shows a steamboat with sails (the hybrid engine of the 1800’s) approaching a shoreline upon which stands, barefoot, a lone native woman, wearing a floral blouse over a long skirt.
A common misconception about scientific proofs runs as follows. The theory — say, evolution — involves too many ideas and relationships that are still unproven to be considered proven. Therefore, we’re free to entertain all kinds of rival theories that are also unproven.
But a theory is a worthwhile claim (meaning one that is not obvious or tautological) about a set of coherent connections and relationships among concepts, ideas, and the like. How does one measure that? One can only measure empirically testable predictions extrapolated from the theory; only these predictions are proved or disproved, not the theory itself.
In short, though theories themselves are by definition not provable, that is a red herring. The significant difference is between theories capable of generating testable predictions (e.g., the theory of evolution) and those not so capable (e.g., creationist theory). Moreover, such “unprovable” theories as biology’s theory of evolution and physics’ Standard Model have generated impressive numbers of successfully confirmed empirical predictions.
Those who claim that these, or other productive scientific theories are unproven are simply playing with words they don’t understand. They should be gently corrected and enlightened, and, if resistant to reason, they should be left to play in their pseudo-intellectual sandbox with their peers. As Upton Sinclair said “It is difficult to get a man to understand something when his salary depends upon his not understanding it.” I would amend this to include his self-image and his hopes, along with his salary.
Now, of course, a theory that generates many empirically provable predictions looks like a true theory, and is certainly a fruitful one, even if it is eventually disproven, or rather, exposed as a useful approximation that falls short of the truth itself. This is what happened to Newton’s theory of gravity , one of the most productive theories in the history of science, and even now, in its “disproven” state, a close enough approximation to the truth that it is used for moon shots, since Newtonian mathematics are much easier to calculate than their Einsteinian replacements.
Still, if we send a space mission outside our little solar system, we must use Einsteinian calculations if we want to hit our target, but for short hops like those to the planets within our solar system, Newtonian calculations are good enough!
By open-ended interpretation and analysis I mean examining arguments (that support positions or claims) in a way that carefully distinguishes between those parts of an argument that follow from its premises and assumptions (or from facts or evidence acknowledged by consensus) and those that do not follow necessarily from those premises and assumptions.
In other words open-ended interpretation and analysis seeks to determine what arguments and positions are both coherent with and follow necessarily from their premises. It’s like logic, but argumentative logic applied to disputable claims (the ones that matter), as opposed to the formal logic devoid of content, an abstract discipline closely related to formal mathematics.
I’m going to simplify this phrase to dialectical interpretation and stipulate its definition as the examination of arguments with a view to what follows necessarily from their premises from what does not follow necessarily from those premises.
Why all this fuss about a name? Because an unwieldy phrase of several words, some of them multi-syllabic and abstract, gets left by the wayside. So, shorten it.
But words like technical terms in philosophy have histories that are often contradictory and confusing, and dialectic is just such a technical philosophical term. If you are familiar with the confusing history of this term, or curious about it, follow this link: A Brief Note on the History of the Technical Philosophical Term “Dialectic”
Dialectical interpretation is required to examine any argument making any claim to its foundations and hidden assumptions. But it is laborious, it requires lengthy training, and it is not lucrative when viewed as a profession. So, few people bother. (Irony alert) Go figure.
Examples always help. I’ll use a simple example, so we can focus on the procedure rather than the content.
John Locke, in his Second Treatise on Government, routinely justifies his more democratic and republican take on Hobbes’s schema of the social contract in parallel arguments draw from both Reason and Revelation.
This leaves open the possibility that either one or the other of the two parallel sets of justifications is superfluous. But if Locke is careful to consistently state his case using both forms of justification, a reader using dialectical interpretation must remain open to three possibilities, none of which are excluded by the form of Locke’s arguments: 1) that Locke was more firmly committed to Reason, 2) that Locke was more firmly committed to (Christian) Revelation, or 3) that Locke viewed them as mutually intertwined and mutually supportive.
Incidentally, my own interpretation is that Locke is pious, yet deliberately fashions his arguments to appeal to distinct audiences, pious and secular; this probably means that he supports the third option.
But the argument, as Locke presents it, supports any of the three interpretations, while excluding none of them, and therefore requires the reader to maintain an open mind (whatever their personal inclinations); this is as close to bias-free interpretation as one can get.
Setting aside the third possibility, that the two differing interpretations merge into one, we may say that if both interpretations fit with the uncontested “facts” of the text — that is, the stated positions, definitions, distinctions, arguments and principles — then a particular interpreter’s preference for one or the other is arbitrary and unsupported.
Nonetheless, either of these interpretations can, if provided with independent support, be argued as a “schismatic” departure from the orthodoxy of the text as advocated by the author being interpreted.
But what if one’s preferred interpretation is the one the author really intended? How can one not claim the “right” to insert that interpretation into the text?
This begs the question, assuming that the interpreter has privileged access to the author’s true intent, even though the text fails to decisively exclude the unwanted interpretation. Rather than thus assuming that the interpreter has direct access to the author beyond the text, the honest (i.e., dialectical) interpreter will say, even if that were so, the author has failed to make it necessary within the text, and it is only by the text that the author’s thought can be judged. In short, no interpreter can claim privileged access to the author’s intent not supported decisively by the text. For the text is the only “empirical” basis for consensus between divergent interpreter’s; only the text, as given, makes those interpreters answerable to one another. See Rules (Rule III.2: Distinguishing Core Structure from Authorial Preference)
So, dialectical interpretations are open-ended in this sense, like flow-charts with forking junctures at decision-points.
And these interpretive forks often lead to further forking junctures, creating downward-branching decision trees.
The dialectical interpreter must be able to keep all these possibilities in play until they are eliminated, converge, or remain indefinitely as rival possibilities. See Rules (Rule II.5: Probing for Seams)
But argument junctures aren’t always simple binary choices. Sometimes there is a range of interpretations from the more conventional to the more radical. In these cases, one must define the range of possible interpretations by these extremes. This is what I call weak form to strong form interpretations. See Rules (Rule I.1: Exposing Weak Form/Strong Form Equivocation).
Dialectical arguments, thus, and their interpretations, are like flow-charts, with stretches where the author decisively channels thinking into a single path, but also with junctures where the argument cannot foreclose alternative choices, whether by authorial design or inadvertently.
From this perspective, reading a seminal work like Locke’s is not a chore of servitude — it is an exploration and an adventure in thinking!
The term dialectic as used nowadays typically refers to either its Marxist or Hegelian uses. I need hardly tell you who uses it in its Marxist sense. It is used in its Hegelian sense in Continental philosophy, which, as the name implies, is taught primarily on the European continent, as opposed to Anglo-American philosophy (as it is referred to on the continent), but which we here in Anglo-America call Analytic philosophy.
Analytic philosophy dominates American university and college philosophy departments, except for Catholic ones (e.g., Notre Dame, Georgetown, Boston College, etc.), which prefer Continental philosophy because it leaves room in the universe for God. For Analytic philosophy is essentially the handmaid of empirical science, and is concerned with establishing the foundation of empirical science against any form of metaphysics, and with establishing the precedence of the knowledge about observable objects produced by the empirical sciences (or by mathematics) over any disputable claims of knowledge produced by subject-centered disciplines (e.g., social sciences, history, the humanities, or Continental philosophy), that is, the liberal arts as opposed to what are now called the STEM studies. This distinction is roughly captured in the common distinction between soft sciences and hard sciences (one guess which side of the dispute coined those terms).
In non-sectarian American universities and colleges, you’re likely to find Continental thought, not in the philosphy department, but in the more culture and language focused departments: languages and literatures, regional studies, feminist studies, African American Studies, and other studies focused on cultural minorities, but also mainstream sociology, psychology, political science, history, and so on. It usually appears in the form of some particular school of thought, such as postmodernism, existentialism, or structuralism.
I am using dialectic in its original sense, as used and practiced by Socrates, Plato and Aristotle. The key to undestanding that orginal sense of the term is that in Greek dialektike means “related to dialogue,” which in turn comes from dia logos meaning “through words”. It means: 1) the form in which Socrates conducted his teaching, 2) the literary form in which Plato recorded — and likely transformed somewhat — Socrates’s teaching, and 3) the procedural form in which Aristotle examined opposing ideas, treating them as theses and antitheses to be compared interactivley against one another.
This meaning is not to be confused with Hegelian dialectic, which Hegel conceived of as conflicting conceptual paradigms working themselves out through human history and its reflection in the history of philosophy.
Georg Wilhelm Friedrich Hegel (1770-1831)
It is also not to be confused with Marxist dialectic, which Marx conceived of as conflicting modes of production, and of the social classes they give rise to, bloodily working out their class conflict through changes in the technology of production and subsequent political, cultural and historical conficts.
When All Is Said and Dunn 