Colloquium talk from March 27, 2009

I've neglected this space for too long.

Here's a talk I gave at the colloquium Mary Joe put together in the 2008-2009 academic year. It is a discussion about the role of thinking in mathematics. Note to the reader: these are my notes, so there are a few indications where I would need to digress to explain some topic or other. Try not to let that trip you up :)



This talk concerns the following question: What is the relationship between mathematics and mathematical thinking. We must begin by striking out to understand what mathematics is and what it is not. We shall attempt to find such an understanding by first considering the notion generally attributed to Georg Cantor known as the or an actual infinite. Following a presentation of the mathematical thinking underlying this notion and the particular mathematics utilized in its defense we shall return to our guiding question in order to consider what we can say. We will then present the critique of this notion of an actual infinite produced by Ludwig Wittgenstein. We will find in Wittgenstein a mode of thought that places great care in the meaning of mathematical thinking and its rightful relationship with mathematics. Then, we return back again to our question of the relationship between mathematics and mathematical thinking. Following our return we shall consider the thought of Heidegger with respect to mathematics, which is nothing more than a questioning of the meaning of the mathematical.

Mathematics is taken by many to be quite difficult in character, obtuse in its progression, and in a transcendental relationship to ordinary thought. None of these qualities subsist in mathematics, though, for mathematics is nothing more than the constructions of mathematicians. Mathematics is the life-work of people thinking mathematically, which is just thinking in such a way that what is found is nothing more than that which is given. To see this in a simple exercise, consider the usual algebraic question: what number added with itself sums to four? Clearly the answer is two, and only two. Why only two? To answer this question is to transcend the place where mathematics happens and to enter the origins, the provenance of mathematics. One might respond, because we concern ourselves here only with natural, counting numbers, and these numbers are ordered such as to bring into effect the law of trichotomy (discuss). Among such numbers, those greater than two, when added with itself is bound to be greater than four, and, in the alternative non-two case, one sums with itself to two. Clearly two taken together with itself is four, and since no other such number qualifies as a natural number that can sum with itself to four, we have shown that only two can satisfy our question. Mathematical thinking requires, it is often said, that if we understand the calculus of natural numbers, and if we understand the rules of inference within a particular mathematical framework (i.e. that the natural numbers are well-ordered, and what that entails), we must come to this conclusion.

So we see at least preliminarily that mathematics is, in a sense, the result of mathematical thinking, and that mathematical thinking, in a sense, shapes mathematics.

I promised, though, a discussion of the topic of infinity. So let us turn, then, to Cantor’s actual infinite. On many occasions philosophers and mathematicians have concerned themselves with the infinite. Some of you are familiar with at least one of Zeno’s paradoxes (I can think of only the one of Achilles and the Tortoise at present, and if anyone wants to hear about that please feel free to ask). Since Aristotle there has been a distinction made between a potential infinite (a sequence of numbers that is never completed) and an actual infinite (a sequence of infinitely many elements that is completed). Aristotle thinks that infinity, rightly considered, is only potential. For Cantor, there is not only the possibility of a sequence of numbers continuing forth without end, but also the existence of a set of numbers that can be taken as a whole yet cannot be listed, even without end. We are “familiar” with a particular set of this kind that we call (because Cantor called them this) Real numbers. For Cantor, the Real numbers are real. He conceives of the real numbers as actually existing.
The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.

So Cantor says that the actual infinite is a mathematical magnitude graspable in abstracto.

Why does he say this? Because he believed that he had a proof that the Reals are not, in principle, listable and so, of a different order (cardinality) than the infinite set of rational numbers (or counting numbers). Cantor’s proof for the existence of un-listably many numbers (that is, more than one can list, even in principle) is known as the diagonal argument. Here is the idea:

We can write down a sequence of numbers that have infinite decimal expansions where not every number appearing in the expansion of any of the numbers from our sequence is all 9’s (this is because we take .99999… = 1). Then we can enumerate a table of numbers,

0.d11d12d13d14d15…
0.d21d22d23d24d25…
0.d31d32d33d34d35…
0.d41d42d43d44d45…
0.d51d52d53d54d55…
. . . . . .
. . . . . .
. . . . . .


Then we can construct a real number that is not in this enumerations by choosing 0.c1c2c3c4… where c1 is not equal to d11, c2 is not equal to d22, c3 is not equal to d33, c4 is not equal to d44, and so on. The resulting number is distinct in at least one place of its expansion from every number in our infinite sequence. Thus the real numbers cannot be enumerated. That is, according to the generally accepted conclusion, the cardinality of the Real numbers is “bigger” than that of the natural numbers (two sets of numbers have the same cardinality if they can be places in 1-1 correspondence).


So we have at least two “sizes” of the infinite: the smaller size corresponding to the natural numbers and the larger size corresponding to the real numbers. Recall, though, that we enter this discussion of the infinite purposefully. We have at the outset defined our investigation to have as its guiding concern the relationship between mathematical thinking and mathematics. We had preliminarily seen that mathematics is the construction of mathematical thinking. But Cantor’s conclusion is taken to be a part of mathematics. Consider the question: is Cantor’s a claim one on mathematics, or a claim on mathematical thinking? Does mathematical thinking require commitment to the notion of an actual, existent infinite set as the result of the diagonalization method for constructing numbers?

To address this, we turn now to Wittgenstein. Wittgenstein holds two axioms about mathematics. The first is that mathematics is a human construction. The received view is that mathematical truths are discovered and that such truths already existed before their discovery. This is the view that Cantor holds. Wittgenstein rejects this “Platonic” view as a fundamental misunderstanding of the ontology and existence of mathematics. The second principle that determines most of Wittgenstein’s philosophy of mathematics is that mathematics consists entirely of intentions, extensions, and calculi. A mathematical calculus is a set of rules for calculating (such as calculating an angle of a triangle). Extensions are finite compositions of symbols, so finite lists of numbers as well as (genuine) propositions of finite length are mathematically extended things. Intentions are rules for constructing “infinite” sequences, substitution, translation, and other rules of inference (induction).

These two principles inform Wittgenstein’s other views, and five important consequeces can be drawn from these principles. The first is a rejection of infinite mathematical extensions. Extensions are sequences of symbols that have their meaning entirely syntactically and exist as they are constructed within the activity of mathematicians. Any sequence of symbols must, then, terminate after only finitely many places or steps according to a rule. So an infinite mathematical extension is, in principle, impossible. The extended is always only known to a finite extent, though constructively within such a domain.

The second consequence we can draw is a rejection of unbounded quantification, i.e. quantification over an infinite domain. The only mathematical objects that exist are extensions and these we said are necessarily finite. Since the only mathematical objects we may quantify are finite (and so can be taken as a whole), we cannot quantify over an infinite set (because it is never whole).

The third consequence is that all genuine mathematical propositions are algorithmically decidable or undecidable. That is, a finite string of symbols can only constitute a genuine mathematical claim if it has been shown to be true (i.e. there is a proof) or false or how to decide the proposition with a known decision procedure. This follows from the fact that propositions are, generally, finite conjunctions of sub-statements, each of which is itself decidable (because it is characteristic of mathematical propositions that they stand as decidable).

The fourth consequence is an anti-foundationalist account of irrational numbers, which holds that irrational numbers are rules that when carried out, yields a finite decimal expansion which is itself a particular extension of the rule.

The fifth consequence is a rejection of definite infinite cardinalities. The last consequence constitutes Wittgenstein’s rejection of Cantor’s conclusions regarding the diagonalization method.

To see why Wittgenstein holds this view, consider what Wittgenstein is getting at in the fourth consequence I listed. Irrational numbers are rules. Rules are categorically distinct from extensions. That is, irrational numbers are not numbers. They are not objects, but an unending computation of some particular rule. Let us return to Cantor’s diagonal. We are to construct a number of infinite extensions that does not occur on our list. The irrational thus constructed is taken to be a number, an object in its own right that may be considered whole and quantifiable. But if Witgenstein is right, Cantor has not constructed a mathematical object, but instead defined a rule. We have not extension, then, in this case; just intention.

Moreover, on Wittgenstein’s view, so long as we have constructed a number using Cantor’s method, that number has a finite decimal expansion followed by an indicator that we’re not yet done. Since any entry of the table will have n terms in its expansion, for some n, we know that there are n terms in the table, and we are free to add Cantor’s constructed number to our list. So long as we heed Wittgenstein’s distinction and do not conflate extension with intension, then we can assert that an actual infinity is never actualized. We can maintain the single notion of a potential infinite.

Let us step back, at this point, and consider what Wittgenstein is objecting to in Cantor’s assumptions. Both men see the diagonlization method as being a series of valid moves, as good mathematics. What Cantor and Wittgenstein disagree about is what this method tells us. In other words, what have we learned from what we have constructed? Since we have indeed constructed something, we can say that there is disagreement between the two as to what is learned from what we have been given (albeit via our own construction?).

This is a disagreement about the mathematical. According to Heidegger, the mathematical is what can be learned from what one already has. Wittgenstein and Cantor disagree about what diagonalization tells us because they do not share the same presuppositions. Cantor is eager to have the absolute show itself. Wittgenstein determines that the finitude of mathematics derives from its being a human endeavor. These are radically different origins or foundations from which mathematics unfolds. From these varying axioms or guiding principles, Wittgenstein and Cantor (with is robust set theory) diverge on what is the same tactile situation. The evidencing of diagonalization occurs in drastically different ways, and the evidence, informed by the mathematical project of both men, becomes something other than itself. The evidence is something other than a method for constructing a sequence. It becomes operational in the correctness of both mathematical projects, and its truth is covered over by the initial projection of the essential determinations of each projector. Which is just to say: the truth subsists and is carried along in the praxis of projecting. Truth, though, always subsists. Within the project, correctness reigns, and truth lies underneath, covered over the by the work of the mathematical thinker.

All of this is simply to repeat a thought from Heidegger, but not for the sake of repetition. The repetition has its place for us vitally, for what is more important at our present stage then the necessity of understanding genuine, foundational disagreement? What have we shown ourselves, what have we seen, other than the damnation of not being honest about the origins of our reasoning and of not striking out to, at bottom, reconcile the pace between origins? Reason-giving is itself a practice whose force and validity comes from itself. Its power, beyond this, is instituted in history, but this should not be confused with some kind of necessity. For, where reasons do not come forth there is not, then, “merely nothing”. For in reasons’ stead may come a more primordial force that we take care not to name, an immutable groundswell… Yet this we must attend, and we fail to do so at great peril. We must attend the groundswell, in our way, which is, at least in one way, by giving reasons, by setting forth to find that which was already there, but whose disclosure and existence must be worked into place.

And here we return to our guide, the connection of mathematics and mathematical thinking. The connection is this: that mathematics is worked into place. This, not out of nothing, but out of the fundamental projection of a determination of the essence of beings as such, from something given. A connection set in place with thanks to a resolve to give reasons. This is a connection of the happening of mathematics, and its particular manner of coming to be.

Stanley Fish

Here is a link to Stanley Fish's column in today's New York Times.

The column seems to be a concise recounting of Terry Eagleton's "Reason, Faith, and Revolution", which, if Fish's portrayal is accurate, is a religious apologetic.

The column mostly veers into predictable arguments that our current "liberal", science-centric world is so off-base precisely because we've failed to ask (this is where the blood boils) the appropriate theological questions, such as "Why is there anything at all?", "Why what we do have is intellectually intelligible to us?", and "Where do our notions of explanation, regularity and intelligibility come from?"

These questions are clearly not theological if one asks them vitally, i.e. in their questionableness. For both Eagleton and Fish feel they may confront (and answer) these questions through religion or some other form of "spirituality" rather than appreciating these questions in their questionableness (i.e. as philosophical quanderies that are only answerable by accepting them as real grounds in the first place, i.e. by presupposing their truthfulness). These are questions that don't have answers. These are questions that reveal the terra firma underlying people qua communal beings. Our notions of explanation, regularity and intelligibility "come from" the same well as our notions of truth, power, and agency (another terrible parenthetical remark from Fish: "although how there can be agency before there is being and therefore an agent is not explained[?]" Being imply agency! Ontology implies economic rationality! But its those damn "rationalists" that are the confused/wrong-headed ones!).

The part that really stands out, though, is:

When Christopher Hitchens declares that given the emergence of “the telescope and the microscope” religion “no longer offers an explanation of anything important,” Eagleton replies, “But Christianity was never meant to be an explanation of anything in the first place. It’s rather like saying that thanks to the electric toaster we can forget about Chekhov.”

Then

And as for the vaunted triumph of liberalism, what about “the misery wreaked by racism and sexism, the sordid history of colonialism and imperialism, the generation of poverty and famine”?

So Eagleton (and presumably Fish) overlook the persecution of Galileo that was conducted in order to preserve the rule of the fiath that God's people were the physical center of the universe (along with countless meddling by the church into religion and the current politicization of science because of religious influence) is to be overlooked while we ascribe "the misery wreaked by racism and sexism, the sordid history of colonialism and imperialism" and "the generation of poverty and famine" to be consequences of "liberalism"? All of these phenomenon regularly appearing throughout human history points to the following question: Are these guys shitting me?

Junta

Lori and I held a discussion the other day centered on the meaning of the term "junta". Lori thought junta was a word the roughly had to do with countries and their militaries. I said it was the ruling military class in a class system ruled by militaries (?informative?).

I sat down this afternoon to read before I go to bed and I stumble across some newshoggerMyanmar where, to say the least, very terrible conditions continue to be endured. As a small detail necessary for any column, one must note that Myanmar is run by a group that ended Democratic rule in 1962 by a coup d’état. This group is commonly and, as we will see, correctly referred to quite often as a junta.

At any rate, I saw the word and I looked it up on my dashboard dictionary, just to make sure everyone knew what was up.


jun*ta ([this is me trying] howntuh, juhntun)
noun
1. a military or political group that rules a country after taking it over by force : the countries' ruling military junta.
2. HISTORICAL a deliberative or administrative council in Spain or Portugal.

ORIGIN early 17th century. (sense 2) : from Spanish and Portuguese, from latin juncta, feminine past participle of jungere 'to join'.


Hmmm... juncta is the feminine past participle of the term 'to join'. This, at its inception, had to do with deliberative or administrative councils in Spain, but were of a kind as bound to a military-political complex ruling an entire country. Moreover, junta is a "joint", it is something that "joins together". The joint that brings-together is very much a theme of Heidegger, and it is interesting that the Spanish have the term as it is used in the manner that they do given their general affection for his philosophy (which is quite understandable).

But just look at the HISTORICAL part. The Iberian Peninsula always intrudes upon the world as the goto for rough and rowdy.

"A Preliminary Sketch of Being-in-the-World, Section I: Being-in" by Hubert Dryfus

The opening section of the third chapter of Hubert Dryfus’ treatise on Being and Time concerns itself with explaining the “being-in” of Dasein. Dryfus tells us “Heidegger calls the activity of existing, ‘being-in-the-world.’” (40) He then sets out distinguishing “being-in-the-world” as an existential concept rather than the more common reading, one might suspect, of the term as referring to a metaphysical concept. As one might suspect, a good chunk of this explanation turns on his discussion of the “being-in” part of “being-in-the-world”.

Dryfus begins by noting the priority with which we treat “in” as it refers an object contained within another object in space. He goes on to juxtapose this common interpretation of the preposition against the “primordial sense of ‘in’ [which] is ‘to reside’, ‘to dwell’.” (42) To dwell and reside are active ways of being, Operating with this understanding we are faced with the problem of being first, and objective in-ness only afterwards. Thus the usual manner of understanding “in” as a relationship between objects is not basic but subsidiary to being-in-the-world.

This distinguishing exercise is supplemented by a discussion on the distinction between what Dryfus calls the metaphorical/literal distinction in language. One reading of Heidegger’s concept of being-in has it that “in” is used metaphorically. This view has it that “being in trouble” is metaphorical since one cannot literally be in trouble: trouble is not one thing in which another can occupy space. This reading misses the take-home message regarding being-in as relating to residing and actively engaging. “Heidegger wants us to see that at an early stage of language the distinction metaphorical/literal has not yet emerged.” (42) Being in trouble is not only metaphorical, it is also contextually definable in the being which finds itself troubled in the world.

From here Dryfus turns towards relating being-in to being-in-the-world. “Being-in as being involved is definitive of Dasein.” (43) As such, Dasein is by defined by its involvement in a world. Further, “Dasein alone can be touched, that is, moved, by objects and other Daseins,” and it is due to the involvement of Dasein with objects and other Daseins that Dasein comes to acquire know-how. “Not only is Dasein's activity conditioned by cultural interpretations of facts about its body, such as being male or female, but since Dasein must define itself in terms of social roles that require certain activities, and since its roles require equipment, Dasein is at the mercy of factual events and objects in its environment. (44)” This last point is what pushes us from being-in to being-in-the world, what we have translated as being-alongside and what Dreyfus calls being-amidst: “What Heidegger is getting at is a mode of being-in we might call "inhabiting." When we inhabit something, it is no longer an object for us but becomes part of us and pervades our relation to other objects in the world.“ (45) This pervasive feature of my objects as they relate to my being and other beings has been obscured by the tradition, even though it “is Dasein's basic way of being-in-the-world.” (45)


"A Preliminary Sketch of Being-in-the-World, Section I: Being-in" by Hubert Dryfus in Being-in-the-world : A Commentary On Heidegger's Being and Time, Division I, Cambridge: MIT Press, 1991.

Response to Husain Sarkar

Question: Taking into account sections II and III of Chapter I as well as the prisoner’s dilemma, how would you as an individual scientist structure your society of scientists?


In practice the crux of this question turns not only on the structure I choose for my group of scientists but also on the capacities of a single individual scientist amongst his fellow scientists. I assume we are to overlook this practical problem.

In theory the answer to the question depends on the approach: one approach would be to give my (rather naïve) particular appreciation for what constitutes the best structure for a society of scientists. Another approach, the one I am inclined toward, is to do a study of many individual structures and combinations of those structures against one another in some principled manner in order to discover the optimal structure for a group of scientists. My preference rests with my inclination that what I fancy as the best structure of a society will in all likelihood differ from one of my colleague’s favorite structure, and so going about things in that route will bring me to a political impasse. That is, the former approach is subjective in the worst sense.

The problem with the latter approach lies in analyzing structures. The intuitive method would be to lay out each structure in some list, choose one and compare it against the very next structure on the list. Whichever structure turns out to be better is next compared to the third structure on the list, then the victor of the this comparison is compared against the fourth structure, and so on until we have finished the list. One potential problem with this approach lies in the assumption that there are finitely many structures (and hence combinations of these structures) in principle (since this approach is theoretical). If there are infinitely many structures then our work is never done: for any finite list of structures our analysis will give us a victorious structure (or group of structures), but we must then compare it to one of the infinitely many structures that did not appear on our list, and again ad infinitum, in order to arrive at the best structure. So there’s one hitch. Another, as I alluded to above, is the problem of choosing between two structures that yield equal scientific results. This problem may never arise (note: the previous problem cannot arise in its principled form; it is more-or-less a specter over the entire method), but it remains a problem for the theoretician.

The big assumption of this approach, though, is that one can compare scientific structures, which according to Kuhn is impossible. This assumption is a playing field for further problems: what is to be compared in scientific structures, results or principles (or norms if you prefer)? If we answer “results” then how are we to valuate the results of any member of a set of theories? Here the answer is always “against some standard”, but implicit in this response lies a presupposed standard. I will leave this regress here, but I trust one will note that we need not. Alternatively if we answer “principles” then we are left with the same problem as we encountered when we responded “results”: what is the standard to which we evaluate principles? Again, this is necessary if we are to compare structures. That given two structures an individual scientist can respond to the question “Which structure is best?” with “theory a” or “theory b” does not get us out of this theoretical impasse. Unless there is an underlying principle or method for determining which response the scientist should make we are left with the problem of comparing scientific structures theoretically.

So the answer to the question for my part has to be that at the moment I am completely unable of choosing a structure for scientists in a non-ad hoc way. So let me choose the option I am most fond of right off of the top of my head: structure the group in accord with the principle that each scientist should practice in their own specialized field only as much as their scientific principles that are reconcilable across scientific domains allow, and to otherwise work with one another to resolve conflicts between principles that are irreconcilable over scientific domains. This principle intends for a homogenous scientific edifice.

The Tragic Sense of Life; Ch. IV

"And it was around this dogma, inwardly experienced by Paul, the dogma of the resurrection and immortality of Christ, the guarantee of the resurrection and immortality of each believer, that the whole of Christology was built up." pp. 56

"And the end of redemption ... was to save us from death ... or from sin [only] in so far as sin implies death." pp.57

"[W]hich is the real Christ? Is it, indeed, that so-called historical Christ of rationalist exegesis who is diluted for us in a myth or in a social atom?" pp.57

"Athanasius had the supreme audacity of faith, that of asserting things mutualy contradictory..." pp. 58

"In truth, [Catholic dogma] drew closer to life, which is contra-rational and opposed to clear thinking. Not only are judgements of worth never rationalizable -- they are anit-rational." pp.58

"Fundamentally [the Sacrament of the Eucharist] is concerned with ... the eating and drinking of God, the Eternalizer, the feeding upon Him." pp.59 This passage comes off as hellishly metal to me.

Funny quote from St. Teresa (one of 'em; not ours): "for I had told him how much I delighted in Hosts of a large size. Yet I was not ignorant that the size of the Host is of no moment, for I knew that our Lord is whole and entire in the smallest particle." pp.59-60

"It was from Kant, in spite of what orthodox Protestants may think of him, that Protestantism derived its penultimate conclusions--namely, that religion rests upon morality, and not morality upon religion, as in Catholicism." pp. 60

"For my part, I cannot conceive the liberty of a heart or the tranquillity of a conscience that are not sure of their perdurability after death." pp. 62

"[T]he highest artistic expression of Catholicism, or at least of Spanish Catholicism, is in the art that is most material, tangible, and permanent ... in sculpture and painting, in the Christ of Velasquez, that Christ who is for ever dying, yet never finishes dying, in order that he may give us life." pp. 62-3 I love that last part.

"No modern religion can leave ethics on one side." pp. 63 How many contemporary ethical theories have left religion to the side?

A doubtlessly honest yet disturbing comment: "And Christ said: "Father, forgive them, for they know not what they do," and there is no man who perhaps knows what he does. But it has been necessary, for the benefit of the social order, to convert religion into a kind of police system, and hence hell." pp. 63, emphasis added

"The gravest sin is not to obey the Church, whose infallibility protects us from reason." pp. 64, yet who will save us from the Church if that is where our faith lies? No one. No some. Only natural destruction in such a case, and we are bound to experience that!

Wow: "The Church defends life. It stood up against Galileo, and it did right; for his discovery, in its inception and until it became assimilated to the general body of human knowledge, tended to shatter the anthropomorphic belief that the universe was created for man." pp. 64, is there any wonder why conceptual and technological advances were in such short supply during the Middle Ages and the reign of the Church? And this is a terribly intelligent theological perspective.

"Do not the Modernists see that the question at issue is not so much that of the immortal life of Christ, reduced, perhaps, to a life in the collective Christian consciousness, as that of a guarantee of our own personal resurrection of body as well as soul?" pp. 65, I think Harry Frankfurt has some pertinent material concerning what is playing out in the background of this comment.

"Here you have the Catholic hall-mark--the deduction of the truth of a principle from its supreme goodness or utility. And what is there of greater, of more sovereign utility, than the immortality of the soul?" pp. 66

"[Religion] feared the excesses of the imagination which was supplanting faith and creating gnostic extravagances. But it had to sign a kind of pact with gnosticism and another with rationalism; neither imagination nor reason allowed itself to be completely vanquished. Amd thus the body of Catholic dogma became a system of contradictions, more or less successfully harmonized." pp. 68

Tragic Sense of Life; Ch. V

"The truth is ... that what we call materialism means for us nothing else but the doctrine which denies the immortality of the individual soul, the persistence of personal consciousness after deat." pp. 71

"And there is nothing that remains the same for two successive moments of its existence. My idea of God is different each time that I conceive it. Identity, which is death, is the goal of the intellect. The mind seeks what is dead, for what is living escapes it; it seeks to congeal the flowing stream in blocks of ice; it seeks to arrest it. In order to analyze a body it is necessary to extenuate or destroy it. In order to understand anything it is necessary to kill it, to lay it out rigid in the mind. Science is a cemetery of dead ideas, even though life may issue from them. Worms also feed upon corpses. My own thoughts, tumultuous and agitated in the innermost recesses of my soul, once they are torn from their roots in the heart, poured out on this paper and there fixed in unalterable shape, are already only the corpses of thoughts." pp. 80

"To think is to converse with oneself; and speech is social, and social are thought and logic. But may they not perhaps possess a content, an individual matter, incommunicable and untranslatable? And may not this be the source of their power?" pp. 80, perhaps this is a key feature of natural, as to compared to, say, formal, language.

"Hence it follows that the theological or advocatory spirit is in its principle dogmatical, while the strictly scientific and purely rational spirit is sceptical, that is, investigative." pp. 81-2