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Understanding in a Flash and the Mastery of Technique

Eternity and Know-how
I’ve been reflecting on the concurrences between the excellent passage from Wittgenstein’s Philosophical Investigations below, and the Spinoza point made, which follows. Wittgenstein’s brilliant, economically, though quite gnomically put point has always been “there is no rule for how to follow a rule”. Simply so, when following a rule for action, the rule itself (nor the appeal to any other meta-rule) is not sufficient to justify the application of the rule, at least from any foundational conception of knowledge. In the passage below Wittgenstein picks up the simple mathematical example of rule-following, which he attempts to sculpt down to its most essential aspects. What is most telling is they way in which he rejects any temporal mental action/experience, what he calls a “mental process” as the source for what is called “understanding”. That is, anything that has what Spinoza named “duration” cannot be the origin of our ability to comprehend.
I’ve always looked at this passage with amazement, as we hear Wittgenstein’s mind ticking in self-conversation in his usual style, feeling a surgeon’s hands being placed right in the meat of what the much philosophically pursued “understanding” is. Never though had I connected the passage, either in influence or just in terms of content, to the similar analogy used by Spinoza to point out the difference between “rational” knowledge given by the appeal to reasons (the art of reasoning) and the preferred “intuitional” knowledge that comes out of a union with God, Substance, Nature.
Here is the respected passage from Wittgenstein. If you are not familiar with his method, keep track of the different Socratic voices:
145. Suppose the pupil now writes the series 0 to 9 to our satisfaction. – And this will only be the case when he is often successful, not if it does it right once in a hundred attempts. Now I continue the series and draw his attention to the recurrence of the first series in the units; and then to its recurrence in the tens. (Which only means that I use particular emphases, underline figures, write them one under another in such-and-such ways, and similar things.) – And now at some point he continues the series independently – or he does not.- But why do you say that? so much is obvious! – Of course; I only wish to say: the effect of any further explanation depends on reaction.
Now, however, let us suppose that after some efforts on the teacher’s part he continues the series correctly, that is, as we do it. So now we can say he has mastered the system. – But how far need he continue the series for us to have the right to say that? Clearly you cannot state a limit here.-
146. Suppose I now ask: “Has he understood the system when he continues the series to the hundredth place?” Or – if I should not speak of ‘understanding’ in connection with our primitive language-game: Has he got the system, if he continues the correctly so far? – Perhaps you will say here: to have got the system (or, again, to understand it) can’t consist in continuing the series up to this or that number: that is only applying one’s understanding. The understanding itself is a state which is the source of the correct use.
What is one really thinking of here? Isn’t one thinking of the derivation of a series from its algebraic formula? Or at least of something analogous? – But this is where we were before. The points is, we may think of more than one/ application of an algebraic formula; and any type of application may in turn be formulated algebraically; but naturally this does not get us any further.- The application is still a criterion of understanding.
147. “But how can this be? When I say I understand the rule of a series, I am surely not saying so because I have found out/ that up to now I had applied the algebraic formula in such-and-such a way! In my own case at all events I surely know that I mean such-and-such a series; it doesn’t matter how far I have actually developed it.” –
Your idea, then, is that you know the application of the rule of the series quite apart from remembering actual applications to particular numbers.  And you will perhaps say: “Of course! For the series is infinite and the bit of it that I can have developed finite.”
148. But what does this knowledge consist in? Let me ask: When do you know the application? Always? day and night? or only when you are actually thinking of the rule? do you know it, that is, in the same way as you know the alphabet and the multiplication table? Or is what you call “knowledge” a state of consciousness or a process – say a thought of something, or the like?
149. If one says that knowing the ABC is a state of the mind, one is thinking of a state of a mental apparatus (perhaps of the brain) by means of which we explain the manifestations/ of that knowledge. Such a state is called a disposition. But there are objections to speaking of a state of the mind here, inasmuch as there ought to be two different criteria for such a state: a knowledge of a the construction of the apparatus, quite apart from what it does. (Nothing would be more confusing here that to use the words “conscious” and “unconscious” for the contrast between states of consciousness and dispositions. For this pair covers up a grammatical difference.)
150. The grammar of the word “knows”  is evidently closely related to that of “can”, “is able to”. But also closely related to that of “understands”. (‘Mastery’ of a technique.)
Footnote, bottom of page 50: a) “Understanding a word” : a state. But a mental/ state? – Depression, excitement, pain, are called mental states. Carry out a grammatical investigation as follows: we say
“He was depressed the whole day.”
“He was in great excitement the whole day.”
“He has been in continuous pain since yesterday”.-
We also say “Since yesterday I have understood this word”. “Continuously”, though? – To be sure, one can speak of an interruption of understanding. But in what cases? Compare: “When did your pains get less?” and “When did you stop understanding that word?”
b) Suppose it were asked: When do you know how to play chess? All the time? or just while you are making a move? And the whole of chess during each move?- How queer that knowing how to play chess should take such a short time, and a game so much longer!
151. But there is also this use of the word “to know”: we say “Now I know!” – and similarly “Now I can do it!” and “Now I understand!”
Let us imagine the following example: A writes series of numbers down; B watches him and tries to find a law for the sequences of numbers. If he succeeds he exclaims: “Now I can go on!” – So this capacity, this understanding, is something that makes its appearance in a moment. So let us try and see what it is that makes its appearance here. – A has written down the numbers 1, 5, 11, 19, 29; at this point B says he knows how to go on. What happened here? Various things may have happened; for example, while A was slowly putting one number after another, B was occupied with trying various algebraic formulae on the numbers when had been written down. After A had written the number 19 B tried the formula a (subscript) n = n² + n -1; and the next number confirmed his hypothesis.
Or again, B does not think of formulae. He watches A writing his numbers down with a feeling of tension, and all sorts of vague thoughts go through his head. Finally he asks himself: “What is the series of differences?” He finds the series 4, 6, 8, 10 and says: Now I can go on.
Or he watches and says “Yes, I know that series” – and continues it, just as he would have done if A had written down the series 1, 3, 5, 7, 9, – Or he says nothing at all and simply continues the series. Perhaps he had what may be called the sensation “that’s easy!” (Such a sensation is, for example, that of a light quick intake of breath, as when one is slightly startled.)
152. But are the processes which I describe here understanding? “B understands the principle of the series” surely doesn’t mean simply: the formula “an =…” occurs to B. For it is perfectly imaginable that the formula should occur to him and that he should nevertheless not understand. “He understands” must have more in than: the formula occurs to him. And equally, more than any of those more or less characteristic accompaniments/ or manifestations of understanding.
153. We are trying to get a hold of the mental processes of understanding which seems to be hidden behind those coarser and therefore more readily visible accompaniments. But we do not succeed; or, rather, it does not get as far as a real attempt. For even supposing I had found something that happened in all those cases of understanding, – why should it be the understanding? And how can the process of understanding have been hidden, when I said “Now I understand” because I understood?! And if I say it is hidden – then how do I know what I have to look for? I am in a muddle.
154. But wait – if “Now I understand the principle” does not mean the same as “The formula…occurs to me” (or “I say the formula”, “I write it down”, etc.) – does it follow from this that I employ the sentence “Now I understand…” or “Now I can go on” as a description of a process occurring behind or side by side with that of saying the formula?
155. If there has to be anything ‘behind the utterance of the formula’ it is particular circumstances, which justify me in saying I can go on – when the formula occurs to me.
Try not to think of understanding as a ‘mental process’ at all. For that is the expression which confuses you. But ask yourself: in what sort of case, in what kind of circumstances, do we say, “Now I know how to go on,” when, that is, the formula has occurred to me? –
In the sense in which there are processes (including mental processes) which are characteristic of understanding, understanding is not a mental process.
(A pain growing more or less; the hearing of a tune or a sentence: these are mental processes.)
156. Thus what I wanted to say was: when he suddenly knew how to go one, when he understood the principle, then possibly he had a special experience – and if he is asked: What was it? What took place when you suddenly grasped the principle?” perhaps he will describe it much as we described it above – but for us it is the circumstances/ under which he had such an experience that justified him in saying in such a case that he understands, that he knows how to go on.
Philosophical Investigations

There are several aspects of comparison between Wittgenstein and Spinoza, but most signficantly are Wittgenstein’s “nothing is hidden” assertion, as well as the notion that the learning of a rule is very much like the “mastery of a technique”. Evocatively he places “know” to be very close to the use of “can” or “is able to”, moving us very close Spinoza’s metaphysical point that knowing isn’t a representational state, but is an action.

Here are the related example from Spinoza’s Short Treatise, followed by the same example from two other works, concerning the ability to follow a rule:

Part II, Chapter I

Someone has merely heard someone else say that if, in the rule of three, you multiply the second and third numbers, and divide the product by the first, you then find the fourth number, which has the same proportion to the third as the second has to the first. And in spite of the fact that the one who told him this could be lying, he still governed his actions according to this rule, without having had any more knowledge of the rule of three than a blind man has of color. So whatever he may have been able to say about it, he repeated it, as a parrot repeats what it has been taught.
A second person, of quicker perception [more active intelligence, Shirley], is not content in this way with report, but tests it with some particular calculations, and finding that these agree with it, give his belief to it. But we have rightly said that this one too is subject to error. For how can he be sure that the experience of some particular [cases] can be a rule for him for all.
A third, being satisfied with neither with report, because it can deceive, nor with the experience of some particular [cases], because it cannot be a rule, consults true reason, which has never, when properly used, been deceptive. Reason tells him that because of the property of proportionality in these numbers, this is so, and coud not have been, or happened otherwise.
But a fourth, who has the clearest knowledge of all, has no need either of report, or of experience, or of the art of reasoning, because through his penetration he immediately sees the proportionality in all the calculations….
Chapter II
We call the first opinion because it is subject to error, and has no place in anything of which we are certain, but only where guessing and speculating are spoken of.
We call the second belief (opinion), because the things we grasp only through reason, we do not see, but know only through a conviction in the intellect that it must be so and not otherwise.
But we call that clear knowledge that comes not from being convinced by reasons, but from being aware of and enjoying the thing itself. This goes far beyond the others…
 the Short Treatise
…Suppose there are three numbers. Someone is seeking a fourth, which is to the third as the second is to the first. Here merchants will usually say that they know what to do to find the fourth number, because they have not yet forgotten that procedure which they heard from their teachers, without any demonstration.
Others will construct a universal axiom from an experience with simple numbers, where the fourth number is evident through itself – as in the numbers 2, 4, 3, and 6. Here they find by trial that if the second is multiplied by the third, and the product is divided by the first, the result is 6. Since they see that this produces the same number which they knew to be the proportional number withou this procedure, they infer that the procedure is always a good way to find the fourth number in the proportion.
But mathematicians know, by the force of the demonstration of Proposition 19 in Book VII of Euclid, which numbers are proportional to one another, from the nature of proportion, and its property, viz. that the product of the second and the third. Nevertheless, they do not see the adequate proportionality of the given numbers. And if they do, they see it not by the force of the Proposition, but intuitively, or without going through any procedure.
Emendation of the Intellect (23)
…Given the numbers 1, 2 and 3, no one fails to see that the fourth proportional number is 6 – and we see this much more clearly because we infer the fourth number from the ratio which, in one glance, we see the first number to have the second.
Ethics, IIp40s2iv
Tool Use and Eternity
After all this quotation I want to create the picture that for both Spinoza and Wittgenstein the rules of reason were to be seen as tools (and not knowledge in their own right), techniques of action. What both thinkers shared was a engineer’s sensitivity towards the way that parts fit together. Spinoza of course was a master lens-craftsman and Wittgenstein an amateur architect and mechanical engineer (I believe he designed his own aircraft engine, if I recall correctly). What is most germane, as both thinkers reflect back upon each other, is I think the way in which Wittgenstein’s “circumstances” which act as the criteria for our justification of the rule-following of others, opens out the historical immanence for Spinoza’s view towards eternity. This is to say, or argumentation about the truths of actions find their criteria in the real, historical milieu to which they are immanent. I think that this is very much in keeping with Spinoza’s articulation (as long as we remain under the question of “justification”). In this sense, reason has a historically contingent, natura naturata manifestation and horizon. But as well, Spinoza’s window unto eternity, aside from the question of justification, makes more clear Wittgenstein’s own attachment to a denial of durative processes as the source of understanding. Instead, while historically contingent criteria, all immanent to their circumstances, may lead us to the intuitive grasp of wholes, these are mere tools in a certain, transpiercing of history through intuitive grasp (in which the issue of justification falls away). Ultimately these are human and abiotic bindings, constitutive of their mutuality, in the end causal relations of perception. In all perceptions, including those of animals and inanimate objects, nothing is hidden, nothing lies beyond.
I think that this goes onto explain Spinoza’s own reticence towards the technological innovations of Christiaan Huygens in the area of lens grinding. Huygens and his brother were brilliant experimenters with the art of lens grinding, an area of craft which Spinoza specialized in. What is of note is that Spinoza did not find the Huygens’ shiny devices of much interest, just the sorts of machines that were arriving on the scene which seemed to manifest cleanly the powers of mathematical knowledge itself. Mechanical instruments seemed to express abstract truths without the stain of human hand. But Spinoza found this the least bit interesting:

The said Huygens has been a totally occupied man, and so he is, with polishing glass dioptrics; to that end a workshop he has outfitted, and in it he is able to “turn” pans – as is said, it’s certainly polished – what tho’ thusly he will have accomplished I don’t know, nor, to admit a truth, strongly do I desire to know. For me, as is said, experience has taught that with spherical pans, being polished by a free hand is safer and better than any machine. 

Spinoza in his letter to Oldenburg refers actually to the supposedly lowest level of knowledge in his trinity, experience, as the reproof of the importance of the semi-automated machine that he saw. I don’t believe that this is a small dispositional point. Rather, just as Wittgenstein refuses any grounding of “understanding” in rules (which are tools), and as Spinoza puts “intuitional knowledge” above the truths found in the Art of Reasoning, I believe that Spinoza found the appeal of instruments themselves as abstract devices and powers (be they rules, theorems, machines) to be utterly secondary to the intuitional revelation of God and Nature itself, through those tools. His preference for the “free hand” over the automated gear turn, expressed above, is not simply a pragmatic issue (most of Huygens devices did not seem to produce usable lenses), but also a question of just what the human/technological interface involved, and the powers of its action. Instrumentality, like that would mark much of scientific pursuit, was a fetish. Maths and Science were tools used for the transformational intuition of truth, a strong de-centering of the subject, and were not truths about the world themselves.

“By mathematical attestation”: Spinoza’s Epiphantic Calculation

Just How Mathematical Was He?

I have been having an interesting conversation over time with Eric Schliesser at Leiden University who holds a minority position on the role mathematics plays in Spinoza’s position on what can be known. He strongly interprets Letter 12 towards a skepticism for just what mathematical calculation (and its attendant applied empirical observation) reveals. As Letter 12 attests, Spinoza regarded mathematics as a “product” of the imagination, come from our imaginary classifications of objects as wholly bound things – cut out from the cloth of Substance, if you will. I wrote on my agreement with this here Spinoza and Mechanical Infinities, but Eric really would like to push the interpretation so far as to restrict all of mathematical endeavours to the realm of the imaginary (the lowest forms of knowledge for Spinoza, the other two being the “rational” and the “intuitional”). This of course comes up against Spinoza’s rather obvious and profound use of mathematics as a model for philosophical investigation and even the higher forms of knowledge as both his method more geometico and his illustrations of higher knowledge both make use of mathematical forms as exemplary.

In large part I really am in agreement with Eric, in the most common Spinoza interpretations the mathematical has somehow risen far above the frame in which Spinoza intended it, but it makes very little sense for me to regard the “products” of imagination as imaginary itself – in the Emendation Spinoza speaks of the production of tools of intellection out of imaginary relations as a positive progression. Clearly mathematical description provides distinct causal understanding of the relations between things, and it is exactly in this vein that the empirical science observations of several centuries does provide a substantive remove from mere superstition, something that Spinoza firmly places himself against. It would seem that Spinoza’s true position lies somewhere in-between, not accepting Galileo’s thought that Nature is written in the language of mathematics, but also grasping that in mathematics (and observation, experiment) there are genuine increases in understanding, freedom, power, joy,  and ultimately for Spinoza, Being.

In this way mathematics is seen as:

1. Both a product of the imagination, and an aid to the imagination [Auxilia imaginationis] (Letter 12).

2. As such they are qualifiable as ens rationis which is what Spinoza calls them in letter 12 [eius modi Entibus rationis], something he is elsewise careful not to be blurred with ens imaginationis (E1App). 

3. It should be admitted that Spinoza does not help things by just prior to distinguishing “number” as a  “things of this reasoning” in letter 12 to Meyer he refers to it as  “nothing more than thinking’s, or better imagining’s, modes” [nihil esse praeter cogitandi, seu potius imaginandi, modos] – he wants to straddle the line here and a little confusedly so.  

4. But reading numbers as ens rationis (distinct from ens imaginationis), these, as Spinoza notes in the all important E2p49 asserting the collapse of volition into the concept of “idea”, are not ens reale (Cor. dem. [III. B (iii)], accept when the latter is understood as an operant, an affirmation and an action. 

5. The real and the rational abstraction that describes it are not to be confused. It is this final distinction that provides the skeptical element towards an ultimate mathematical reduction of Nature.

6. One has to live with the exegetical problem that while Spinoza in the Appendix of Ethics part I makes a strong distinction between things of the imagination, and things of reason, in letter 12 he oscillates, even within sentences, between things of (this) reason and something he undefinedly calls aids (auxilia) of the/to the imagination, never sure how he wants to describe Number.

Are Maths Only Imaginary? What Would that Mean?

By my understanding Eric places increases of power in mathematical description solely within the a “usefulness” category, all the while restricting them to the “imaginary realm”. While I really enjoy this outlier position, the very substantive nature of these increases in usefulness simply remains unexplained. And though this in part may be due to some inconsistency in how Spinoza treats the imagination (and the concept of order itself), I do think that Spinoza saw in mathematics (and scientific testing) genuine increases in the understanding of things, without acceding to the thought that mathematics genuinely reveals the eternal essences of things. For Spinoza we are, even the most scientific among us, like a “worm in blood”, not comprehending entirely the body and indeed the Universe we live in.

In this discussion there is an interesting, and indeed I think problematic sentence that at least provisionally I would like to retranslate. It is found in the Appendix of the first part of the Ethics, and in it Spinoza appeals to the very mathematical attestation by which we perceive or understand things of the world. He has just finished rebuffing two stages in thinking: addressed are those that feel that astronomically it is the motions of the heavenly bodies themselves that somehow compose [componere] a  harmony, a harmony that God delights in; and then those that from this notion of celestial harmony then find that it is the disposition of the brain alone from which human judgment comes, something which results in a skepticism of knowing in general. One is left with either a physiognomic theory of differences of perceptions (there are as many different kinds of brains as there are palates), or presumably on the other end the “veil of ideas” of proto-idealism.

In contrast to this physiognomic skepticism, Spinoza turns to the very discernment of things [res]:

Res enim si intellexissent, illae omnes teste mathesi, si non allicerent, ad minimum convincerent.

I translate this literally because there is some disagreement in the major English translators on the meaning of Spinoza’s sentence (and I think that both of them are somewhat wrong):

For if things they would have been able to discern, those all by mathematical attestation, if they were not allured, at minimum they would have been convinced.

The two counter translations I provide here. Curley in some rather convoluted restructuring, attempts to emphasize the “all” as an accusative. All these persons would be convinced if they merely discerned things correctly. The things themselves would convince everyone. While Shirley, I think more correctly, places emphasis upon both Spinoza’s mode of argumentation, and mathematical attestation. Here they are:

“For if men had understood them, the things would at least convinced them all, even if they did not attract them all, as the example of mathematics shows.” (Curley)

“For if men understood things, all that I have put forward would be found, if not attractive, at any rate convincing, as Mathematics attests.” (Shirley)

There really is no support for Curley’s inventive transformation of “omnes” into a universal emphasis of agreements, though that may be implied. Really what Spinoza is saying is that indeed contrary to merely the physionomic understanding of judgments (and also a celestial orderliness model), distinct discernments of things has come via the testament of mathematical treatments. While Shirley’s translation grasps the general thought of this, Curley captures the very epiphantic nature of such intellection, it through mathematical treatments that the very nature of “things” appears.

And the little caveat on the nature of how such men will be affected by such discernment is telling. Such fellows will be convinced, though they may be “attracted” to such an interpretation. Here Spinoza seems to be putting his thumb in the eye of those that disagree. There are it would seem libidinal investments in seeing the world other than the way in which it is most arguably so. There is also perhaps a commercial connotative association of liceo, “to buy, to put a price on, to value” which may not be far from Spinoza’s intention.

Teste Mathesi

So what are we to make of this “by mathematical attestation” [teste mathesi]. Clearly, it is by reason of mathematics that philosophers (and scientists) arrived at the notion of a harmony composed of celestial bodies in motion, a sense of harmony that for Spinoza ultimately lead to viewing the brain as the source of all human judgment; so it cannot be by mathematics alone that we come to discern things properly. And Spinoza has in turn used the geometic method in such a way that he seems to feel that he has, via such a mathematical attestation, produced a discernment of things. As Spinoza in Letter 12 strongly calls into question the ultimate knowledge available by mathematical measurement and calculation, there would seem to be only one more meaning remaining. Mathematical attestation is for Spinoza a revelatory one, one in which the coherence itself (what is calls elsewhere a different “standard of truth”) provides the conviction of discerment, but also one in which any mathematical description always remains merely an approximation, a rounding off of the edges. And these are edged through which the epiphany of perception itself shows through. This is in keeping with my general sense that in that all the propositions found in the Ethics are linguistic expressions, none of them actually are adequate ideas. It is rather that the interaction with the Ethics itself, its real, material and ideational body, is to provoke, is to cause, a real material and ideational change in the reader, one which cannot be reduced to the arguments themselves.

In a certain sense, Spinoza’s very intellectual and physical experiences as a craftsman, the precise use of calculation when applied to physical substances in the service of gaining the desired effects is the very thing that would preclude any minimization of mathematics or the testing of experimentation. Craftsmanship is after all where abstracted calculation and experiential rule of thumb come most closely together. And by all testament, Spinoza was a superb and devoted craftsman.

In a modern sense, we might want to say that for Spinoza the Universe is not a linear mathematical thing, but that the coherences of cognitions and communications between things is at best brought out by linear mathematical treatments (those only known of the day), treatments that in the end must also then be compared with man’s own finitude as a creature. As a craftsman perhaps he not only understood the way in which calculation and figure could be used to control and shape material, but also understood the often unexpected, unique and eruptive form of material itself, the way in which the glass, bubbled and fogged as it is, defies the curve of optical imprint of the lens grinding form. For Spinoza there are always non-linear magnitudes within magnitudes, beyond any one boundary-making, linear abstraction. But this does not prevent mathematics itself to produce reductive epiphanies unto the relationship between things.

Some follow-up thoughts: Spinoza “Following the Traces of the Intellect”: Powers of Imagining