Digital Humanities: What’s in it for us (philosophers)?

With the raise of Digital Humanities (DH), computational tools and techniques became available to support researchers in many areas of the Humanities. Philosophers, however, don’t seem too keen to embrace this development. Luckily, there are exceptions, and digital philosophy is getting off the ground. This is a very exciting development, because the first projects aimed at developing DH tools for philosophy showed that already with very basic computational methods, philosophers can gain a lot. In this post I will tell a bit about such a project in which I worked during my Master’s and after, which clearly showed that there is a big potential for digital methods in philosophy.

The project was called Phil@Scale, and in it philosophers teamed-up with computer scientists to develop computational tools for philosophers. Accordingly – and this was for me a very interesting aspect -, my primal task in this project was to reflect on my job as a philosopher. For in order to be able to develop computational tools which are to aid philosophers in their research, we have to know how can computers aid us. Thus, to begin with, I had to answer the question: what do I do, when I do philosophical research?

An important part of philosophical research, of course, is text-based. Philosophical research consists for a large part in close-reading of texts, which then serves as a basis for analyses and hypotheses on a conceptual level. This dependence on close-reading makes, first, that philosophical research is very slow and thus can be done only on a small scale, and second, that a lot depends on the interpretation of the text by the individual philosopher, which makes it a very qualitative and subjective area of research. It were those two aspects which we aimed to improve by means of computational tools in this project.

To aid philosophers in text-based research we developed a text-mining tool called SalVe. Text-mining is – roughly – a technique in which a text (the `corpus’; this can also be a collection of texts) is regarded as a data set, and by means of statistical methods certain kinds of information are extracted from it. SalVe has several functionalities and allows researchers to determine faster (compared to traditional methods) which parts of the textual corpus are relevant for their research, and moreover offers statistical (and thus in some important sense quantitative and objective) evidence for or against hypotheses.

Compared to other DH tools, SalVe is very basic, in the sense that it uses relatively simple techniques. This was a deliberate choice, since for philosophers it is very important to keep track of what the tool is doing exactly, and it is important that the tool does not do any interpretation of the text that philosophers want to do or should do themselves. In other words, we wanted SalVe to facilitate, rather than to replace, the work of a philosopher.

The first test-case for SalVe (which was part of my Master’s thesis) was Bernard Bolzano’s Wissenschaftslehre (Theory of Science). This is a work of about 2500 pages, so any help to master those is more than welcome. In fact, SalVe proved itself more than useful in answering a pretty technical interpretative question, namely the question whether according to Bolzano all analytic truths are grounded in a corresponding synthetic one (an hypothesis of De Jong (2001)). Not only did SalVe enable me to decide quicker which parts of the Wissenschaftslehre were relevant for my research question, and did it point me towards parts of the Wissenschaftslehre which I had not found using traditional methods, it also gave statistical evidence in favor of De Jong’s hypothesis.

A technical description of SalVe and a report of the results of the project Phil@Scale can be found in this article. The project showed clearly, in my opinion, that digital tools such as SalVe offer a welcome complementary improvement to the traditional way in which philosophical research is done. SalVe makes it possible to do philosophical research faster and makes this research more quantitative and objective, at least for corpora of a certain size. For sure within the history of philosophy, such tools can be highly useful, but I think other branches of philosophy can profit as well. What is needed in order to develop more, and more fancy computational tools for philosophy is that we, philosophers, think about the following: what we do, when we do philosophical research, and how could digital tools be of help? Any thoughts on this are warmly invited in the comments!

The desire to be puzzled

A prominent professor in the philosophy of mathematics once told me that the key to writing an attractive philosophy paper is to present the reader with a puzzle. “Give me a puzzle, and I’ll be interested”, he said. As I was surrounded by mathematicians and philosophers of mathematics which were steadily exchanging puzzles, I had no doubt that he was right: mathematicians and philosophers of mathematics like puzzles. But then, mightn’t it be the case that this fondness of puzzles influences much more than just our judgment of a philosophy paper (and our conversations over dinner)? Here’s a crazy idea – or maybe not so crazy – does our desire to be puzzled affect our judgement of a certain foundational mathematical theory?

The foundational mathematical theory which I have in mind is, of course, Cantor’s transfinite set theory. Given its general acceptance nowadays, it is easy to forget that in order to generalize arithmetic from the finite to the infinite, Cantor’s theory is not inevitable, and in fact is based on an objectionable conceptual choice: it requires that we give up the principle that a whole is always bigger than each of its proper parts. Since the notion of `size’ of a collection (or: set) is in Cantor’s theory defined in terms of one-one correspondence, the collection of natural numbers has the same size, i.e. is  `just as big’, as, for example, the collection of all even numbers, even though the latter is a proper part of the former.

As is finely described in Mancosu’s (2009) paper Measuring the size of infinite collections of natural numbers, throughout history two basic intuitions have been at play concerning the ‘size’ of collections:

PW (Part-Whole, or Euclid’s axiom): every whole is strictly bigger than each of its proper parts;

HP (Hume’s principle, or Cantor’s axiom): two collections have the same size iff there is a one-to-one association between their elements.

For finite collections, PW and HP are both obviously true; the problem is that for infinite collections they turn out to be inconsistent. We can only hold on to both intuitions if we deny that the relations of equality, less than, and greater than, apply to infinite collections (this was Galileo’s solution), or that infinite collections can be taken as a whole to which a size can be attributed (as held Leibniz); and this is nowadays commonly considered to be too high a price.

Cantor himself acknowledged that there are two intuitions at play concerning the notion of `size’ of collections. He wrote that in some sense the collection of natural numbers is bigger (he calls it `richer’) than the collection of the even numbers (because, for example, the collection of even numbers is a proper part of the natural numbers), but in another sense, the collection of natural numbers is just as big as the collection of even numbers (because every natural number has an even number corresponding to it). Nevertheless, as is well-known, Cantor opted for abandoning PW and adopting HP as the basis for the notion of `size’ in his transfinite theory.

Cantor’s choice engendered a beautiful and powerful mathematical theory, which seems to have led us to believe that dropping PW is the only way to generalize arithmetic to infinite sets. But, as is pointed out by Mancosu, this is in fact not the case: mathematical theories have been developed which generalize finite arithmetic in such a way that they preserve the part-whole principle for infinite sets. Examples are F.M. Katz’s Class Size theory and Benci, Di Nasso & Forti’s theory of numerosities.

Thus, Cantor’s theory is commonly accepted even though it forces us to let go of the highly intuitive part-whole principle and there are alternatives which do not force us to do so. What makes Cantor’s theory so attractive?

A possible answer to this question, and indeed Kitcher’s (1984) answer, is that Cantor’s theory is superior to its alternatives for its explanatory power. Kitcher, as quoted in Mancosu, writes that the real advantage of Cantor’s theory is that

“we do not even need to go so far into transfinite arithmetic to receive explanatory dividends. Cantor’s initial results on the denumerability of the rationals and the algebraic numbers, and the non-denumerability of the reals, provide us with new understanding of the differences between the real numbers and the algebraic numbers. Instead of viewing transcendental real numbers (numbers which are not the roots of polynomial equations in rational coefficients) as odd curiosities, our comprehension of them increased when we see why algebraic numbers are the exception rather than the rule.” (Kitcher 1984, p. 221)

Thus, according to Kitcher, the benefit of Cantor’s theory is that while generalizing arithmetic from finite to infinite sets, it yields new insights which are not linked to this generalization, namely, new understanding of the differences between the real and the algebraic numbers.

But to which extent did Cantor’s theory provide us with new understanding of the differences between the real numbers and the algebraic numbers? Is it really true that our comprehension of the transcendental numbers increased with Cantor’s theory?

For sure, Cantor provided us with a new way to look at transcendental numbers. His theory employs new concepts (such as set, equinumerosity, denumerability, non-denumerability) and a new proof methodology (diagonalization). Cantor’s diagonal method allows us to construct transcendental numbers, and indeed infinitely many of them. A different method to construct infinitely many transcendental numbers was already given by Liouville in 1844, but on the assumption of Cantor’s transfinite theory, there are not simply infinitely many transcendental numbers, but uncountably (or non-denumerabily) many of them. Importantly, Cantor’s proof that there are more transcendental than algebraic numbers is one by contradiction, and rests upon the assumption that the reals are uncountable.

It thusly seems that we might as well argue, contrary to Kitcher, that Cantor’s transfinite theory upgraded the status of transcendental numbers from being “odd curiosities” to an outright mystery. Today, several classes of transcendental numbers have been identified, but still we have found only countably many of them (see the Wiki). This means, in Cantor’s framework, that there are uncountably many that we are missing. It seems to me that on the basis of a theory which does preserve the part-whole principle (such as the two theories mentioned before), even if it can be proved that there are more transcendental than algebraic numbers, then there will be `more’ of them in a much less interesting, that is, less puzzling, way.

The factors which eventually lead to the acceptance or rejection of a theory are for sure not always transparent, nor need they be completely rational. For all I know, it adds to the allure of a mathematical theory if it provides us with a nice new puzzle.

Thanks to Eric Wawerczyk for helpful discussion.

Bolzano on merely possible objects – A reply to Schnieder

– This post has been updated December 5th 2015 –

In the paper Mere Possibilities: A Bolzanian Approach of Non-Actual Objects, Benjamin Schnieder examines, as he puts it himself, Bernard Bolzano’s views on merely possible objects. Beyond doubt, Schnieder is right in arguing that Bolzano held that there are more objects than those inhabiting the actual world. Bolzano held for example that there are propositions, numbers, geometrical figures, etcetera, which will never enter into actuality. But Schnieder argues that Bolzano also accepts what he calls merely possible objects: objects which are not in the actual world, but could become actual.

Now, there is no passage in any of Bolzano’s works in which he plainly and unquestionably argues that there are merely possible objects in Schnieder’s sense. And in order to defend the view that Bolzano holds this, Schnieder claims that there are several passages in which Bolzano is contradicting his own theory: Schnieder argues that Bolzano sins against his own insight that properly understood, ‘merely possible’ is a modifying predicate by using it in a determining way (p. 537), and he argues that Bolzano misunderstands his own notion of impossibility (p. 541). These are, in my opinion, already reasons enough to doubt that Bolzano indeed had the view that Schnieder ascribes to him. For this reason I’d like to find out: isn’t there another viable interpretation of Bolzano’s writings on the possible, that does not require that we attribute such mistakes to him? Here’s an attempt.

My reconstruction of Bolzano’s views on the possible will start from his theory of ideas. Bolzano answers many metaphysical questions by means of his theory of ideas, and notably also formulates his account of necessity in terms of it. Therefore it seems to me that his theory of ideas is a good place to start.

As rightly explained by Schnieder, the question whether there are such-and such objects, comes down for Bolzano to the question whether the idea of such-and-such is objectual, i.e. has (an) object(s) falling under it. Ideas, in Bolzano’s theory, are abstract entites which are the parts of propositions that are not themselves propositions. Ideas make up the matter (Stoff) of their counterparts in the physical world, subjective ideas, which in turn are constitutive of our judgments and linguistic utterances. Importantly, for every object in Bolzano’s universe there is an idea that has this object in its extension, but not every idea that there is has an object in its exension. Some ideas have objects falling under them which are concrete, i.e. inhibiting the physical (actual) world, others have objects which are abstract, i.e. not inhabiting the physical world, but a popperian World 3-like abstract realm instead. To give an example saying that there is the number 2, means for Bolzano that the idea of the number 2, which is following a common use in Bolzanian scholarship denoted by [2], is objectual. Objects like the number 2 do in Bolzanian terms not exist, i.e. they do not inhibit the physical world. I will use the term abstract, as Schnieder does, for those objects that cannot exist, like propositions, ideas, and mathematical objects. Phrased in these terms, the question that we want to answer is whether for Bolzano there are non-existing objects which are not abstract; in particular, whether there are in Bolzano’s view what Schnieder calls merely possible objects.

Schnieder defines a merely possible object in Bolzano’s view (henceforth MPO) as follows (enumeration is mine):

(Df. MPO) x is a merely possible object ↔df.

  1. x is non-actual &
  2. x is possibly actual. 

We know what it means for an object to be (1.) non-actual: an object is non-actual iff it is there but is does not exist. Abstract objects are non-actual objects, for they are not in the physical world. Importantly however, abstract objects cannot exist. Thus, abstract objects are not (2.) possibly actual, and are consequently not MPO’s. A presumed example of an MPO would be a golden mountain, because there is no actual golden mountain (1.) but a golden mountain could be actual (2.).

Now, did Bolzano indeed hold that there are MPO’s? Are there in his view merely possible golden mountains? Let’s look a bit further into his theory of ideas.

Ideas can have other ideas as parts (i.e. be complex) according to Bolzano, and some complex ideas are such that their parts attribute properties that contradict each other, which means that they cannot be united in an object (WL §70). Consequently, such ideas are not objectual, i.e. they are objectless (cf. WL §352). An example of such an idea would be [triangle which has four angles]. Bolzano writes about these contradictory ideas that:

“Hitherto it was common to call them empty [leere], impossible [unmögliche], or imaginary [imaginäre] ideas. All the others one gave the name possible [möglich], actual [wirklich], or real [real] ideas.”  (WL §70)

And Bolzano adds that he will stick with this common use of the terms. Thus, possible seems to be in Bolzano’s view whatever is not impossible (cf. WL §182); impossible, in turn, simply means contradictory. Importantly, possibility, etc., is here predicated of ideas and not of the objects that (are supposed to) fall under them. Later we will come back at the question whether it is correct to see possibility as a property of ideas only in Bolzano’s view. I can already say that this will be exactly the interpretation that I propose: when Bolzano is attributing possibility, etc., to things, then he does this not at the level of the objects themselves, but at the level of the ideas that these objects (are supposed to) fall under.

The important thing to understand now is Bolzano’s view on contradictory ideas. Determining whether an idea is contradictory helps us in Bolzano’s view to determine whether or not this idea is objectual, as he describes in WL §352:

“Is the idea that we are considering complex, and has the form [something] a + b + c + d + …, then we should investigate whether the properties a, b, c, d, which are united here do not contradict each other, that is, whether one of the propositions: No A is B, No B is C, No C is D, etcetera, is provable from purely conceptual truths. If we can find such propositions, then our idea is imaginary; if we do not find them, then it is real.” (WL §352)

We will focus here on complex ideas considered as concretum (cf. WL §60), so that  the ideas that Bolzano discusses in the above quote are the relevant ones for us (extending the following to complex ideas of other forms will be easy; extending it to simple ideas will be highly complicated, since Bolzano’s views on simple ideas are so far hardly understood, so I’ll leave that aside here). According to this quote, these ideas are real (i.e. possible, cf. WL §70 above) if and only if they do not have ideas as parts that attribute properties which contradict each other (cf. WL §138). It is important to note that contradictory ideas are in Bolzano’s view not (only) ideas of the form [A, which is not A] or [something, which has b and not b]. Rather, as he puts it in WL §70, contradictory ideas are of the form [A, which is both B and P], where B and P are such that there is a property M such that all B are M and all P are not M. As an example, Bolzano gives the idea of a triangle which is equilateral and has a right angle. From pure concepts (i.e. from solely a priori truths) one can prove that all equilateral triangles do not have a right angle, and that all triangles with a right angle are not equilateral (WL §70).

Thus, so far we have:

(Df. POS) [x] is possible (actual, real)  ↔df.

it cannot be proven from purely conceptual truths that the properties that [x] attributes cannot be found in one object simultaneously.

Note that for all ideas (thus, also for empirical ideas), purely conceptual truths settle the question whether or not they are possible (cf. WL §182). Note further, that to say that [x] is possible is not yet to say that [x] is objectual (cf. WL §352). In order to establish whether [x] is objectual, we need to take into account whether [x] is what Bolzano calls an actuality-demanding (Wirklichkeitsfordernd) idea. Actuality-demanding ideas are such that if they have an object falling under it, then this object is in the physical world. An example of such an idea would be the idea of a horse: if [horse] is objectual, then a horse exists in the physical world.

Now, for not actuality-demanding ideas according to Bolzano it holds that if they are possible, then they are objectual (WL §352). This is not the case for actuality-demanding ideas: if they are possible, they can still be objectless because they contradict empirical truths (WL §§70, 352). 

Thus, we have:

(PI) If [x] is a possible idea, then exactly one of the following holds:

  1. [x] is not actuality-demanding and [x] is objectual;
  2. [x] is actuality-demanding and [x] is objectual;
  3. [x] is actuality-demanding and [x] is objectless.

(For simplicity I ignore here the temporal aspect; it will not be problematic to incorporate this.) Ideas that satisfy 1. are for example ideas of mathematical objects; ideas that satisfy 2. are ideas of objects in the physical world. Ideas that satisfy 3. are ideas such that if they are objectual then their object is in the physical world, but there are empirical truths that prohibit these ideas to be objectual.

Back to MPO’s, specifically to golden mountains. As might have been expected, Bolzano claims that the existence of a golden mountain is not prohibited by purely conceptual truths (WL §67, cf. WL §§70, 352). Notably, this means by (Df. POS) that the idea of a golden mountain is possible. Further, it seems that Bolzano would have agreed that [golden mountain] is actuality-demanding (since a golden mountain should be in the physical world), and that it is objectless (since no golden mountain is in the physical world). This means in terms of (PI) that 3. holds.

Note that at this point, we have all we need to account in a Bolzanian manner for the role of golden mountains in our worldly and scientific activities: we have [golden mountain] which enables us to think and speak about golden mountains, which is at the logical level constitutive of any truth that concerns golden mountains (cf. WL §67), and which can become objectual if for example at a certain point God decides to enrich this planet with a golden mountain. What we so far do not have, however, are merely possible golden mountains.

If there is a merely possible mountain in Bolzano’s universe, then, in all likelyhood and also according to Schnieder, Bolzano would have taken it to fall under the idea [merely possible golden mountain]. Let’s assume this for the moment, and let’s examine this idea. Importantly: in which respect is [merely possible golden mountain] different from [golden mountain], i.e. what does the [merely possible] do?

If we plug in “golden mountain” in (Df. MPO), we should get a merely possible golden mountain in Schnieder’s sense. Thus, [merely possible] seems to attribute to the golden mountain that it is not actual but could be actual. But as was noted above, this is already true of [golden mountain]: since it is actuality-demanding and possible there could be an actual golden mountain, and since it is actuality-demanding and objectless there is no actual golden mountain. This means that [merely possible] in [merely possible golden mountain] is wholly redundant.

One might even wonder whether Bolzano does not take [merely possible golden mountain] to be a contradictory idea. For [golden mountain] is an actuality-demanding idea, and [merely possible] attributes non-actuality to the object. As I see it, it might very well be that [merely possible golden mountain] is not a possible idea in Bolzano’s view at all, and for that reason there are no and cannot be merely possible golden mountains in Bolzano’s universe. 

My proposal is to understand Bolzano’s theory of the possible in the following manner:

(BP) Being possible is a second-order property.

Interpreted like this, being possible in Bolzano’s view thus closely resembles being there in his view. Just like according to Bolzano, sentences of the form “there are F‘s” expresses a proposition of the form [[F] is objectual], which means that the idea of an F is an objectual idea, according to (BP) in his view, sentences of the form “F‘s are possible” express a proposition of the form [[F] is possible], which means that the idea of an F is a possible idea.

Now, if a proposition of the form [[F] is possible] is true, then by (PI) [F] can be either objectual (option 1. and 2.) or objectless (option 3.). In this spirit I propose that we should interpret merely possible in Schnieder’s sense as expressing that option 3. holds: 

(Df. MP) [x] is merely possible ↔df.

0. [x] is possible &

1. [x] is actuality-demanding &

2. [x] is objectless.

Thus, merely possible according to (Df. MP) expresses that the idea is possible (0.), and furthermore not abstract (1.) and does not have an actual object falling under it (2.). Note that in virtue of (2.), (Df. MP) implies that there are no merely possible objects

Against my argument based on Bolzano’s theory of ideas, and in favor of Schnieder’s account, it could be argued that it might be that we do not have epistemic access to the merely possible and therefore fail to understand under which idea a merely possible golden mountain falls, and that therefore my argument is not decisive to show that in Bolzano’s universe there are no merely possible golden mountains. In reply to this, I would like to point out that with the exception of 1 passage, all the passages that Schnieder quotes to show that Bolzano does hold that there are merely possible objects, can be interpreted in such a way that Bolzano simply argued that there are non-actual objects (which is beyond doubt true, for Bolzano held that there are logical and mathematical objects, but these qualify as abstract objects). The only passage in which Bolzano indeed seems to argue that there are non-actual objects besides abstract objects is WL §483. But in my view this is meager evidence to attribute to Bolzano the view that there are non-actual and non-abstract objects. Especially since, as I have shown, he does not need these objects to be there: already the fact that the idea under which these objects are supposed to fall are there makes that his theory can account for all our worldly and scientific activities that concern objects that are not actual but could be actual. Furthermore, Bolzano developed and then employed his theory of ideas in order to deal with virtually all metaphysical questions. This makes it highly likely that if he had held that there are merely possible objects, we would have found somewhere in his works an account of the ideas under which these merely possible objects fall. Bolzano is way too precise a thinker just to “forget” to deal with merely possible objects.


Generality in explanations

A favorite example in the debate about the applicability of mathematics and mathematical explanation is the one concerning the life cycle of cicades (cf. Ginammi 2014, p. 109; Mancosu SEP): the (biological) fact that cidades have a prime life cycle is to be explained by the (mathematical) fact that prime periodes minimize the intersections with other periodes (and therefore make the cycades better protected from predators).

However, as Ginammi explaines when he discusses this example in his disseration about the applicability of mathematics, it is pointed out by Pincock (2012, ch. 10.2) that in the explanation of the prime life cycles of cicades, we could replace the proposition

(p) prime periodes minimize intersections with other periods


(p*) prime periods less than 100 years minimize intersections with other periods.

In this case, the weaker (p*) seems to have as much explanatory value as (p), because both explain the actual life cycles of cicades equally well. This example suggests that in Biology we can adopt propositions of different strenght to explain the same conclusion, and we have no decisive reason to choose one over the other (cf. Ginammi 2014, p. 110).

However, do we really not have a reason to prefer one of (p) and (p*)? According to an ideal of science that originated from Aristotle’s Analytica Posteriora, proper scientific proofs should state their premisses in there most general (hence strongest) form (cf. Betti & De Jong 2010). In accordance with this tradition, in Bolzano’s view for example

(t) Equiangular triangles have angles that together equal two right angles

should never be deployed in a proper scientific proof. The reason for this is that having angles that together equal two right can be truthfully predicated of all triangles, and not only of those that are equiangular. Therefore, as Bolzano argues, in a proper scientific proof we should use the maximally general

(t*) Triangles have angles that together equal two right angles (cf. Bolzano 1837, §447).

Pincock argues, as Ginammi quotes him (2014, p. 110), concerning the different propositions that we can adopt to explain the prime life cycles of cicades:

I do not believe that the ability to explain nonactual instances of these phenomena should heighten the explanatory power of these explanations

At first sight, Pincock seems to have a point here. Why should we adopt premisses that seem way to strong to explain the fact that we are concerned with?

However, I would like to pose the opposite question: why should we not adopt the stronger premise?

Evidently, both (p) and (p*) are truths that are themselves in need of proof. How do we prove these truths? It seems that in both cases, the most straightforward choice would be a proof that appeals to the concept of prime number. Moreover, it seems that the proof of (p*) just is the proof of (p), where the conclusion is narrowed down to prime numbers less than 100. But now suddenly it looks artifical to use (p*) in the proof of our biological fact: the proof of the mathematical fact that it is based on gives us the more general (p) “for free”.

Maybe this points us to an essential difference between proofs in mathematics and proofs in the empirical sciences: whereas the former are proofs about concepts and for this reason are naturally stated in their most general (strongest) form, the latter are proofs about actual things, and can for this reason be narrowed down to cover just the actual instances.

Bolzano was not sure whether his ideal of proper science that he explicated for the deductive sciences should also hold for the natural sciences. And maybe this is an interesting question to raise concerning the problem of applicability of mathematics: given that we appeal to mathematical concepts in our proofs of empirical facts, does this imply that our ideal of generality that we have for proofs in mathematics give us a reason to prefer stronger premises over the weaker in a proof of an empirical fact?

Read Michele Ginammi’s blog.