David Hilbert on Unification

At the end of David Hilbert’s Mathematical Problems, Hilbert goes into the details for his motivations for what we may call unity of science thesis. These reasons are as poignant today in my view as they were in his own time. Motvations could be summarised thusly:


  • Divergences/fracturing mathematics into subdisciplines will mean specialised areas will not engage with other areas outside their specialism

  • The most important innovations are driven by simplicity, more refined tools and less complication.


The first thesis is a problematic of overspecialisation and genrefication of any kind of academic research. Becoming so niche that one is essentially writing for a peer group that is too specific and few. Perhaps this is inevitable in the world of industrial research and constant innovation. If we are to believe that subdisciplines and specialisation are a necessity, then we cannot understand Hilbert’s second thesis, of parsinomy. Granted, more needs to be elaborated if such a unification thesis were to work. Unification has its own problems, but there is a bonus to clarity and it is a matter of fact that many great scientific innovations are of the sort that unify and simplify seemingly irrelevant areas (Maxwell Equations or Relativity for example).

 The conclusion of Hilbert’s lecture is as follows:

The problems mentioned are merely samples of problems, yet they will suffice to show how rich, how manifold and how extensive the mathematical science of today is, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection becomes ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separate branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but only manifests itself the more clearly.

But, we ask, with the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and cast aside older more complicated developments. It is therefore possible for the individual investigator, when he makes these sharper tools and simpler methods his own, to find his way more easily in the various branches of mathematics than is possible in any other science.

The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfil this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples! [David Hilbert, 1900]


People should need a license to use the word ‘logical’

‘Logical’ is a word that I would almost never use. Why? There are two reasons.

1. I do not really know how to use the term. Michael often seems to think that he does know the meaning of the word logical; which is essentially a synonym for the term ‘categorial’ iff such categorial properties can be formalised viz. logic.

So, what is a categorial property? The categorial property links to Michael’s interest in Systematicity. Systematicity is a thesis concerning the ordered organisation of phenomenological and intellectual matter. In short, our organisational scheme of the world. This notion goes back to Aristotle’s metaphysics, who imposed a notion of fundamental categories that construe everything and anything in the world.

Such a systematic ordering of reality, following Kant’s categories and going up to Frege; suggest the formalised scheme in which particularised (and perceived) reality may be known and knowable. Categorials include things like


We may call such categorial features ‘logical’ features. To think logically is to think categorially (although they are not exact synonyms. A further caveat is to say that categorial properties do not exhaust the concept of the logical, by no means at all do they do as such! I think, however, that a good conception of the logical would be to account for those things that are part of our conceptual understanding of the world.

2. People in common usage use the term ‘logical’ in a similar way that they may speak of ‘objectification’; namely, without defining it. To define a word means to suggest of the conditions of its legitimate usage. Logical is often a synonym of modus ponens. But it is also such a muddled and confused common term, that people seem to associate causal reasoning with propositional logic; or cannot comprehend worldhood. Such fuzzy use of terminology is unhelpful to people’s conceptual schemes. Their stupidity limits their own conditions of apprehension.


Everything must have a class

I take this proposition to be almost axiomatic, a truism, a necessary truth. The domain of everything consists of all concievable things; even things such as the set of inconceivables (where inconcievables are undefined); all objects, propositions, truthmakers, logical atoms and operators, and the space in which any logical system may subsume.

We may distinguish between ‘everything’ and any such thing that may be other than such a term ‘everything’ could capture. If the latter could be an object of genuine distinction; without some absurdity (such as: ‘everything’ captures those objects and referents that may be impossible either logically or conceivable, or absurd in some other way), we may introduce the notion of a Noumenon; but this is not necessarily needed for such a constructible set ‘everything’ to maintain, even eliciting the objects within such a set are not necessary, merely that it is there would suffice.

Kant maintained that the experience and knowledge could only be grasped by means of an active part of human nature (or conscious nature) to operate in such a way where all things are forced under the principle of seeking categories; what these categories may be are not terribly important (for this post) but may be debated. It would seem to me hard to argue for something so basic and primal as the phenomena that is immediately present to us. When we feel an immediate pang of pain, or that capture of laughter; we hardly come to the first thought or experience of questioning whether we have such an experience at all, or whether there is such a thing as a quale; but we come to reside in that experience. Humean nature is also true in that there is a brute fact-ness of how human nature is enslaved by experience.

It is a misnomer to deride the notion of categories; to deride any given notion of categories would be fine, but categories as a wholesale notion would be harder, insofar as any correlation system may be made. Association is but inherent to conscious experience. Whether such an experience is veritable, is immaterial to that brute insight.

It is however, presumptuous to pose that human experience is objective experience. If we are to speak of objective knowledge (in a putative sense of the term); we may note it as being in some way indexical to the principles or conditions of experience itself. Talk of ‘subjective’ and ‘objective’ can be muddied in that respect.


Dawkins on the ontological argument

There is a section in the God Delusion (Dawkins, 2006) early on, where Dawkins’ addresses a certain confusion about how Russell had claimed in his early philosophical career that the proof of God’s existence by means of appeal to essence (namely, the Ontological argument) is actually valid. Dawkins then gives an anecdote where he gives a flippant variant of the ontological argument to prove that some trivial fact is necessarily true and he ends with the punchline: “They felt the need to resort to Modal Logic to prove that I was wrong.”

While the remarks that Dawkins makes on the chapter concerning A priorist arguments treats the enterprise as fruitless and a joke; he does make a half-serious point. Gaunillo, as Dawkins rightly attributes, gives the case of the ‘most perfect island’ that must necessarily exist, if we are to accept the inference that Anselm wants to use to prove God. One common response is to give add some caveats to the thing that we are trying to argue into existence. Adding caveats like ‘most perfect being’ cannot refer to contingent beings and are necessary by definition may attempt to exclude. Anthony Kenny, in his exegesis of Descartes’ ontological gives a hearing to this view with some comparison cases; firstly, Russell’s notion of the Gold mountain and secondly the more thorny issue of non-existent objects.

There is a case, if we accept the theorems of S5 logic where we might say that the ontological proof of Good is valid (which banks on the S5’s Rule of Necessitation). We can say that a maximally perfect being exists possibly. However, valid as it is; it does require some metaphysical steps to infer that it is the man with the beard. Dawkins addresses the so-called ‘Einstein’s God’ which is mischaracterised by theists, and he states from the offset that this is not the notion of God that he’s attacking. It is, however, this very ‘Spinozan’ notion of God (deus sive natura) that the ontological argument apparently proves, and not the Christian deity. Most Christian apologists who argue with the ontological argument in public debates always use the religious experience/testimony argument concerning the historicity of Jesus, because the S5 compatible proof allocates a non-religiously-affiliated God. What I find most interesting about this proof is the metaphysical world that it would entail, given S5 modal conditions (such as the issue of natural kinds, worldhood and perhaps the issue of universals).

In closing. Modality is no flimsy subject matter but one that has serious implications in term of systematic philosophy. I suppose my irk with Dawkins concerns a distinction between the protestant and catholic atheist which is so-joked about. My Thomistic tendencies would emphasise the role of reason and the a priori in terms of structuring reality. Aquinas, in his conciliatory effort to bring Aristotle to Christianity, believed that the Aristotelian method and reason itself must have a place in elaborating our view of the universe. Later theologians debated between the notions of analogia entis and analogia fides; whether our comprehension of the divine comes exlusively through scripture (and its empirical connotations) or our rational attempt to structure ultimate reality. The comment made my Dawkins seems insincere to the a priorism that supercedes its theological origins. I find it amusing that there would be such a distinction between a ‘protestant’  (empiricist) and ‘catholic’ (rationalist – or, reason + experience) atheist.


Utility and ‘paradise’

I’ve been reading a bit of David Lewis recently, one point he makes early on in his work “On the Plurality of Worlds”, is that the utility of a theory is a reason for accepting that it is true. The non-philosophical example he gives is in set theory, construct your sets and ontology however way which you want, and you get out what you want, by determinedly deciding one’s axioms and the conditions in which one may establish a theorem or establish some proof or impossibility by way of reductio.

Lewis quotes a phrase (apparently) from Hilbert, which goes something like “Set theory is a mathematician’s paradise”; likewise, we may also have a metaphysician’s paradise by way of thinking within the jargon of modal realism; of possibilia, logical space, closeness of worlds, while many object to the proposal of modal realism, the very fact that people still talk about it, and use the terminology of worlds, counterparts, and so on; is a testimony to the influence and power of this thesis.

While one has critical thoughts about this thesis (concerning isolation and the knowledge of worlds); there is an underlying appeal which must be taken seriously. Philosophy considered as establishing theories that balances a strength of a theory against its weakness. What are strong aspects to a theory? By theory, I mean not just metaphysical theories, but scientific theories, or even moral or empirical hypotheses as well.

A theory may have strengths in virtue of the following things:

1. Theoretical unity, interconnectedness
2. Parsimony
3. Explanation
4. Confirmable predictions
5. Upgrading past theories
6. Refuting contemporamous competing theories
7. Being formalised, mathematicised

A theory, while emphasising one of these things, may also have a cost:

1. Being empirically false
2. Being non-empirical
3. Being inapplicable to higher genera concepts
4. Invoking weird ontologies
5. Violating parsimony
6. Incommensurability (ie. an incompatibility with other theories, or no address of corollorary issues)
7. Not having any predictive power

The spirit of utility arguments is that they are not so much arguments but appeals to truth. With utility one does not argue that something is or is not the case, as such, as in a formal deductive argument, but one appeals to the truth of something by its utility; to force the in another way: no other theories explain as much as this does; or if it does, then it is a better theory.

If we pursued this kind of line of metaphysics, we would have two implications:

1. Arguments are not so easy to knock down (hopefully); if we establish a thesis by many prongs, taking one prong away does not take away the thesis. This means we can concede to criticisms without complete abandonment and philosophy becomes essentially a concilatory project of theories and different metaphysical topics.
2. Philosophy would truly work in the old way of being systematic; we can show how one discourse relates to another; we may simultaneously be doing philosophy of mind and metaphysics; epistemology and philosophy of science; philosophy of mind and metaethics. Perhaps this issues would be more muddied up, admittedly, but we may find the age of the big systems arriving again. Lewis, in one’s eyes, is a systematic philosopher; but not one par excellence.


Confused Questions

1. Does it make sense to construe norms into the discourse of epistemology?
2. What are the status of the theoretical norms?
3. What theoretical norms are there?

We could have:

i. Ontological unity (naturalism – strong)
ii. Methodological unity (naturalism – weak)
iii. Conceptual unity (Transcendental)
iv. Systematicity (Transcendental)
v. A set of peacemeal norms, induction, parsimony etc.

4. Question-begging, how is induction set into a norm? This relates to the following question

5. If we assume inductive behaviour is inevitable (which, it kind of is), then there is a fact of the matter about the fact that we do use it; further, there is an inevitability about our use of it. Given its inevitability, is there an ought implies can consideration to be made? I see contrary tendencies as to the question of the rationality of questioning the epistemic practice that we deem inevitable (Cf. Stern 2000)

6. We may have epistemic norms of differing graces: strong norms like induction, or systematicity is stronger still, but we may have rules of thumb like parsimony; it may seem that the image is far from systematic, but Quinean-web-like

Destre (and Michael)

The ship of Stratovarius (on ideology and semantics)


Anyone who is familiar with the metal scene of Finland knows about the recent spat between Timo Tolkki and the rest of the members of Stratovarius. In a previous post, I reported the news that Stratovarius broke up; but then came a whole barrage of replies from two parties; Timo Tolkki, and the rest of Stratovarius. These recent events are much like the whole open letter affair with Nightwish and their former singer. Has Finnish heavy metal become so big, that it has taken on the mechanics and suave of modern bands, of having official fan clubs, official merchandise, PAs, photoshoots and open letters? It seems long from the harked days of underground bands playing in California who were known by their audiences bootlegging their gigs, but that’s a whole other point at hand…

The heart of Stratovarius

I can engage in a suitably philosophical discussion about the semantics and modality of ‘Stratovarius’; but I want to address a more human point.

Stratovarius is a band that, for me, and a lot of people I know, represents a mindset. It is, I thought quite clearly, until recently, a band that was in tune with a lot of the heavy metal scene in Europe; trying to come to terms with the bleakness, superficiality, conformism and fostered attitude of normative-heterogeneity, by replying either by an expression of despair [such as EToS]; fantasy; or perseverence. Stratovarius represented the most noble of these responses: perseverence, the strength to keep fighting on in a world of superficiality. How ironic, and how disturbing I find it that Stratovarius engages in this kind of dispute. Not to take any sides on the issue, but when a band that for me, represents perseverence and a way of coping with the modern world, has infighting, one kind of loses hope in the message they once represented.

Now, for a rather odd analysis of ‘Stratovarius’….

The semantics of ‘Stratovarius’

Timo Tolkki, de re, was not the original founder of the band, contrast this to Tuomas’ role in Nightwish. It is Tuomas’ baptism of the band, that makes him the essential feature of the band; the necessary condition for ‘Nightwish’ to refer is that Tuomas is in it. Can we say the same for Timo and Strato? The short answer is yes (because he is the lyrical and musical direction of the whole band since 1984); but the long answer is no he fails to fulfill the de re necessity Kripke designator.

There have been many bands (my first thought on this is the Norweigian band Mayhem) which have none of the de re original members present in their current lineup, yet the name of the band still refers. This is obviously like the philosophical problem of identity, the Ship of Theseus; if you replace every plank, is it still the same ship?

In the case of Mayhem; some of the original members have left, and then returned; much like Ozzy Osbourne in Black Sabbath (replaced by Dio, Tony Martin, etc.); however, unlike Mayhem, Black Sabbath maintained the essential feature, the conponent of Tommy Iommi; who has, rather significantly, maintained throughout the whole career of Sabbath; being the creative force behind it, despite how most people associate it with Ozzy (or, as some of the fan discourses argue, Dio, but that very fact points out the finitude of the lead singer as being core to the band).

Is it possible, further, is it legitimate, semantically, for a band to have changed its whole membership and yet still refer by its original name? What of any organisation for that instance. Is the philosophy department of Cambridge still legitimate to claim heritage of Russell, Wittgenstein and Moore, even though they have long gone?


Commitments (Arithmetic)

A thought without a resolution; a question without an answer I propose in the following:

Consider the following statements:

i. The set of real numbers has a subset of entities which are prime

ii. There is always a higher prime number after any given prime number n

iii. Either there is a highest prime number, or there is not

iv. (It is necessarily the case that) If I entertain any candidate for a ‘highest prime number’; there will always be a higher prime

v. If there is either a highest prime number or there is not, and it is the case that there will always be a higher prime, then it is not the case that there is a highest prime number

vi. If the number line were finite, then perhaps there would be a highest prime number

These are all different kinds of statements:

i. Is a claim about a set of mathematical objects, an assertoric claim about the number line; but is it existential? Is it analytic? is it a priori? is it contingent or necessary?

ii. The expression ‘There is’ intuitively expresses an existential proposition, however, it is an existential proposition about a set (the set of prime numbers), which itself is part of a set (natural numbers); so, to say ‘there is’, in this context is necessarily elliptical upon the committment to the prime number and real number set; do any of these have existential commitment within itself?

iii. Is a disjunctive logical statement; the nature of a disjunction (PvQ) is such that P may hold as an assertoric claim, where Q may be complete nonsense; furthermore, do we have an exclusive (PvQ), where it is the case that (P&Q); or (P&¬(Q)) v ((¬P)&Q)? furthermore, if we address this issue of the inclusive or exclusive disjunction, is this an ontic claim about the metalanguage? Further; what rules can we assert about the construction of the grammar, and what is the status of these rules themselves?

v. Is a modus ponens; to what do I commit to when I assert modus ponens? Do I need to commit to P to assert P–>Q? Furthermore; do I need to assert the truth claim; or tacitly suppose the rule of modus ponens in order to assert any material implicature

iv. Is a necessity claim, and thus, a modal claim; how do we construct necessity claims at all? Do we need token entities to instantiate natural kinds to demarcate the neccessary? Or do we need to posit an Ersatz plurality to ensure the plenitude of possibilities being genuinely assertoric?

vi. This is a counterfactual; is it absurd to posit a counterfactual about a finite number line? For example; if the last number in the line was 100, and there was nothing above it; is this inconcievable, further, is this possible, but inconceivable? or impossible because its inconceivable? Finally, if we were a Lewisian Modal Realist; could we entertain modal claims about the nature of mathematics itself? or modal claims about the rules of modality? (for instance, saying ‘it’s possible to have a world where we are ersatzist, and another world which is Lewisian in logical space’? This sounds ridiculous a claim prima facie.

Destre (and Michael)

Moral Logic?

What if we were to say that a given modal logic could have a mapping relation upon another aspect of reasoning? Or in other words, if we could have a discourse or sets of truth values and operator functions about the world, in this case, modal logic, so our operators here would be at least necessity and possibility; but then to make a further claim that these operators, and moreover, their behaviour, have an isomorphism with another operation. For example…CAN isomorphs to POSSIBLE and OUGHT isomorphs to NECESSARY, we might be able to translate one discourse into another.
My suggestion, or hope, is, we could make a modal analysis of morality in such a way.


Glass shattering, and Marble leftness

Lets say I have before me a glass of wine. If I were to throw it at my wall, the glass would presumably smash. The object (Gw) has the disposition of shatterability. After I shatter the glass, it no longer can shatter, as it no longer fulfills the condition of being a glass. Perhaps this is too metaphysical. Okay, lets give another story.

Let’s say I have three marbles before me in a line in front of me. These marbles have a certain character about them, marble 1 is to the left of 2,3; marble 3 is to the right of 1,2, and marble 2 is in between them. We may say that a term to describe their spatial index is contingent upon their positioning, and those objects besides them. If I remove marble 1, marble 2 becomes the leftmost marble in the line.

What is the nature of the leftness of marble 1. It is a contingent fact, but also an indexical relation. We could say it is much like the glass then, upon which the shatterability is contingent upon the fact, and thus, indexical to the property of, the glass being solid (and cool). Could we not go further and show it is the case that many properties we ascribe are indexical, and fickle; semantic rather than ontic ascriptions.