Glue Code + Pipeline Jungles. Thought of the Day 25.0

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Machine learning researchers tend to develop general purpose solutions as self-contained packages. A wide variety of these are available as open-source packages at places like mloss.org, or from in-house code, proprietary packages, and cloud-based platforms. Using self-contained solutions often results in a glue code system design pattern, in which a massive amount of supporting code is written to get data into and out of general-purpose packages.

This glue code design pattern can be costly in the long term, as it tends to freeze a system to the peculiarities of a specific package. General purpose solutions often have different design goals: they seek to provide one learning system to solve many problems, but many practical software systems are highly engineered to apply to one large-scale problem, for which many experimental solutions are sought. While generic systems might make it possible to interchange optimization algorithms, it is quite often refactoring of the construction of the problem space which yields the most benefit to mature systems. The glue code pattern implicitly embeds this construction in supporting code instead of in principally designed components. As a result, the glue code pattern often makes experimentation with other machine learning approaches prohibitively expensive, resulting in an ongoing tax on innovation.

Glue code can be reduced by choosing to re-implement specific algorithms within the broader system architecture. At first, this may seem like a high cost to pay – reimplementing a machine learning package in C++ or Java that is already available in R or matlab, for example, may appear to be a waste of effort. But the resulting system may require dramatically less glue code to integrate in the overall system, be easier to test, be easier to maintain, and be better designed to allow alternate approaches to be plugged in and empirically tested. Problem-specific machine learning code can also be tweaked with problem-specific knowledge that is hard to support in general packages.

As a special case of glue code, pipeline jungles often appear in data preparation. These can evolve organically, as new signals are identified and new information sources added. Without care, the resulting system for preparing data in an ML-friendly format may become a jungle of scrapes, joins, and sampling steps, often with intermediate files output. Managing these pipelines, detecting errors and recovering from failures are all difficult and costly. Testing such pipelines often requires expensive end-to-end integration tests. All of this adds to technical debt of a system and makes further innovation more costly. It’s worth noting that glue code and pipeline jungles are symptomatic of integration issues that may have a root cause in overly separated “research” and “engineering” roles. When machine learning packages are developed in an ivory-tower setting, the resulting packages may appear to be more like black boxes to the teams that actually employ them in practice.

Bayesianism in Game Theory. Thought of the Day 24.0

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Bayesianism in game theory can be characterised as the view that it is always possible to define probabilities for anything that is relevant for the players’ decision-making. In addition, it is usually taken to imply that the players use Bayes’ rule for updating their beliefs. If the probabilities are to be always definable, one also has to specify what players’ beliefs are before the play is supposed to begin. The standard assumption is that such prior beliefs are the same for all players. This common prior assumption (CPA) means that the players have the same prior probabilities for all those aspects of the game for which the description of the game itself does not specify different probabilities. Common priors are usually justified with the so called Harsanyi doctrine, according to which all differences in probabilities are to be attributed solely to differences in the experiences that the players have had. Different priors for different players would imply that there are some factors that affect the players’ beliefs even though they have not been explicitly modelled. The CPA is sometimes considered to be equivalent to the Harsanyi doctrine, but there seems to be a difference between them: the Harsanyi doctrine is best viewed as a metaphysical doctrine about the determination of beliefs, and it is hard to see why anybody would be willing to argue against it: if everything that might affect the determination of beliefs is included in the notion of ‘experience’, then it alone does determine the beliefs. The Harsanyi doctrine has some affinity to some convergence theorems in Bayesian statistics: if individuals are fed with similar information indefinitely, their probabilities will ultimately be the same, irrespective of the original priors.

The CPA however is a methodological injunction to include everything that may affect the players’ behaviour in the game: not just everything that motivates the players, but also everything that affects the players’ beliefs should be explicitly modelled by the game: if players had different priors, this would mean that the game structure would not be completely specified because there would be differences in players’ behaviour that are not explained by the model. In a dispute over the status of the CPA, Faruk Gul essentially argues that the CPA does not follow from the Harsanyi doctrine. He does this by distinguishing between two different interpretations of the common prior, the ‘prior view’ and the ‘infinite hierarchy view’. The former is a genuinely dynamic story in which it is assumed that there really is a prior stage in time. The latter framework refers to Mertens and Zamir’s construction in which prior beliefs can be consistently formulated. This framework however, is static in the sense that the players do not have any information on a prior stage, indeed, the ‘priors’ in this framework do not even pin down a player’s priors for his own types. Thus, the existence of a common prior in the latter framework does not have anything to do with the view that differences in beliefs reflect differences in information only.

It is agreed by everyone that for most (real-world) problems there is no prior stage in which the players know each other’s beliefs, let alone that they would be the same. The CPA, if understood as a modelling assumption, is clearly false. Robert Aumann, however, defends the CPA by arguing that whenever there are differences in beliefs, there must have been a prior stage in which the priors were the same, and from which the current beliefs can be derived by conditioning on the differentiating events. If players differ in their present beliefs, they must have received different information at some previous point in time, and they must have processed this information correctly. Based on this assumption, he further argues that players cannot ‘agree to disagree’: if a player knows that his opponents’ beliefs are different from his own, he should revise his beliefs to take the opponents’ information into account. The only case where the CPA would be violated, then, is when players have different beliefs, and have common knowledge about each others’ different beliefs and about each others’ epistemic rationality. Aumann’s argument seems perfectly legitimate if it is taken as a metaphysical one, but we do not see how it could be used as a justification for using the CPA as a modelling assumption in this or that application of game theory and Aumann does not argue that it should.

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Weil Conjectures. Note Quote.

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Solving Diophantine equations, that is giving integer solutions to polynomials, is often unapproachably difficult. Weil describes one strategy in a letter to his sister, the philosopher Simone Weil: Look for solutions in richer fields than the rationals, perhaps fields of rational functions over the complex numbers. But these are quite different from the integers:

We would be badly blocked if there were no bridge between the two. And voilà god carries the day against the devil: this bridge exists; it is the theory of algebraic function fields over a finite field of constants.

A solution modulo 5 to a polynomial P(X,Y,..Z) is a list of integers X,Y,..Z making the value P(X,Y,..Z) divisible by 5, or in other words equal to 0 modulo 5. For example, X2 + Y2 − 3 has no integer solutions. That is clear since X and Y would both have to be 0 or ±1, to keep their squares below 3, and no combination of those works. But it has solutions modulo 5 since, among others, 32 + 32 − 3 = 15 is divisible by 5. Solutions modulo a given prime p are easier to find than integer solutions and they amount to the same thing as solutions in the finite field of integers modulo p.

To see if a list of polynomial equations Pi(X, Y, ..Z) = 0 have a solution modulo p we need only check p different values for each variable. Even if p is impractically large, equations are more manageable modulo p. Going farther, we might look at equations modulo p, but allow some irrationals, and ask how the number of solutions grows as we allow irrationals of higher and higher degree—roots of quadratic polynomials, roots of cubic polynomials, and so on. This is looking for solutions in all finite fields, as in Weil’s letter.

The key technical points about finite fields are: For each prime number p, the field of integers modulo p form a field, written Fp. For each natural number r > 0 there is (up to isomorphism) just one field with pr elements, written as Fpr or as Fq with q = pr. This comes from Fp by adjoining the roots of a degree r polynomial. These are all the finite fields. Trivially, then, for any natural number s > 0 there is just one field with qs elements, namely Fp(r+s) which we may write Fqs. The union for all r of the Fpr is the algebraic closure Fp. By Galois theory, roots for polynomials in Fpr, are fixed points for the r-th iterate of the Frobenius morphism, that is for the map taking each x ∈ Fp to xpr.

Take any good n-dimensional algebraic space (any smooth projective variety of dimension n) defined by integer polynomials on a finite field Fq. For each s ∈ N, let Ns be the number of points defined on the extension field F(qs). Define the zeta function Z(t) as an exponential using a formal variable t:

Z(t) = exp(∑s=1Nsts/s)

The first Weil conjecture says Z(t) is a rational function:

Z(t) = P(t)/Q(t)

for integer polynomials P(t) and Q(t). This is a strong constraint on the numbers of solutions Ns. It means there are complex algebraic numbers a1 . . . ai and b1 . . . bj such that

Ns =(as1 +…+ asi) − (bs1 +…+ bsj)

And each algebraic conjugate of an a (resp. b) also an a (resp. b).

The second conjecture is a functional equation:

Z(1/qnt) = ± qnE/2tEZ(t)

This says the operation x → qn/x permutes the a’s (resp. the b’s).The third is a Riemann Hypothesis

Z(t) = (P1(t)P3(t) · · · P2n−1(t))/(P0(t)P2(t) · · · P2n(t))

where each Pk is an integer polynomial with all roots of absolute value q−k/2. That means each a has absolute value qk for some 0 ≤ k ≤ n. Each b has absolute value q(2k−1)/2 for some 0 ≤ k ≤ n.

Over it all is the conjectured link to topology. Let B0, B1, . . . B2n be the Betti numbers of the complex manifold defined by the same polynomials. That is, each Bk gives the number of k-dimensional holes or handles on the continuous space of complex number solutions to the equations. And recall an algebraically n-dimensional complex manifold is topologically 2n-dimensional. Then each Pk has degree Bk. And E is the Euler number of the manifold, the alternating sum

k=02n (−1)kBk

On its face the topology of a continuous manifold is worlds apart from arithmetic over finite fields. But the topology of this manifold tells how many a’s and b’s there are with each absolute value. This implies useful numerical approximations to the numbers of roots Ns. Special cases of these conjectures, with aspects of the topology, were proved before Weil, and he proved more. All dealt with curves (1-dimensional) or hypersurfaces (defined by a single polynomial).

Weil presented the topology as motivating the conjectures for higher dimensional varieties. He especially pointed out how the whole series of conjectures would follow quickly if we could treat the spaces of finite field solutions as topological manifolds. The topological strategy was powerfully seductive but seriously remote from existing tools. Weil’s arithmetic spaces were not even precisely defined. To all appearances they would be finite or (over the algebraic closures of the finite fields) countable and so everywhere discontinuous. Topological manifold methods could hardly apply.

Abelian Categories, or Injective Resolutions are Diagrammatic. Note Quote.

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Jean-Pierre Serre gave a more thoroughly cohomological turn to the conjectures than Weil had. Grothendieck says

Anyway Serre explained the Weil conjectures to me in cohomological terms around 1955 – and it was only in these terms that they could possibly ‘hook’ me …I am not sure anyone but Serre and I, not even Weil if that is possible, was deeply convinced such [a cohomology] must exist.

Specifically Serre approached the problem through sheaves, a new method in topology that he and others were exploring. Grothendieck would later describe each sheaf on a space T as a “meter stick” measuring T. The cohomology of a given sheaf gives a very coarse summary of the information in it – and in the best case it highlights just the information you want. Certain sheaves on T produced the Betti numbers. If you could put such “meter sticks” on Weil’s arithmetic spaces, and prove standard topological theorems in this form, the conjectures would follow.

By the nuts and bolts definition, a sheaf F on a topological space T is an assignment of Abelian groups to open subsets of T, plus group homomorphisms among them, all meeting a certain covering condition. Precisely these nuts and bolts were unavailable for the Weil conjectures because the arithmetic spaces had no useful topology in the then-existing sense.

At the École Normale Supérieure, Henri Cartan’s seminar spent 1948-49 and 1950-51 focussing on sheaf cohomology. As one motive, there was already de Rham cohomology on differentiable manifolds, which not only described their topology but also described differential analysis on manifolds. And during the time of the seminar Cartan saw how to modify sheaf cohomology as a tool in complex analysis. Given a complex analytic variety V Cartan could define sheaves that reflected not only the topology of V but also complex analysis on V.

These were promising for the Weil conjectures since Weil cohomology would need sheaves reflecting algebra on those spaces. But understand, this differential analysis and complex analysis used sheaves and cohomology in the usual topological sense. Their innovation was to find particular new sheaves which capture analytic or algebraic information that a pure topologist might not focus on.

The greater challenge to the Séminaire Cartan was, that along with the cohomology of topological spaces, the seminar looked at the cohomology of groups. Here sheaves are replaced by G-modules. This was formally quite different from topology yet it had grown from topology and was tightly tied to it. Indeed Eilenberg and Mac Lane created category theory in large part to explain both kinds of cohomology by clarifying the links between them. The seminar aimed to find what was common to the two kinds of cohomology and they found it in a pattern of functors.

The cohomology of a topological space X assigns to each sheaf F on X a series of Abelian groups HnF and to each sheaf map f : F → F′ a series of group homomorphisms Hnf : HnF → HnF′. The definition requires that each Hn is a functor, from sheaves on X to Abelian groups. A crucial property of these functors is:

HnF = 0 for n > 0

for any fine sheaf F where a sheaf is fine if it meets a certain condition borrowed from differential geometry by way of Cartan’s complex analytic geometry.

The cohomology of a group G assigns to each G-module M a series of Abelian groups HnM and to each homomorphism f : M →M′ a series of homomorphisms HnF : HnM → HnM′. Each Hn is a functor, from G-modules to Abelian groups. These functors have the same properties as topological cohomology except that:

HnM = 0 for n > 0

for any injective module M. A G-module I is injective if: For every G-module inclusion N M and homomorphism f : N → I there is at least one g : M → I making this commute

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Cartan could treat the cohomology of several different algebraic structures: groups, Lie groups, associative algebras. These all rest on injective resolutions. But, he could not include topological spaces, the source of the whole, and still one of the main motives for pursuing the other cohomologies. Topological cohomology rested on the completely different apparatus of fine resolutions. As to the search for a Weil cohomology, this left two questions: What would Weil cohomology use in place of topological sheaves or G-modules? And what resolutions would give their cohomology? Specifically, Cartan & Eilenberg defines group cohomology (like several other constructions) as a derived functor, which in turn is defined using injective resolutions. So the cohomology of a topological space was not a derived functor in their technical sense. But a looser sense was apparently current.

Grothendieck wrote to Serre:

I have realized that by formulating the theory of derived functors for categories more general than modules, one gets the cohomology of spaces at the same time at small cost. The existence follows from a general criterion, and fine sheaves will play the role of injective modules. One gets the fundamental spectral sequences as special cases of delectable and useful general spectral sequences. But I am not yet sure if it all works as well for non-separated spaces and I recall your doubts on the existence of an exact sequence in cohomology for dimensions ≥ 2. Besides this is probably all more or less explicit in Cartan-Eilenberg’s book which I have not yet had the pleasure to see.

Here he lays out the whole paper, commonly cited as Tôhoku for the journal that published it. There are several issues. For one thing, fine resolutions do not work for all topological spaces but only for the paracompact – that is, Hausdorff spaces where every open cover has a locally finite refinement. The Séminaire Cartan called these separated spaces. The limitation was no problem for differential geometry. All differential manifolds are paracompact. Nor was it a problem for most of analysis. But it was discouraging from the viewpoint of the Weil conjectures since non-trivial algebraic varieties are never Hausdorff.

Serre replied using the same loose sense of derived functor:

The fact that sheaf cohomology is a special case of derived func- tors (at least for the paracompact case) is not in Cartan-Sammy. Cartan was aware of it and told [David] Buchsbaum to work on it, but he seems not to have done it. The interest of it would be to show just which properties of fine sheaves we need to use; and so one might be able to figure out whether or not there are enough fine sheaves in the non-separated case (I think the answer is no but I am not at all sure!).

So Grothendieck began rewriting Cartan-Eilenberg before he had seen it. Among other things he preempted the question of resolutions for Weil cohomology. Before anyone knew what “sheaves” it would use, Grothendieck knew it would use injective resolutions. He did this by asking not what sheaves “are” but how they relate to one another. As he later put it, he set out to:

consider the set13 of all sheaves on a given topological space or, if you like, the prodigious arsenal of all the “meter sticks” that measure it. We consider this “set” or “arsenal” as equipped with its most evident structure, the way it appears so to speak “right in front of your nose”; that is what we call the structure of a “category”…From here on, this kind of “measuring superstructure” called the “category of sheaves” will be taken as “incarnating” what is most essential to that space.

The Séminaire Cartan had shown this structure in front of your nose suffices for much of cohomology. Definitions and proofs can be given in terms of commutative diagrams and exact sequences without asking, most of the time, what these are diagrams of.  Grothendieck went farther than any other, insisting that the “formal analogy” between sheaf cohomology and group cohomology should become “a common framework including these theories and others”. To start with, injectives have a nice categorical sense: An object I in any category is injective if, for every monic N → M and arrow f : N → I there is at least one g : M → I such that

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Fine sheaves are not so diagrammatic.

Grothendieck saw that Reinhold Baer’s original proof that modules have injective resolutions was largely diagrammatic itself. So Grothendieck gave diagrammatic axioms for the basic properties used in cohomology, and called any category that satisfies them an Abelian category. He gave further diagrammatic axioms tailored to Baer’s proof: Every category satisfying these axioms has injective resolutions. Such a category is called an AB5 category, and sometimes around the 1960s a Grothendieck category though that term has been used in several senses.

So sheaves on any topological space have injective resolutions and thus have derived functor cohomology in the strict sense. For paracompact spaces this agrees with cohomology from fine, flabby, or soft resolutions. So you can still use those, if you want them, and you will. But Grothendieck treats paracompactness as a “restrictive condition”, well removed from the basic theory, and he specifically mentions the Weil conjectures.

Beyond that, Grothendieck’s approach works for topology the same way it does for all cohomology. And, much further, the axioms apply to many categories other than categories of sheaves on topological spaces or categories of modules. They go far beyond topological and group cohomology, in principle, though in fact there were few if any known examples outside that framework when they were given.

Diagrammatic Political Via The Exaptive Processes

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The principle of individuation is the operation that in the matter of taking form, by means of topological conditions […] carries out an energy exchange between the matter and the form until the unity leads to a state – the energy conditions express the whole system. Internal resonance is a state of the equilibrium. One could say that the principle of individuation is the common allagmatic system which requires this realization of the energy conditions the topological conditions […] it can produce the effects in all the points of the system in an enclosure […]

This operation rests on the singularity or starting from a singularity of average magnitude, topologically definite.

If we throw in a pinch of Gilbert Simondon’s concept of transduction there’s a basis recipe, or toolkit, for exploring the relational intensities between the three informal (theoretical) dimensions of knowledge, power and subjectification pursued by Foucault with respect to formal practice. Supplanting Foucault’s process of subjectification with Simondon’s more eloquent process of individuation marks an entry for imagining the continuous, always partial, phase-shifting resolutions of the individual. This is not identity as fixed and positionable, it’s a preindividual dynamic that affects an always becoming- individual. It’s the pre-formative as performative. Transduction is a process of individuation. It leads to individuated beings, such as things, gadgets, organisms, machines, self and society, which could be the object of knowledge. It is an ontogenetic operation which provisionally resolves incompatibilities between different orders or different zones of a domain.

What is at stake in the bigger picture, in a diagrammatic politics, is double-sided. Just as there is matter in expression and expression in matter, there is event-value in an  exchange-value paradigm, which in fact amplifies the force of its power relations. The economic engine of our time feeds on event potential becoming-commodity. It grows and flourishes on the mass production of affective intensities. Reciprocally, there are degrees of exchange-value in eventness. It’s the recursive loopiness of our current Creative Industries diagram in which the social networking praxis of Web 2.0 is emblematic and has much to learn.

Without Explosions, WE Would NOT Exist!

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The matter and radiation in the universe gets hotter and hotter as we go back in time towards the initial quantum state, because it was compressed into a smaller volume. In this Hot Big Bang epoch in the early universe, we can use standard physical laws to examine the processes going on in the expanding mixture of matter and radiation. A key feature is that about 300,000 years after the start of the Hot Big Bang epoch, nuclei and electrons combined to form atoms. At earlier times when the temperature was higher, atoms could not exist, as the radiation then had so much energy it disrupted any atoms that tried to form into their constituent parts (nuclei and electrons). Thus at earlier times matter was ionized, consisting of negatively charged electrons moving independently of positively charged atomic nuclei. Under these conditions, the free electrons interact strongly with radiation by Thomson scattering. Consequently matter and radiation were tightly coupled in equilibrium at those times, and the Universe was opaque to radiation. When the temperature dropped through the ionization temperature of about 4000K, atoms formed from the nuclei and electrons, and this scattering ceased: the Universe became very transparent. The time when this transition took place is known as the time of decoupling – it was the time when matter and radiation ceased to be tightly coupled to each other, at a redshift zdec ≃ 1100 (Scott Dodelson (Auth.)-Modern Cosmology-Academic Press). By

μbar ∝ S−3, μrad ∝ S−4, Trad ∝ S−1 —– (1)

The scale factor S(t) obeys the Raychaudhuri equation

3S ̈/S = -1/2 κ(μ +3p/c2) + Λ —– (2)

where κ is the gravitational constant and Λ the cosmological constant.

, the universe was radiation dominated (μrad ≫ μmat) at early times and matter dominated (μrad ≪ μmat) at late times; matter-radiation density equality occurred significantly before decoupling (the temperature Teq when this equality occurred was Teq ≃ 104K; at that time the scale factor was Seq ≃ 104S0, where S0 is the present-day value). The dynamics of both the background model and of perturbations about that model differ significantly before and after Seq.

Radiation was emitted by matter at the time of decoupling, thereafter travelling freely to us through the intervening space. When it was emitted, it had the form of blackbody radiation, because this is a consequence of matter and radiation being in thermodynamic equilibrium at earlier times. Thus the matter at z = zdec forms the Last Scattering Surface (LSS) in the early universe, emitting Cosmic Blackbody Background Radiation (‘CBR’) at 4000K, that since then has travelled freely with its temperature T scaling inversely with the scale function of the universe. As the radiation travelled towards us, the universe expanded by a factor of about 1100; consequently by the time it reaches us, it has cooled to 2.75 K (that is, about 3 degrees above absolute zero, with a spectrum peaking in the microwave region), and so is extremely hard to observe. It was however detected in 1965, and its spectrum has since been intensively investigated, its blackbody nature being confirmed to high accuracy (R. B. Partridge-3K_ The Cosmic Microwave Background Radiation). Its existence is now taken as solid proof both that the Universe has indeed expanded from a hot early phase, and that standard physics applied unchanged at that era in the early universe.

The thermal capacity of the radiation is hugely greater than that of the matter. At very early times before decoupling, the temperatures of the matter and radiation were the same (because they were in equilibrium with each other), scaling as 1/S(t) (Equation 1 above). The early universe exceeded any temperature that can ever be attained on Earth or even in the centre of the Sun; as it dropped towards its present value of 3 K, successive physical reactions took place that determined the nature of the matter we see around us today. At very early times and high temperatures, only elementary particles can survive and even neutrinos had a very small mean free path; as the universe cooled down, neutrinos decoupled from the matter and streamed freely through space. At these times the expansion of the universe was radiation dominated, and we can approximate the universe then by models with {k = 0, w = 1/3, Λ = 0}, the resulting simple solution of

3S ̇2/S2 = A/S3 + B/S4 + Λ/3 – 3k/S2 —– (3)

uniquely relating time to temperature:

S(t)=S0t1/2 , t=1.92sec [T/1010K]−2 —– (4)

(There are no free constants in the latter equation).

At very early times, even neutrinos were tightly coupled and in equilibrium with the radiation; they decoupled at about 1010K, resulting in a relic neutrino background density in the universe today of about Ων0 ≃ 10−5 if they are massless (but it could be higher depending on their masses). Key events in the early universe are associated with out of equilibrium phenomena. An important event was the era of nucleosynthesis, the time when the light elements were formed. Above about 109K, nuclei could not exist because the radiation was so energetic that as fast as they formed, they were disrupted into their constituent parts (protons and neutrons). However below this temperature, if particles collided with each other with sufficient energy for nuclear reactions to take place, the resultant nuclei remained intact (the radiation being less energetic than their binding energy and hence unable to disrupt them). Thus the nuclei of the light elements  – deuterium, tritium, helium, and lithium – were created by neutron capture. This process ceased when the temperature dropped below about 108K (the nuclear reaction threshold). In this way, the proportions of these light elements at the end of nucleosynthesis were determined; they have remained virtually unchanged since. The rate of reaction was extremely high; all this took place within the first three minutes of the expansion of the Universe. One of the major triumphs of Big Bang theory is that theory and observation are in excellent agreement provided the density of baryons is low: Ωbar0 ≃ 0.044. Then the predicted abundances of these elements (25% Helium by weight, 75% Hydrogen, the others being less than 1%) agrees very closely with the observed abundances. Thus the standard model explains the origin of the light elements in terms of known nuclear reactions taking place in the early Universe. However heavier elements cannot form in the time available (about 3 minutes).

In a similar way, physical processes in the very early Universe (before nucleosynthesis) can be invoked to explain the ratio of matter to anti-matter in the present-day Universe: a small excess of matter over anti-matter must be created then in the process of baryosynthesis, without which we could not exist today (if there were no such excess, matter and antimatter would have all annihilated to give just radiation). However other quantities (such as electric charge) are believed to have been conserved even in the extreme conditions of the early Universe, so their present values result from given initial conditions at the origin of the Universe, rather than from physical processes taking place as it evolved. In the case of electric charge, the total conserved quantity appears to be zero: after quarks form protons and neutrons at the time of baryosynthesis, there are equal numbers of positively charged protons and negatively charged electrons, so that at the time of decoupling there were just enough electrons to combine with the nuclei and form uncharged atoms (it seems there is no net electrical charge on astronomical bodies such as our galaxy; were this not true, electromagnetic forces would dominate cosmology, rather than gravity).

After decoupling, matter formed large scale structures through gravitational instability which eventually led to the formation of the first generation of stars and is probably associated with the re-ionization of matter. However at that time planets could not form for a very important reason: there were no heavy elements present in the Universe. The first stars aggregated matter together by gravitational attraction, the matter heating up as it became more and more concentrated, until its temperature exceeded the thermonuclear ignition point and nuclear reactions started burning hydrogen to form helium. Eventually more complex nuclear reactions started in concentric spheres around the centre, leading to a build-up of heavy elements (carbon, nitrogen, oxygen for example), up to iron. These elements can form in stars because there is a long time available (millions of years) for the reactions to take place. Massive stars burn relatively rapidly, and eventually run out of nuclear fuel. The star becomes unstable, and its core rapidly collapses because of gravitational attraction. The consequent rise in temperature blows it apart in a giant explosion, during which time new reactions take place that generate elements heavier than iron; this explosion is seen by us as a Supernova (“New Star”) suddenly blazing in the sky, where previously there was just an ordinary star. Such explosions blow into space the heavy elements that had been accumulating in the star’s interior, forming vast filaments of dust around the remnant of the star. It is this material that can later be accumulated, during the formation of second generation stars, to form planetary systems around those stars. Thus the elements of which we are made (the carbon, nitrogen, oxygen and iron nuclei for example) were created in the extreme heat of stellar interiors, and made available for our use by supernova explosions. Without these explosions, we could not exist.

Conjuncted: Demise of Ontology

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The demise of ontology in string theory opens new perspectives on the positions of Quine and Larry Laudan. Laudan stressed the discontinuity of ontological claims throughout the history of scientific theories. String theory’s comment on this observation is very clear: The ontological claim is no appropriate element of highly developed physical theories. External ontological objects are reduced to the status of an approximative concept that only makes sense as long as one does not look too closely into the theory’s mathematical fine-structure. While one may consider the electron to be an object like a table, just smaller, the same verdict on, let’s say, a type IIB superstring is not justifiable. In this light it is evident that an ontological understanding of scientific objects cannot have any realist quality and must always be preliminary. Its specific form naturally depends on the type of approximation. Eventually all ontological claims are bound to evaporate in the complex structures of advanced physics. String theory thus confirms Laudan’s assertion and integrates it into a solid physical background picture.

In a remarkable way string theory awards new topicality to Quine’s notion of underdeterminism. The string theoretical scale-limit to new phenomenology that makes Quine’s concept of a theoretical scheme fits all possible phenomenological data. In a sense string theory moves Quine’s concept from the regime of abstract and shadowy philosophical definitions to the regime of the physically meaningful. Quine’s notion of underdeterminism also remains unaffected by the emerging principle of theoretical uniqueness, which so seriously undermines the position of modest underdeterminism. Since theoretical uniqueness reveals itself in the context of new so far undetected phenomenology, Quine’s purely ontological approach remains safely beyond its grasp. But the best is still to come: The various equivalent superstring theories appear as empirically equivalent but ‘logically incompatible’ theories of the very type implied by Quine’s underdeterminism hypothesis. The different string theories are not theoretically incompatible and unrelated concepts. On the contrary they are merely different representations of one overall theoretical structure. Incompatible are the ontological claims which can be imputed to the various representations. It is only at this level that Quine’s conjecture applies to string theory. And it is only at this level that it can be meaningful at all. Quine is no adherent of external realism and thus can afford a very wide interpretation of the notion ‘ontological object’. For him a world view’s ontology can well comprise oddities like spacetime points or mathematical sets. In this light the duality phenomenon could be taken to imply a shift of ontology away from an external ‘corporal’ regime towards a purely mathematical one. 

To put external and mathematical ontologies into the same category blurs the central message the new physical developments have in store for philosophy of science. This message emerges much clearer if formulated within the conceptual framework of scientific realism: An extrapolation of the notion ‘external ontological object’ from the visible to the invisible regime remains possible up to quantum field theory if one wants to have it. It fails fundamentally at the stage of string theory. String theory simply is no theory about invisible external objects.