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Maths & physics

‘All science is either physics or stamp collecting’
Ernest Rutherford, British chemist and physicist c. 1900
Many reasons can be found for placing mathematics and physics at the forefront of the sciences. Since at least the time of the classical philosophers of Ancient Greece mathematics has been treated as a model or template for all knowledge including physics, as the mode of thinking towards which all other thinking should aspire. A sign above the entrance to Plato’s Academy in ancient Athens read: ‘Let no-one ignorant of geometry enter here‘.


Artist’s impression of Gravity Probe B orbiting the Earth to measure space-time
This is a four-dimensional description of the universe including height, width, length, and time using differential geometry
Differential geometry is the language in which Einstein’s General Theory of Relativity expresses the smooth manifold that is the curvature of space-time – which allows us to position satellites in orbit around the earth. Differential geometry is also used to study gravitational lensing and black holes
The Riemannian geometry of relativity is a non-Euclidian geometry of curved space
Courtesy Wikimedia Commons
Image sourced from NASA at

In the ancient civilizations of Egypt and Mesopotamia it was mathematics, physics, and astronomy that dominated scientific enquiry as they studied the heavens, recording their observations in tables and charts that later became known to the Greeks. Mathematics had practical application since it provided the precision needed to engineer the magnificent monumental architecture we associate with classical civilization. Numerologists like Pythagoras (c. 570–495 BCE) became cult figures for thinking men. The pre-Socratic philosophers had examined the nature of substance, looking for universal properties and fundamental elements, bequeathing to their successors the idea of four foundational elements – Earth, Air, Fire, and Water – in a tradition that continued into the Medieval world, along with Democritus’s idea of matter being composed of tiny indivisible particles of matter called atoms. The study of living organisms, we believe, did not really get started until the time of Aristotle (zoology) and Theophrastus (botany). Only then do we see the emergence of a critical analytic curiosity in organisms themselves rather than just their utilitarian value as food, medicines, and materials. So biology, it seems, arrived as an afterthought in scientific enquiry as expressed so eloquently in Aristotle‘s ‘Invitation to Biology”.

‘It is not good enough to study the stars no matter how perfect they may be. Rather we must also study the humblest creatures even if they seem repugnant to us. And that is because all animals have something of the good, something of the divine, something of the beautiful’ … ‘inherent in each of them there is something natural and beautiful. Nothing is accidental in the works of nature: everything is, absolutely, for the sake of something else. The purpose for which each has come together, or come into being, deserves its place among what is beautiful’
Aristotle – De Partibus Animalium (The Parts of Animals) – 645 a15

The universality of mathematics
Mathematical knowledge has a unique and appealing beauty: it gives us knowledge that is: certain; incorrigible (it does not undergo revision in the way that empirical facts do); timeless or eternal (we are inclined to think that 2 = 2 = 4 must always be true: it was true before humans occupied the world and it would even be true if no universe existed); and it is necessary (its truths seem to lie outside our world of space and time and yet they can be grasped by our reason, they could not be otherwise). In addition, numbers are not causally interactive.

All this makes mathematical knowledge highly abstract since we are not really sure what it is actually about. The simple answer is of course ‘it is about numbers’, but the concept of number has baffled philosophers from the earliest times. If numbers do not actually exist in space and time and they are causally inert (they are abstract objects) then how can we have any knowledge of them? There is no universally-agreed answer to this question but three broad approaches. Either they are independent abstract objects, or they are in the world, or they are mental constructs. The details need not concern us but if numbers do not depend on experience then perhaps we have some special faculty of numerical perception (say, the intuitive abstract objects of Kant), or we can relate them to set theory, to objects in the world (logicism) or the yare simply mental constructs. Each of these positions has major difficulties and the question still has no universally accepted answer. The fact that mathematics is so abstract means we have every reason to dismiss it as some kind of mental construct, a phantom of our minds. But maths has been applied directly to the world in a practical and economic way that has had an immeasurable impact on human life (see, for example, Gravity Probe B illustrated above). There are the many facts about the material world that were first suggested by mathematics before being empirically proven – for example, the Higgs Boson, gravity waves, the existence of Neptune, and the speed of light.

Because numbers seem to have a special kind of reality (and probably under the influence of the charismatic Pythagoras) Plato postulated his world of Forms, (Plato’s world of forms was a world of timeless truths, of generalities, not to be thought of like a separate place from Earth, like a heaven), it was a realm of ideas that could be accessed and applied by reason. This was a special kind of objective knowledge superior to empirical knowledge which, being derived from experience and sensation, was contingent and corrigible.

But how can we possibly believe in the objectivity of such an abstract realm and, anyway, how could we possibly connect with it?

Aristotle did not believe in Plato’s world of Forms, considering number to exist in the world as a property of objects. But, as philosophers later pointed out, how can number exist in a pair of shoes (one pair or two shoes)? Is the property in such a case 1 or 2? Philosopher Kant believed mathematics to be a form of innate intuition, an expression of our human sense of space and time. Arithmetic expressed, through number, our linear and sequential experience of time, while geometry was a way of representing our sense of space. For Kant then mathematics was an abstraction that came from our heads, it did not exist objectively in the world.

The subjectivity or objectivity of number (whether numbers are real) remains a matter for intense intellectual debate. The impact of mathematics on the world cannot be questioned, and the security we feel as a consequence of its necessity, universality, and certainty have given it a special place in the scientific vision of reality . . . so it is hardly surprising that it has been emulated by other disciplines. In physics we see its universality reflected in the laws of physics.

Modernity has maintained its reverence for the application of mathematics to scientific theories and concepts but with the recognition that maths, at its core, is logical not empirical, it is founded on subjunctive statements (if … then): if X (this may be an axiom) then Y. As philosopher David Hume expressed it, maths is about ‘relations of ideas’ not ‘matters of fact’ … it is not empirical.

Principle 1 – Mathematics was inherited from the ancient world as the most secure form of knowledge. As it provided certain, necessary, timeless and universal truth it was a form of knowledge against which the statements any science could be measured and to which all science should aspire

The explanatory regress
Nothing can explain itself. Reductionism, like all science, is a form of explanation: it gives a clarification, simplification, reasons, or justification. Aristotle noticed that when we provide an explanation, to be logically consistent, the explanation itself in all likelihood requires explanation itself. At some stage in this process we might think that one particular explanation is enough for our purposes – but that does not mean that logically the questioning could not continue. Every time we provide an explanation for a natural phenomenon we can, like a child, keep asking ‘But why?’ – there is usually yet another question to ask.

Explanations, like philosophical justification, can threaten an infinite regress. Sometimes explanations can seem sufficient for purpose but they can also continue indefinitely, or become circular, or find some foundation. This foundation serves as an unquestioned base that consists of either self-evident or unjustified truths and axioms (sometimes called primitive or brute facts). A good example of a scientific brute fact is a law of physics. We feel a compulsion to be as fundamental as possible in our explanations: if further questions can be posed then the problem has not been adequately addressed. But science stops at physical laws – even though we cannot explain why these laws are as they are or, indeed, why there are any laws at all.

So, one effective way of bringing a regress to a halt is to find an explanation that does not need justification, one that is beyond question, primitive, self-evident, or a brute fact. At this point it becomes futile looking for further definitions, explanations or proofs because such foundational concepts presuppose the things they are meant to be explaining. In mathematics basic assumptions are known as axioms and they form the foundational structure on which everything else rests. If the axioms are unreliable then the entire edifice comes crashing down. This mode of thinking may be called foundationalism. Aristotle used this principle to underpin his logic of scientific demonstration, the famous deductive syllogism which passed from a universally reliable foundational principle to a particular instance such that the premise necessarily entails the conclusion (e.g. All swans are white (foundational or universal principle), this is a swan (particular instance), therefore this swan is white). (e.g. This is a swan, all observed swans have been white, therefore this swan is probably white). The conclusion of a deductive argument appears certain while that of an inductive argument has degrees of probability that depend on the quality of evidence.

Principle 2 – Foundationalism – is the search for secure assertions that can be taken as the underpinning for other statements and assertions

Physics, in investigating the material world proceeds analytically by breaking up matter into ever smaller parts, a process that, over the years, has always found (albeit different) an apparent material ‘rock bottom’. In 1947 this was electrons, protons and neutrons, later it became quarks and other sub-atomic particles, today we have fermions and bosons. In this way all our explanations of matter have brought us to an end point, what we might indeed call the ‘fundamental reality’ of matter and existence . . . the smallest scientifically acceptable units as described by physics.

Principle 3 – Smallism – physics explains matter by proceeding analytically and experimentally to discover its smallest indivisible constituents, its fundamental particles

We can refer to our intuition that the small units of physics and chemistry are fundamental to both matter and material explanation as ‘scientific fundamentalism’. From this flows the sense of what has also been called ‘generative atomism’, the belief that, like a childs Lego set any whole can be built out of its fundamental building blocks. To understand the whole we must start with the parts. Small units, it might seem, somehow have greater scientific credibility; they are more authoritative and reliable; they provide better explanations; they are less complicated and therefore more easily understood and they are objects studied by physics.

Principle 4 – Fundamentalism – is the assumption that all scientific explanation of matter must ultimately reduce to explanation of the smallest known particles of matter and their interactions

In arriving at the smallest or fundamental constituents of matter we have a feeling of finality: being fundamental we might feel that these constituents are in some sense more real than the wholes of which they were a part. But this is clearly some kind of mental trickery, a cognitive illusion. There is nothing more ‘real’ about a fundamental particle than an elephant. Indeed, because we can see, touch, and hear an elephant we might argue that the elephant is more empirically real than an invisible fermion or boson (which has a smaller wavelength than that of light). We regard small thigs as special not because of their mere existence (their ontology or being) but because of their role in analysis and explanation (their significance is epistemological). They are part of our habitual explanation of wholes in terms of their components and the relations between these components. Following Aristotle’s explanatory regress our explanations must therfore bottom-out at the smallest particles we know at any point in history.

Sometimes referred to as ‘ontological reduction’ this principle asserts that no physical object ‘exists’ more or less than any other. Smaller units of matter are no more ‘real’ than larger units of matter, nor are more inclusive or less inclusive units, or even more or less complex units. In terms of existence or reality atoms, rocks, bacteria, and humans are equals.

Principle 5 – All matter exists equally: no physical object ‘exists’ more or less than any other. Smaller units of matter are no more ‘real’ than larger units of matter, nor are more inclusive or less inclusive units, or more complex or less complex units (principle of flat ontology)

Reduction, organization, explanatory power
What is controversial in reductionism and science today is not the matter itself (ontological reduction) – but the nature of its organisation, the relations between its parts (epistemological reduction) – especially the parts of living organisms. We must therefore look for other reasons for our prioritization of one domain of knowledge over another, for the intuition that explanations in one domain are in some way superior (have greater explanatory power) than those in another: why, for example, we might consider it useful to think of biology in terms of physico-chemical processes. Why does scientific fundamentalism have such persuasive power over our general attitude to science. If all matter is ontologically equivalent then it is our cognitive focus that is making a distinction between different domains or scales of existence (the physicochemical, biological, social, psychological and so on). Analysis has explanatory power but this does not make the parts under consideration, either their size or inclusiveness, more ‘real’. On reflection we realise that no sort of matter is more real or fundamental in itself. Matter is just matter: small matter is just smaller than big matter, it does not have properties that make it existentially privileged in any way. So, in terms of material reality or existence (ontology) a bison is just as real as a boson.

When we take an overview of all the sciences is it true that ‘Particle physics is the foundational subject underlying – and in some sense explaining – all the others‘?[1] Could this be simply a comment on the way analysis is a habituated mode of explanation? To investigate the regress of scientific explanation to foundational particle physics we need to look at different kinds of explanation.

Explanatory rock bottom and adequate explanation
We might assume that, of necessity, the explanatory regress passes to ever smaller and ‘more fundamental’ material objects. But this is not inevitable: sometimes one particular kind of explanation is sufficient. Sometimes we feel no need to enter an explanatory regress. One particular answer is adequate.

Here are a couple of everyday examples of explanation. First, if asked ‘Why did the chicken cross the road?’ we could call on answers from scientific specialists such as a chicken biochemist, a neurologist, an endocrinologist, and an animal psychologist. But what if we were told that the chicken was being chased by a fox. This, surely, for most of us, is a satisfying and sufficient answer to our question. We do not need or desire to be told anything else. Does this mean that in this case scientific answers were incorrect or inferior in some way? No, only that their explanations were not the most appropriate for the circumstances under consideration. Statements like ‘polar bears hibernate in winter’, ‘inflation can be managed by adjusting interest rates’, ‘evolution is replication with variation under selection’, or even ‘e = mc2’ appear sufficient in themselves: their veracity may be challenged but we do not think they need reformulating or reducing to improve or clarify what is being expressed.

Practical incoherence
Firstly, there is the logical absurdity of trying to explain all phenomena in terms of the smallest workable scientific particles. What is to be achieved by explaining many biological facts in this way, like the fact that polar bears hibernate in winter? Examples become more ludicrous as we consider wider scientific contexts. How could we possibly explain a rise in interest rates in terms of fundamental physical particles and the laws of physics? What would such an explanation possibly look like? It is not that such a situation is logically impossible. We can imagine a supercomputer of the future that could enumerate the many causal factors at play in such a situation but we simply do not think this way, and nor do we need to. Explaining the causes of an interest rate rise in physicochemical terms would not simplify matters and give greater clarity, it would entail an explanation so complex as to be barely imaginable.

What then constitutes a satisfactory scientific answer to a scientific question?

Principle 4 – The principle of sufficient explanation: explanations are fit for purpose, they do not need to be circular, foundational, or part of an infinite regress

This example demonstrates the multi-causal nature of many occurrences – like car and plane crashes. Questions about cause(s) in such situations are not abandoned because of their complexity since they must achieve a resolution in a court of law. In many instances, in spite of the apparent complexity, rulings are readily made.

Our intuitive desire for foundational explanations creates several difficulties.

The primacy of analysis – generative atomism
If someone asks you ‘What is a heart and how does it work?’ we might answer analytically by treating the heart as a whole and explaining the parts and how they interact. Alternatively we might answer synthetically by treating the heart as a part and explaing how it interacts with other organs to contribute to the functioning of the body as a whole.

Much of science proceeds by explanatory analysis, breaking down physical entities into their constituent parts. But here too Aristotle’s dictum applies as we are inclined to proceed in a regress to ever smaller parts until we feel we have reached rock bottom, the world’s fundamental particles. There has, in the course of history, been a variable rock bottom. If the future continues as the past then there is nothing absolute, necessary, or certain about the particles that make up rock bottom. Democritus defined atoms as indivisible particles but physics has split the atom again an again with today perhaps fermions and bosons approximating the foundational bricks out of which the universe is constructed.

Scope – universality of physical constants
Physics approaches mathematics in the (near) universality of of its physical constants. Since it has a universal scope it also has an all-embracing character that is not shared by other sciences: its principles, theories, and laws are of such generality that they encompass all matter excepts under the most extreme situations. A falling stone and a falling monkey both conform to Newton’s laws of gravitational attraction. Physics tries to explain the world at not only the smallest scale as the behaviour of fundamental particles but also at the widest scale as constants or constraints that apply universally to all matter.
The foundations of science are generally taken to lie in mathematics and physics because their basic assumptions have universal application in two important ways: firstly, physics works with the stuff of the universe at its extremes – from the smallest particles to the cosmos in its entirety; secondly

Principle 3 – Physics combines with mathematics to formulate constants and constraints that apply to not only the smallest known particles but to the universe as a whole and therfore its scope is wider than that of other sciences

So what have we decided constitutes something being more scientific or less scientific?

Arguing that that one is ‘more scientific’ than another requires an extended justification. So far we might claim, for example, that physics encompasses all matter, while biology only deals with living matter. Physics deals with generalities and regularities that apply throughout the universe while biology only deals with the subset of generalities that relate to living orgnisms. Whatever principles and generalities we can establish in relation to life appear to lack the scope and reliability that we see in physical laws.

Because both a rock and an elephant conform to the same effects of gravity does not automatically mean that physics is more fundamental.

Adding value
We might intuitively feel that the objects of an explanation (the explanans) are more fundamental than the object being explained (the explanandum)

True science, special science, hard and soft science
Has this account so far established a clear distinction between fundamental or foundational science and other science? Can we distinguish between hard and soft sciences, or indeed between science and non-science – or are such distinctions just a matter of semantics? The term ‘special sciences’ is generally used to denote those sciences dealing with a restricted class of objects as, say, biology (living organisms), and psychology (minds) while physics, in contrast, is kown as ‘general science’. Reductionism would maintain that the special sciences are, in principle, reducible to physics or entities that may be described by physics.

Can we establish a clear benchmark using criteria of certainty, necessity, universality, corrigibility (falsifiability), certainty, or predictive capacity by which to rank in order the following areas of study: mathematics, physics, astrology, genetics, biology, psychoanalysis, psychology, history, political science, sociology, and economics. Would this establish a reliable table of scientific merit? Are such ranking criteria appropriate or should other factors be considered and, if so, what would they be?

In spite of many historical attempts, the philosophy of science has failed to establish uncontroversial necessary and sufficient conditions that would satisfy a definition of ‘science’ (see Science and reason). At present it appears that what we call science is, more or less, our most rigorous application of reason to an assemblage of theories, principles, and practices that share a family resemblance as a means of enquiry. It is this that has proved our most effective way of organising the knowledge we use to understand, explain, and manage the natural world.

In at least a practical and intellectual sense the special sciences are autonomous, their explanations, methodologies, terms, and objects of study are perceived as self-sufficient without any requirement or benefits flowing from translation to another scale or ‘lower level’ in spite of assumptions about successful reductions in the past and the causal completeness of physics.

Fundamental can be ontic (that out of which everything is made – microphysics) or epistemic (that to which everything conforms).

When we reduce are we suggesting a relation of identity between the reduced and reducing entities that justifies the elimination of the reduced entity: or are we merely referring to differe3nt modes of describing the same thing?

Method & subject-matter

It is a characteristic of explanation that it abstracts: it considers one particular aspect of the natural world to the exclusion of a more general context. In general our focus is on the explanation, not the context, the context being assumed or taken for granted. When a biologist gives an explanation of the way a heart pumps blood, it is assumed that the laws of physics are in operation – this does not have to be stated. Thus all explanations we provide have two key characteristics: firstly, abstraction – that is, they abstract from a greater whole, they focus on a particular situation or object while ignoring the context; secondly, they enter a potential analytic or synthetic regress. Explanations thus resemble our perceptive and cognitive focus by paying attention to a particular set of circumstances (foreground) while ignoring the wider context (background). In providing an explanation there is a kind of unspoken rider … something along the lines … ‘assuming the uniformity of nature, and other things being equal (ceteris paribus)’.

Principle 3 – Explanations abstract information from a wider context

Proximate & ultimate explanation
Is sex for recreation or procreation?

A proximate explanation is the explanation that is closest to the event that is to be explained while an ultimate explanation is a more distant reason. In behaviour a proximate cause is the immediate trigger for that behaviour: the proximate cause for running might be a gun shot, the ultimate cause being survival. Biology itself divides in its approach to proximal and ultimate causes. Ultimate causes usually relate to evolution and adaptation and therefore function, answering the question of why selection favoured that trait – and the answers tend to be teleological. Proximal causes deal with day-to-day situations and immediate causation. Proximate and ultimate explanations are complementary, they are not in opposition with one being better or more explanatory than the other, both have their place. This is a trap for the unwary since proximate answers can be mistakenly given to ultimate questions.

So, one possibility is that there is no privileged perspective that entails all others, each is equally valid and the explanation that is most appropriate will depend on the particular circumstances. In all this we are abstracting and studying certain factors while ignoring others. When we study the genetic code we do not consider it appropriate to think about electrons and quantum mechanics: when we study the heart we do not worry about gravity or consult the periodic table.

Principle 9 – Satisfactory explanations generally depend, not on the size of the units under consideration or the inclusiveness of the frame of reference, but the plausibility, effectiveness, or utility of the answer in relation to the question posed.

So, sex is for both procreation and pleasure.

(Is the explanation contingent on our human interests and limitations or is it a full causal account?)

4. Unity of science, spatiotemporal boundaries, scope & scale
As science progressed it provided increasingly elegant summations of knowledge about the physical world. Apparently disparate phenomena were united under common laws that could be expressed using mathematical equations: the motion of the planets, the behaviour of fluids, electricity, and light. The integration of physics and mathematics had such explanatory and predictive power in relation to so many phenomena that there seemed no end to what they might achieve. Gravity was a universal force that treated falling rocks and falling monkeys with absolute equality. Physics embraced space and time, matter and energy – and that was mighty close to everything. Its explanatory breadth and predictive power was, and still is, thoroughly demonstrated through its spin-off technology. Today our GPS systems integrate space flight and complex electronics with relativity theory and quantum physics to provide flat earth maps on our car navigation systems. There was a vision of physics as a fundamental discipline incorporating all other knowledge. Physics was universal in scope and scale while other scientific disciplines dealt with only sub-sets of the physics enterprise. So, for example, physics encompassed all matter, biology only living matter, animal behaviour all sentient living matter, sociology humans as they interact in groups, anthropology human beings, human psychology human brains and behaviour. This characterization of science presents us with a metaphysical monism: there is one scientific truth for one reality based on one set of underlying principles (scientific laws). This vision is generally referred to as the ‘unity of science’.

Principle – Scientific fundamentalism is a metaphysical monism: there is one scientific truth for one reality based on one set of underlying principles (scientific laws). This monistic vision is generally referred to as the ‘Unity of Science’

All the convoluted complication of complexity – the mess of multiplicity of objects – their properties, relations, and aggregations – can be simplified and reduced by analysis as the adoption of a philosophy approximating monism as a description of the many in terms of the few. Scientifically we do this by means of the elementary particle, generalization to principles and laws, and systematization.

For some physicists there is a goal like a ‘unified field theory’: when quantum mechanics is reconciled with relativity then our account of the physical world will be complete.

Principle 10 – The unity of science (metaphysical monism) – there is one scientific truth for one reality based on one set of underlying principles (scientific laws)

Does this universal character of physics give some kind of precedence to physics: does it make physics more ‘fundamental’?

Principle 10 – Because physics is broad in scope it seems to encompass or absorb other disciplines of more limited scope.


The unity of science
We can define scientific fundamentalism as the view that the smallest particles of matter and the principles and theories of physics and chemistry underpin all other science. There are at least five reasons why this view has appeal.

1. The analytic process of explaining wholes in terms of their constituent parts has explanatory weight that suggests parts are in some way more real or fundamental than wholes.
2. Second, analytic explanation, like the philosophical requirement for rational justification or causal origin, leads to an explanatory regress seeking ever more ‘fundamental’ solutions and suggesting that there must be rock bottom or ultimate explanation that can only lie within physics
3. The explanation of the complex in terms of the simple reduces causal complexity
4. The scope of physics (the universe, space, time, and matter) suggests that it must incorporate or subsume all other scientific disciplines
5. Fifth, both philosophers and scientists when explaining natural phenomena employ the metaphorical hierarchical language and imagery of levels of organisation. Though a convenient mental device hierarchical thinking suggests that the natural world is itself ranked from high to low (with physics as a foundation) . Talk of hierarchical organisation is better replaced by the language of scale.
6. It is a consequence of the historical tradition coming to us from antiquity wjereby the physics of astronomy and mathematics both preceded and received greater attention than biology although subjects that today we might call political and social science were regarded a very important.

Aristotle’s gave science its foundation in reason through deductive logic while scientists of the early modern period emphasized inductive logic and the importance of an emphasis on the world itself, on experiment and observation. Up to the 1960s there was a hope and belief that science could be defined and unified under a common set of principles. Today this ambition is meeting strong opposition because it seems that we have no conclusive criterion clearly demarcating science from non-science. That does not mean that astrology is science: robust scientific explanation entails many demanding criteria that astrology fails meet. But the distinction between the sciences of physics, chemistry, biology, the social sciences, history, and everyday reasoning is one of family resemblance or degree, not necessary and sufficient demarcation. Foundationalism with its insistence on science as a unique and special form of knowledge grouded in physics has been replaced by coherentism or pragmatism, the view of science as a coherent system of justified belief, a system of shared ideas that work.

The view that physics somehow expresses ‘reality’ more effectively than other disciplines (scientific fundamentalism) comes from the general impression given by these factors. However, parts do not have some special quality (ontological privilege) or are more ‘real’ than wholes. Scientifically credible units of matter have no intrinsic (ontological) precedence over one-another based on size or inclusiveness alone. Smaller units of matter (molecules) are no more ‘real’ than larger units of matter (dogs and cats). However they might have utility in explanation and large units may be more complex in terms of their causation and our conceptual understanding of them.

All explanation abstracts from a wider context and, in this sense, it is reduction. Though it is in the nature of explanation to ‘reduce’ by looking at constituent parts the adequacy of the explanation does not depend on the size of the units under consideration, but the plausibility, effectiveness, or utility of the answer in relation to the question posed. Using parts to explain wholes gives parts explanatory value but does not make them more ‘fundamental’ in any meaningful physical sense.

Though the objects of physics are no more real or fundamental than those of biology, it is evident that adaptive complexity (life) involves intricate systems of causality that increases the difficulty of prediction at smaller scales. The greater complexity (causal relations) of the domain units under consideration, the greater the difficulties in prediction, communication, and translation into other domains.

Aristotle’s Objection
Pre-Socratic natural philosophers were materialists who regarded nature as consisting only of matter (Earth, Air, Fire, and Water in some combination). Aristotle criticised this view because matter is always changing. Any functional structure such as an organism can have all or some of its matter replaced by different matter and yet retain its identity as a particular organism. That is, continuity is maintained, but not through the matter of an organism but through its arrangement or functional structure; although we have a concept of ‘dog’, each individual and kind of dog consists of different matter; matter is just ‘stuff’, when an organism grows it grows in a particular structured way, it does not simply add to what is already there by simply getting larger.

For Aristotle an, organism’s form rather than its matter is its nature. To understand an animal or plant we need to know not only its constituent matter but the way it is structured and why it is structured in a particular way. Matter is necessary to create form but it is subordinate to it.

Biology is not just molecules, it is molecules of certain kinds integrated in ways that give rise to unique properties. A living organism (life) is very different from a rock (inanimate matter). Every physical thing is physical, but not every physical thing is biological. There is no privileged bottom level or a universe consisting of one stuff: all representations are partial.

In science ‘black box’ refers to a system whose inputs and outputs are known but not the inner workings. We really need a corresponding term ‘white box’ to indicate the explanation of the inner workings of a system that ignores or takes for granted the context outside the system which, in much of science, might be expressed as ‘the uniformity of nature’.

LaPlace’s demon
Scientific explanation is steeped in the culture of causation and hence determinism. Lurking in the background there is always the figure of Laplace’s demon, the claim that someone (the demon) who knows the precise location and momentum of every atom in the universe (to infinite precision) at a given time should, in principle, be able to calculate all past and future states of the universe.

Explanation by analysis & synthesis
All explanations abstract certain features from a wider circumstance and in this sense they are reductionist.

When we wish to explain the structure and/or function of a particular physical object, as we have seen, we do not explain it in terms of itself but either in terms of the structures out of which it is composed or the role that it plays within a greater whole (or both). Which option we choose (analysis or synthesis) depends to some extent on the particular object that we choose to explain and understand. If, say, the object is gold, Au, then I tend to proceed by analysis, looking for the atomic number, density, bouiling point and so on. It is true that I gain a better understanding of gold if I see where it fits in the periodic table in relation to other elements but my focus of interest is on the element itself and the method of analysis. In contrast, if I want to understand and explain the heart then, although I can explain its division into auricle, ventricle, valves and so on, but it is difficult to just rely on such factors without explaining the role that the heart plays within a body, that is its relation to the other organs within a greater whole. In this case we proceed by both analysis and synthesis.

This is the methodology of explanation but, also as already considered, the success of the outcome depends on the purpose for which the explanation was given.

Science has always fought over what appear to be these alternative or opposing methodologies. On the one hand knowledge and understanding is to be gained by placing an object in its full and natural context (synthesis). On the other hand we try to understand the same object by isolating it from its natural context in order to better understand its unique features (analysis).

Scientific utility
We may simply choose the explanations, terms, definitions, laws, and assumptions (categories) that provide answers to the particular questions that concern us.

Principle 14 – there is no unequivocal criterion that distinguishes science from non-science

If we assume that science proceeds by the constant critical scrutiny and refinement of our scientific categories (which include theories and generalisations, principles, names, definitions, laws, phenomena, and so on) as we map our concepts onto the natural world itself (reality). The better we can explain and understanding the world the better we can manage it. And of course science has extended our senses through technology like microscopes and telescopes which have allowed us to experience the world that lies beyond our natural biology and sensory input.

Principle 15 -Scientific categories help us to organise the knowledge we use to understand, explain and manage the natural world.

hierarchy operates like the ‘stacking’ or subroutines in computer programming as nested subroutines are completed, returning to the primary routine: it also resembles the nesting and trees that occur in generative grammar (Chomsky hierarchy).

As we establish new cognitive frames of reference with the macro-microscope so causation appears to occur between the different cognitive categories. A molecule causes W, a leg causes X, a body causes Y, a colony causes Z. Molecules do not cause legs, legs do not cause colonies, colonies do not cause biomes. We have to ask whether causation can occur in this way according to each frame (do causal ‘levels’ make sense?). Is there a nested hierarchy of causation. And does any particular kind of causation take priority? How does this relate to material, formal, efficient, and final cause of Aristotle. Adaptive significance deals with ultimate or teleological causes while mechanistic and developmental analysis deals with proximate causes.Explaining bird song

Principle 10 – The analytic process of explanation of large and complex in terms of small and simple persuades us that parts have some ontological privilege (are more real) than wholes – but parts can also be explained synthetically by considering their role within a greater whole

The use of the word reduction emphasises the size of the units under consideration rather than the actual source of the process which is based in the abstractive process of cognitive focus on scale.

A cognitive dissonance arises when we realize that we can think of such a grouping in two ways – either as progressive division (analysis) or progressive addition (synthesis) depending on whether we begin our thinking with the most-inclusive or least-inclusive category. The dissonance seems to arise in part because we think of groups as ranks and it is then difficult to think of ranks as being of equal status, we find it very difficult to resist our impulse to create rank-value: we also find it difficult to think of a particular system in terms of analysis and synthesis at the same time, and for similar reasons.

Principle 10 – Nested hierarchies can be understood in two ways as being either progressively inclusive or progressively divisive – to understand and describe the objects within the hierarchy we can proceed either by analysis or by synthesis (or both)

Explore top-down and bottom-up. Is the world nested?

Life is not just stuff but the dynamic constraints operational within dynamic structural relations that are inherited by the work of negentropy.

Senses in which all science is grounded in physics:
1. It aspires to the certain, necessary, timeless and universal truth of mathematics

The nature of explanation see [2]
We might regard metaphysics as the study of ‘what there is’ and/or the study of ‘what depends on what’. The latter refers to the way the human mind struggles to find order and the slippery relation between our mental ordering processes and the order of the world. Explanations proceed by ‘grounding’, by providing reasons. One ‘thing’ can be grounded in many ways and we can express grounding in many ways – as a means of justification, a reason, a cause, a foundational axiom, ‘because’ etc. So, for example, we explain wholes in terms of their parts.

The position argued here is that this grounding is illusory but it cannot be simply removed (eliminativism) because it serves valuable role (fictionalism)[3]. The grounding relation maybe between, say, facts and material objects.

Grounding also relates to our intuitions about the structure of reality – say, for example, that facts about biology, depend on facts about, chemistry, which depend on facts about physics etc. Dependence relations thus present us with structure which facilitates thought and further explanation. ‘Grounders’ (people who support the idea of grounding) support their views in several ways: that grounding is asymmetric higher level scientific facts depend on lower level scientific facts (if biology is explained by chemistry, then chemistry cannot be explained by biology); it is irreflexive (it cannot ground itself); it is transitive (if biological facts depend on chemical facts and chemical facts depend on physical facts then biological facts depend on physical facts); there is a fundamental or foundational ‘level’ of explanation where the process of grounding must stop and that this foundation explains everything else. Philosophers use the idea of supervenience to try and come to grips with grounding.

Examples of possible ‘grounds’ might be: for morality – non-moral properties like happiness, pleasure or pain; for material objects – the smallest possible particles; for logic – the way true propositions are based in the world (that the proposition ‘snow is white’ is true if in fact snow is white); the logically complex is grounded in the logically simple.

Grounding talk expresses our intuitions about dependence relations in reality – that some things are less ‘real’, or less significant than others.

Realists and eliminativists
Realists hold that grounding relations hold independently of what people may think or say; they are are independent of conceptual and linguistic schemes and people. They are discovered, not created. Eliminativists hold that grounding talk is incoherent or unintelligible and should be abandoned. Perhaps there are composition relations in tables but that is all etc., there is no role for grounding talk.

Reductionism is not just a thesis about the way the world is, iyt is also a thesis about what the mind is like as well.

Ultimate reality
The question of ultimate reality is a metaphysical question. Consider the following: any understanding of the universe can only be established from a particular point of view. There must be, as it were, an independent interpreter of whatever there is – a ‘point of view of the universe’. No such point of view exists with the unlikely exception of God who, arguably, must have a God’s-eye view. Secondly, it is reasonable to claim that the only worthwhile, relatively reliable, or non-controversial answer to such a question posed in human terms must ultimately rest on empirical evidence. With this tacitly agreed, ultimate reality then translates into the best that science has to tells us. For some reason many people then interpret the question as one about the nature of matter and its relations. Falling back on our predilection for analytic explanation and the mistaken conviction that the smallest is the most real the discussion of ultimate reality falls into a debate about fundamental particles, waves, fields, and the like. But the boson is no more real than a bison, or a human being.

[1] Ellis 2005
[2] see Naomi Thompson and Fictionalism about grounding
[3] Fictionalism can apply across many domains. So, for example, we can be fictionalist about numbers (i.e. numbers have no referents, but they are useful) and morality (there is no objective right or wrong, but the notion of right and wrong, good and bad serve an important role in human life)

Ellis, G.F.R. 2005. Physics, complexity and causality. Nature 435: 743

Physical reductionism is possible but explanatory reductionism is not.
Supervenience of th emental on the neuralogical was an idea introduced by Donald Davidson as a dependence relationship.

The article on reality and representation also discussed the way our minds, that is, our cognition based on the objects of our perception, attempt to put order into the confusing complexity of mental categories that make up reality. Working on the scientific image can improve the categories we use to describe the nature of reality but it does not give is an overall structure. We give structure to reality by applying metaphors that generally work well for us in daily life – by distinguishing between: what is bigger and what is smaller; what is contained in or is a part of something else; what is simple and what is tied to other factors in a complex relationship; and by what can be ranked or valued in relation to something else.

It was also noted that when we describe the physical world we do so from different perspectives: we can give different accounts and explanations of the same physical state of affairs. So, for example, we can give physical, chemical, biological, psychological, sociological accounts of what is the same physical situation.

The question than arises as to whether any one particular mode of explanation and description should have priority over others and, if so, for what reason? That is the topic of this article.

The problem of reduction in science brings together a web of ideas, beliefs and assumptions about the world. To help connect some of the threads of this story I have organized the discussion into a set of principles that can be used for easy reference.

So far we have considered cognitive segregation, the way our minds divide the world into meaningful categories of understanding, our percepts and concepts, and the way that our cognition allows us to, as it were, look beyond the world of our biologically-given human perception (the manifest image) to a less anthropocentric world that allows us to not only investigate the way other sentient organisms perceive the world but to investigate the composition and operation of the external world itself.

What about the world of solid objects around us? Our curiosity about substance stretches back to at least Democritus’s and his world of fundamental indivisible particles called atoms. This was not an observed world but a postulated metaphorical world. By the 1940s it was thought that we had reached the truly fundamental constituents of matter when atoms were split into protons, electrons and neutrons. The metaphor was still of ‘particles’ like billiard balls rotating in a solar-system-like way around a nucleus. The world of particles would be transformed into one of forces made up of fields. Since the 1940s the metaphor has been changed again. Particles have been replaced by waves: so the world outside our minds is perhaps best characterized as space which consists of interacting vibrating fields. The Higgs field explains where ‘particles’ get their masses.

Humans are, nevertheless, special. Our unique mode of representation and comprehension (our reasoning faculty and the capacity to communicate and store information using symbolic languages) allows us to look beyond the world of direct experience (the manifest image) towards the way the world actually is (the scientific image).

This is amply demonstrated by the time-honoured deference to ‘hard’ sciences like maths, physics and chemistry when compared to a ‘soft’ science like biology.

A force is due to a field and a field is something that has energy and a value in space and time like a magnetic field, temperature, and wind speed.
With increasing complexity comes greater difficulty in predicting outcomes. As a consequence biological principles and patterns seem to lack the precision and universality that we see in the laws of physics. Biological principles are derived from highly complex organisational and causal networks and open systems with a vast number of variables in which no two organisms are structurally identical. We might think that physics in accounting for the behaviour of the planets in the solar system has achieved much but the impressive and universal predictive laws of celestial mechanics can be derived relatively simply from the positions and momenta of planetary bodies in a relatively closed system. The number of variables is few.

Because the physical world ‘contains’ living organisms as a part, does it follow that the the universal laws of physics ‘contain’ those of biology it is tempting to assume that its scope is universal and that other realms of knowledge are simply sub-sets of physics. For example, biology is spatiotemporally bounded,[8] it takes the laws of physics as given; it is answering different questions in a different realm of thought. We could conceive replacing biology with the physical sciences thus making biology part of a system of strict universal laws but even if that were possible ‘we would not have explained the phenomena of biology. We would have rendered them invisible’.[9] Some laws apply over the whole range of scales.

Perhaps an explanation at one level does not require an explanation at another – or, at least, not at a level that is distant from it? We might explain chemistry in terms of physics but biology is conceptually more distant. We can feel cognitive focus at work here … atomic numbers, Maxwell’s equations, or the theory of relativity are not directly relevant when we work within the biological domain, or at least they are taken for granted as background. Hence the absurdity of explaining sociological phenomena in terms of physics and chemistry.

For example, since the large is explained analytically in terms of the small, we intuitively place greater value on the small giving it ontological precedence simply by virtue of size (but see Principle 3). Biologists no longer claim, as they once did, that living matter is quite different in kind from inanimate matter but this is a matter of perspective (all matter is physical matter but not all matter is biological matter). Many people once believed that the mind was inhabited by a spirit or soul and that, in a similar way, bodies were also inhabited by some special kind of spirit or vital force (elan vitale, entelechy). This general view, known as vitalism, is now discredited. The existence of such forces is not only implausible but, since they cannot be detected and studied, are of no explanatory value. They are best ignored.

Physics has its own problems with scale as it wrestles to reconcile the behaviour of matter at the small distances of quantum physics and the vast scale of cosmology, the break-down of laws in at the Big Bang or the singularities of Black Holes. Whether we look at the patterns in nature described by Newton, Einstein, Joule, Faraday, Maxwell or the various laws of thermodynamics the link to biology frequently seems tenuous. Of course the physics of matter is important to know about when studying nerves and macromolecules, but much of this is incidental to many biological questions.

In its most basic form foudationalism regards matter as the only reality but even the mechanistic philosophers of the Scientific Revolution recognised that this matter was in motion and today we realize the sigificance of not just matter but its mode of organization.

A flat ontology removes the necessity for the grounding of an object in something other than itself. There is no need for the Principle of Sufficient Reason. Explanations and reasons do not provide underlying truth or get closer to reality, they simply express or ,reduce, one scale or mode of existence in terms of another.

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