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Ernst Mayr in his last book titled "What Makes Biology Unique?" argues that many of the theories in biology do not need any mathematical support. He says that much of biology is only conceptual and cannot be describe by mathematical formulations. In the meantime, he also argues that these properties of biology does not prevent this field of knowledge to be considered as a science.

Here are a couple of relevant quotes:

Mathematics remained the earmark of true science. Kant certified this opinion by saying “there is only that much genuine [richtig] science in any science, as it contains mathematics.” And this greatly exaggerated evaluation of physics and mathematics has dominated science until the present day. What would be the scientific status of Darwin’s Origin of Species (1859), which contains not a single mathematical formula and only a single phylogenetic diagram (not a geometric figure) if Kant had been right? And this was also the philosophy of science of the leading philosophers (e.g., Whewell, Herschel) that affected Darwin’s thought (Ruse 1979). Yet several recent philosophers of science have published science and sciences a Philosophy of Biology strictly based on the conceptual framework of the classical physical sciences (e.g., Kitcher 1984, Ruse 1973, Rosenberg 1985) while ignoring the autonomous aspects of biology (chapter 2).
...
The philosophers of logical positivism, and indeed all philosophers with a background in physics and mathematics, base their theories on natural laws and such theories are therefore usually strictly deterministic. In biology there are also regularities, but various authors (Smart 1963, Beatty 1995) severely question whether these are the same as the natural laws of the physical sciences. There is no consensus yet in the answer to this controversy. Laws certainly play a rather small role in theory construction in biology. The major reason for the lesser importance of laws in biological theory formation is perhaps the greater role played in biological systems by chance and randomness. Other reasons for the small role of laws are the uniqueness of a high percentage of phenomena in living systems as well as the historical nature of events.

Owing to the probabilistic nature of most generalizations in evolutionary biology, it is impossible to apply Popper’s method of falsification for theory testing because a particular case of a seeming refutation of a certain law may not be anything but an exception, as are common in biology. Most theories in biology are based not on laws but on concepts. Examples of such concepts are, for instance, selection, speciation, phylogeny, competition, population, imprinting, adaptedness, biodiversity, development, ecosystem, and function.

(sections from pages 14 and 28)

Is it possible that a scientific theory cannot be supported by any mathematical formulations? Or stated differently: Can a field of knowledge be a field of science if it does not contain any mathematical formulations?

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Biology is hardly unique in having conceptual theories which may be expressed without formal mathematics. Physics, for instance, has them; its just that these can also be further elaborated by mathematics. Some examples:

  • Reductionism — the principle that not only can he behaviour of compound objects be reduced to that of its contituent parts, but in fact just to the interaction of pairs of those parts. In short: no essentially different behaviours arise when larger numbers of components are added; and if the whole system has complex and difficult-to-predict behaviour, this is borne out of what one might expect from an analysis of pairs of components. A broad, quasi-quantitative principle which is related to this is the principle of superposition: that the total influence of upon one component of its environment can be taken simply as the sum of the influences of individual sub-components of the environment.

  • Symmetry — the principle, essentially in the tradition of Newton's Third Law, that in an interaction between two systems, both systems are comparably affected; and that any apparent asymmetry (e.g. the lack of any obvious gravitational acceleration of the Earth towards individual falling apples) is due to the relative magnitudes of the objects involved, i.e. in the possibility of the effect of the interaction to be distributed widely among several sub-components, so that the interaction is not between two systems but between many. (Here reductionism appears again: apparent asymmetry of interactions is explained as symmetric interactions between many systems, where some large collection of the subsystems appear to act in concert as a single large system whose internal forces are ignorable.) This is witnessed in a quantitative fashion in various conservation laws; the particular conservation laws are also explainable by other symmetries by Noether's theorem, though the notion of precise conservation is difficult to express without appealing to a notion of quantity.

  • Causality & Locality — the principle that the world is intelligible in terms of a network of causes and effects; and subsequent to Einstein's qualitative contemplation of the relativity of simultaneity, the principle that causal influences can only be due to spatio-temporally nearby objects, i.e. that despite the success of Newton's theory of gravity, his critics were right to intuit that spatial mediation/separation of interactions are significant.

The actual success of physics is founded on these principles; the historical success is that the fact that these principles are simple enough to be easily mathematised, which lent them a weight of authority inherited from the apparent certainty of Euclid's work. The fact that they were easily mathematisable also allowed them to be easily verified in cases of interest, such as the orbits of moons and planets and the ability to calculate the tides. As these applications were surprising at the time, the success seemed remarkable and a standard to achieve in the future.

It seems to me that the problem with mathematising biology is that it would be akin to studying number theory having only random samples of numbers on the order of 1024 or larger to deal from expressed as a string of 1s. Any two are going to seem different; the probability that you will notice that you've seen the same number before will diminish rapidly with the size of the number concerned; and relationships between them apart from relative size will not be easy to formulate. It would be difficult to find out about prime numbers, for example, if one could never reliably obtain the same number twice, let alone any number less than a trillion. So it is unsurprising that biology appears to lack (or if we take 'biology' to be the human activity of studying living systems and their components, actually does lack) so-called universal laws, because it is difficult to isolate anything whose behaviour is regular enough that it possibly could give rise to one — and it is also unclear that in a complex system, that the regular behaviour of such simple subsystems could be evident in the behaviour of the whole. Any such regularity (one might imagine the conservation of energy in the form of sunlight and chemical potential as an example) might simply be deferred to chemistry or physics.

This is not to say that it is either necessary or fruitful to mathematise biology; only that it is not impossible in principle — and that the nature of the obstacle amounts to one of sheer computational complexity, as with the solutuon of the three-body gravitational problem. From the outside, biology seems to do quite a good job at tackling extremely complex phenomena using qualitative principles. It is not difficult to imagine that an accurate mathematisation of biology would either be so coarse or ad-hoc as to provide no special benefit over qualitative reasoning, or so complex as to be actually impossible for a human mind to grasp. If so, what benefit would be obtained through its mathematisation?

As long as it is possible to formulate theories with enough precision (with or without mathematics) that one may discern whether or not it seems useful in practise (neither too vague nor too inaccurate), it is not a necessary feature of science that it be mathematical. Mathematics is merely one tool of ascertaining precision and accuracy, superlative though it may be; there is no reason to believe that it excludes the possibility of all other means of judging these qualities.

  • Niel, I'm not sure I agree with the implicit equation of "mathematical formulation" and "quantitative analysis". Concepts of symmetries and permutations get occasionally invoked in biological theorising; wouldn't we say that abstract Algebra has some informative things to say about these notions, how to use them and what consequences follow from them? – Paul Ross Oct 10 '13 at 18:07
  • @PaulRoss: that's very interesting. Of course, I've already described simple (bilateral) symmetry above as non-mathematical; while admittedly in principle as someone with formalist sympathies I should include under the rubric of 'mathematics', anything which one can express formally enough to be quite precise. But never mind; the boundaries of the quantitative and the mathematical are in principle not crisp. Would you be able to give me an example of permutations in reasoning with biology? If group theory has traction in biology in some way I've missed, I would be quite intrigued. – Niel de Beaudrap Oct 12 '13 at 0:38
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    (disclaimer: IANAB) It turns out "Permutation" means something more specific in genetics than in group theory, so that was maybe a bad example, but group theoretic permutations seem to be used to explore the possible protein structures of the coatings of viruses - see, for example, Biomathematics at the University of Durham: dur.ac.uk/mathematical.sciences/biomaths/events/iop08 – Paul Ross Oct 15 '13 at 12:53
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(I am sorry if Niel's answer should entail mine. I was not able to decide.)

First off, "does not need mathematical support" is not the same as "cannot be supported by mathematical formulations" or "can there be science without mathematical formulas" (the latter is only loosely quoted from the question).

Whatever your stance on "what science is" is, it will probably entail the following:

  • A (systematic) approach to
  • generate hypotheses that are
  • tested for their falsification

None of this presupposes mathematics per se. If this is right, then the answer is: there can be science without mathematical models.

However, a science can benefit from mathematical models as they provide a widely accepted and immensely useful framework for (essentially) dealing with certain concepts and their relations.

  • I'd like to add that some basic mathematics, most notably statistics, is very useful for testing hypotheses. This applies to any science. Besides specifically hypotheses testing, Biology has utilized mathematics in formulation of Mendel's laws (probability theory), in deducing existence of certain enzymes (topology), in analysis of x-rays and other images (signal processing), in analysis of population dynamics (ODEs), etc. – Michael Oct 9 '13 at 16:09
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One way to think of mathematics is "the study of patterns"1. So if I may reformulate Ernst Mayr's claim:

He says that much of biology is only conceptual and cannot be described by mathematical formulations patterns.

Is there a sense in which something can be 'conceptual' and yet have no patterns? In my view, no. What sets biology apart from physics (although increasingly less so) is that there is much more unity to the patterns in e.g. physics, than biology. The theory of evolution provides a kind of unity, but it is more like a cohesive narrative than an equation. One thing is for sure: evolution is full of patterns.

Consider what it would be like for a science to not depend on patterns. That's like saying that we haven't found a way in which things are alike. But if we haven't found that, how can we predict, or explain in a way that is different from a just-so story?

Something abstract to look at is this answer to "What are the philosophical implications of category theory?":

Thus, it's apparent that category theory is relevant to and has implications for both mathematics and philosophy, and is not just semantic. From the perspective of mathematics, category theory is very significant because "doing mathematics in a categorical framework is almost always radically different from doing it in a set-theoretical framework."

 

1 http://en.wikipedia.org/wiki/Mathematics, notes 9 & 10, accessed 2013-10-11 22:56 GMT

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First, other sciences can have their concepts explained without math. There is something like physics without math. It is called conceptual physics. However, the aim of it is not to explain the whole field without math, just to introduce new students to physics and to provide them with a solid conceptual foundation. Conceptual physics is by no means less scientific than the rest, but it is very limited, and if you want to solve any problems, math will be your tool of choice.

Second, biology can imply lots and lots of math. Just start sequencing the human genome or analyzing interactions between proteins. You'll be nearer computer science than to traditional biology.

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