Despite the widespread use of mathematics as a language for science, it is not itself a science. A good exposition of science and the scientific method is given by Steven Hawking in A Brief History of Time:
“A (scientific) theory is a good theory if it satisfies two requirements: It must accurately describe a large class of observations on the basis of a model that contains only a few arbitrary elements, and it must make definite predictions about the results of future observations. You can never prove (a theory, but) you can disprove a theory by finding even a single observation that disagrees with the predictions of the theory.”
In contrast, P.A.M. Dirac, an equally famous theoretical physicist in his own time, said of mathematics:
“Mathematics is the tool specially suited for proving abstract concepts of any kind and there is no limit to its power in this field.”
To sum up:
Science=real; mathematics=abstract.
Science=disprove; mathematics=prove.
Since abstract ideas are difficult to hold in one’s hand, and proving a logical conclusion differs fundamentally from disproving a scientific hypothesis, there seems to be little overlap here.
However, things aren’t necessarily that simple. (When are they ever?) Science uses mathematics as a tool to codify descriptions of the universe. The mathematics, however, must be combined with definitions that describe real world phenomena (mass, velocity, energy,…) in terms of the abstract mathematical concepts. The use of mathematics to codify scientific knowledge does not make mathematics a science. To put this point into a more familiar context, the fact that logs are often cut with chain saws does not mean that a chain saw is a type of log!
But if science deals with real-world objects, how can mathematics, which only deals with abstract concepts, be a science? There is a potential loophole. A commonly-held viewpoint within the philosophy of mathematics is that of mathematical realism. In mathematical realism, abstract concepts are held to be real – as real as the floor or the roof. In this view, mathematicians do not invent mathematical concepts – rather they are discovered. Hmmm…discovery. Somehow, mathematics does begin to look more like a science…
There still remains a fundamental difference between science and mathematics, however. Science is carried out using inductive reasoning while mathematics uses deductive reasoning, which leads to dramatically different approaches to seeking and confirming knowledge.
What does it mean to say that mathematics uses deductive reasoning? A mathematical argument is a logically valid deduction if and only if it is impossible for its premises to be true and its conclusion to be false. The following is a famous example:
1 – All men are mortal;
2 – Socrates is a man;
Therefore
3 – Socrates is mortal.
Here the premises establish certain information about the memberships of the set of mortals and of the set of men. Symbolically, one can say that given that S>M (Socrates is a man) and M > MB (all men are mortal), the logical conclusion is that S>M>MB, or by concatenation S > MB (Socrates is mortal). This is called a proof.
One can easily illustrate that logical validity does not depend on what symbols are being manipulated. Replace the symbol ‘men’ with ‘rabbits’, ‘mortal’ with ‘vegetarians’, and ‘Socrates’ with ‘Harvey’. The result is still valid:
1 – All rabbits are vegetarians;
2 – Harvey is a rabbit;
Therefore
3 – Harvey is a vegetarian.
Notice that a logically valid deduction is not necessarily true, as seen in the following example:
1 – All airplanes use propellers;
2 – The SR-71 is an airplane;
Therefore
3 – The SR-71 uses propellers.
Given that some airplanes actually use turbojets, the first premise is factually false. This means that the conclusion, although logically valid, is not necessarily true. As the SR-71 uses turbojet engines fitted with afterburners, this logically valid conclusion is false.
Truth requires not only that the deduction is logically valid, but also that the premises are true, by whatever criteria we determine truth. However, mathematics is only concerned with the validity of the deduction, because the symbols have no meaning outside the context of the mathematical structure to which they belong. There are mathematical arguments whose validity cannot be determined, but if found valid one can have full confidence in the resulting conclusion.
In contrast, inductive reasoning shapes the structure of science differently. Inductive reasoning in science involves developing a general rule (a physical law) from a set of observations such that the general rule is consistent with existing observations and accurately predicts the result of future observations.
Let’s work with a simple physical model. An example would be ‘Because on all recorded occasions in the past the sun has risen in the morning, we predict that the sun always rises in the morning.’ In other words, the prediction of our model is that things will always be consistent with their past behavior.
But this isn’t necessarily the case. ‘Rises’ can reasonably be taken to mean ‘moves from below the horizon to above the horizon’, but what about a definition for ‘morning’? If the definition of ‘morning’ is ‘the time the sun rises’, we then have a tautology, a model which is true no matter what can or does happen in the real world.
Tautologies not being very useful (or interesting), we have to define ‘morning’ in some other manner. In everyday life, mornings come about roughly once every 24 hours. Let’s use this as a definition. Now our model reads ‘Because on all recorded occasions in the past the sun has risen about every 24 hours, we predict that the sun always rises about every 24 hours.’
This improved model still has some problems. For example, location matters. If we try to apply the model on the moon where the sun only rises every 700 hours or so, the predictions of the model are incorrect. Now correct this problem by limiting our model to an observer on the surface of the Earth. Our new model states ‘Because on all recorded occasions in the past on Earth the sun has risen every 24 hours, we predict that the sun always rises every 24 hours on Earth.’
This model isn’t quite correct either. The moon’s tidal influence is slowly lengthening the rotational period of the Earth. In about 400 million years, the Earth will rotate once in about 48 hours, twice as long as at present. This means in the far future there will be periods about 24 hours long when the sun does not rise, thereby contradicting the predictions of the proposed physical model.
This model can again be repaired. Simply limit the scope of the model to times within a million years of the present? This gives us ‘Because on all recorded occasions in the past million years on Earth the sun has risen every 24 hours, we predict that the sun will rise every 24 hours during the next million years on Earth.’ Now the lengthening of the day does not result in false predictions.
Does our physical model now accurately predict the rising of the sun? Well – no. Near Earth’s poles, the sun does not rise and set daily, but only once a year. These regions must be excluded from the scope of the model as well.
Now our model states ‘Because on all recorded occasions in the past million years at locations on Earth between the Arctic and Antarctic circles the sun has risen every 24 hours, we predict that the sun will rise every 24 hours at locations on Earth between the Arctic and Antarctic Circles during the next million years.’ Phew!
The scope of this model is greatly reduced from that of the original version. However, it has the advantage of being more or less correct. Phenomena which could invalidate this model include asteroid impacts that lengthen the rotation period of the Earth, the Earth being removed from the Solar System by the gravity of a black hole, the Earth or Sun being hit by a huge lump of antimatter, or some other unexpected event which would result in contradict of the model predictions.
Note that no version of the model can be obtained from deductive reasoning, unless we already know how to add an additional axiom which describes the relevant physics of the real world. In other words, scientific truth is obtained via a never-ending refining process of comparison of the predictions of the model with real world observations. This is the fundamental difference between mathematics and science, showing that mathematics is not a science.
