THE WRATH OF GODEL : MATHEMATICS AND MEDICINE
There is no doubt that a knowledge of mathematics can intensely benefit the medical professional. Mathematics is everywhere in medicine – in physiology, in biomechanics and anatomy, in pharmacology, in epidemiology and in all medical specialties, in particular the more advanced ones such as cardiology. This is a very obvious fact, in no need of substantiation, though recently there have been some articles published to remind medical professionals of the obvious importance of mathematics to medical specialties. For instance J.H. Bates and B.E. Sobel writing in the journal of ‘Coronary Artery Diseases’ (2003 Apr;14 (2):135-48) stated in their introduction:
“Unfortunately, medical education does not emphasize, and in fact, often neglects empowering physicians to think mathematically. Reference to statistics, conditional probability, multicompartmental modeling, algebra, calculus and transforms is common but often without provision of genuine conceptual understanding... The principles and concepts they address provide the foundation needed for in-depth study of any of these topics. Perhaps of even more importance, they should empower cardiologists and cardiovascular researchers to utilize the language of mathematics in assessing the phenomena of immediate pertinence to diagnosis, pathophysiology and therapeutics”
Now, apart from the obvious practical implications of mathematics to medicine, does it have any other implications? The answer is yes – it most definitely does.
The first and one of the most important implications of mathematics to medicine, or indeed any specialty is that it gives the specialty a degree of authority. It gives the specialty that utilises it an air of authority. I think this has at least to be partially due to the fact that mathematics is deductive, based almost entirely on logic, on proofs – a proof is defined as “a logically deduced argument, not an empirical one. That is, the proof must demonstrate that a proposition is true in all cases to which it applies, without a single exception. An unproven proposition believed or strongly suspected to be true is known as a conjecture”. Because logic is cold and objective, and not affected by our experience, it is the most powerful tool of acquiring knowledge.
Indeed, in mathematics, there is either truth or falsehood. There is none of the unpredictability or ‘perhaps’ that characterises many specialties, including medicine. There is a law in mathematics called the law of the excluded middle, which says that:
“Every statement must be either true of false, never both or none. If it is not true, then it is considered to be false. For example, consider the statement y = x2. In real life's terminology, like with most statements, it is sometimes true and sometimes false. In mathematics it is false since it is not completely true. Take for example y = 1 and x = 2”.
Because of this absolutism, it is obvious to see why many sceptical philosophers, who were out on the hunt for absolute truth, tried to find it in mathematics. We will turn to this subject in a bit.
In any case, the obvious implication of this is that, the more mathematical a field becomes the more objective, powerful and advanced and ‘truthful’ it becomes. It is well known that the scientific revolution that has changed the world began with the work of two mathematicians – Galileo and Isaac Newton. They began the application of mathematics to the study of the cosmos and led to the advancement first of physics, and then other specialties. In can be easily argued that physics, astronomy and chemistry are the most advanced of the sciences, simply because of how ‘mathematical’ they have become. Biology, psychology and medicine are less mathematical, and hence less advanced.
The reason for this simply lies in the fact that we are human beings – we are therefore less able to apply the methods of cold mathematical logic to things like biology, psychology, sociology and medicine, which are so close to our hearts. As explained brilliantly by Raymond Tallis:
“The process of placing medicine on an objective basis in not complete even today. While we do not know how recent medicine-taking is, we do know that scientific therapeutics is little more than a century old…It is hardly surprising that the objective inquiries of Homo scientificus should have been directed rather late to the human body – to the body of the inquirer. Since it is out of our special relationship to our bodies that knowledge has grown, the pursuit of objective knowledge about the body and its illnesses requires a return to the very place where knowledge first awoke. Somewhat less esoterically, we may anticipate that the body ‘we look out of’ should be the kind of object we are most likely to ‘look past’. It is something that we are as well as something we know or use; mired in subjectivity, it was a late focus for systematic objective enquiry. Humans found it easier to assume an objective attitude towards the stars than towards their own inner organs: scientific astronomy antedated scientific cardiology by thousands of years”.
The power of mathematics is such that some, like the great Leonardo da Vinci regard that, “No human investigation can be called true science without passing through mathematical tests”. Some people believe that if things are immeasurable, and cannot be ‘mathematicised’ so to speak, they should be excluded from the world of objective science. A.J. Balfour remarked that, “Science depends upon measurement, and things not measurable are therefore excluded, or tend to be excluded, from its attention”. Robert Hooke remarked that, “measurement is science’s highest court of appeal, pronouncing its
final verdict for or against the meekest and the loftiest ideas alike.” There is an old Latin saying, “Defendit numerus: There is safety in numbers” and an Indian one, “Like the crest of a peacock so is mathematics at the head of all knowledge”.
The implications of this are obvious to medicine – highly mathematical medical specialties, would be far more advanced and effective than the less mathematical ones. Arguably the most mathematical of medical specialties is ITU – intensive care. In fact, I don’t think it is far wrong to define ITU as an exercise in the detection and attempted reversal of extreme abnormalities of electrocardiograms, blood pressure, central venous pressure, pulmonary artery ‘wedge’ pressure (PAWP), cardiac output (measured using the thermodilution technique using a PA catheter), urine output, fluid balance (weighing the patient, fluid balance charts which record inputs (oral, nasogastric and intravenous) and outputs (urine, nasogastric, fistulae, vomiting, diarrhoea, surgical drain, insensible losses), temperature, arterial blood gases, lactate levels, oxygen saturations, lung function tests and capnography – all highly numerical entities.
Endocrinology, cardiology and respiratory medicine, are full of numbers and physical laws too (such as Poiseuille’s and Pascal’s laws, the Bernoulli effect, Ohm’s law), and it is extreme numerical deviations in their disorders that comprise the majority of patients in ITU, the most mathematical of hospital specialties.
And of course, the least mathematical of all medical specialties is psychiatry. By default therefore, if what Leonardo da Vinci and Balfour are saying is true, which I think it is, then psychiatry can be confidently excluded from the realm of science, and come to be regarded as a pseudoscience.
Knowing that mathematics imparts a degree of solid backbone, or almost ‘certainty’ to professions would make them seek to ‘mathematise’ their specialties as much as possible. This would also help provide the specialty with a degree of predictability, which would be most welcome, considering the unpredictability inherent in much in medicine. As brilliantly put by the great Harvey Cushing:
“In a certain sense every drug a doctor administers and every operation a surgeon performs is experimental in that the result can never be mathematically calculated, the doctor’s judgement and the patient’s response being variables indeterminable by any law of averages”.
It may be safely said that the search for certainty is what the doctor and patient want the most – they want to know that their treatments will work, or at the very least know their prognosis. One may safely believe that the more mathematical an area becomes, the more predictable it becomes.
Nevertheless, to those who hope for absolute certainty – they will be left disappointed by knowing the saddest fact in mathematics; that it too is based on unprovable ‘axioms’. Listen to the great philosopher and mathematician Bertrand Russell described his encounter with those axioms in his youth:
“Before I began the study of geometry somebody had told me that it proved things and this caused me to feel delight when my brother said he would teach it to me. Geometry in those days was still 'Euclid.' My brother began at the beginning with the definitions. These I accepted readily enough. But he came next to the axioms. 'These,' he said, 'can't be proved, but they have to be assumed before the rest can be proved.' At these words my hopes crumbled. I had thought it would be wonderful to find something that one could prove, and then it turned out that this could only be done by means of assumptions of which there was no proof. I looked at my brother with a sort of indignation and said: 'But why should I admit these things if they can't be proved?' He replied, 'Well, if you won't, we can't go on.'”
Since the days of Godel, we have come to know that certainty is impossible in mathematics, just like, since the days of Hume, we have known that certainty is impossible in ethics, science and experience, and since the days of Heisenberg’s publication of his uncertainty principle, that certainty is impossible in physics. Faith is necessary in mathematics as it is in other specialties.
And with this – we return to faith. Those who ridicule faith as ‘unscientific’ and ‘irrational’, who believe science and mathematics is everything, do not understand the fundamental fact that both are fundamentally equally based on faith – that the power of reason will never be able to elucidate everything in them to the core. They ought to realise this, and with it, realise one of the most astonishing truths of life; that “there are truths totally beyond the reach of science and reason, even assuming an infinite time for the human mind to evolve”, as Martin Gardner put it. I could not have concluded this essay better than by mentioning that brilliant remark by one of the greatest mathematical and philosophical writers of our time.
 I remember John Larkin mocking endocrinologist and stating, “You don’t need them… Endocrinology is all down to numbers...There is no requirement for any expertise or experience; your sugar's up, you've got diabetes, your thyroxine's up, you've got thyrotoxicosis, your thyroxine's down, 'go to' (ooh tricky - your TSH is up, primary hypothyroidism, your TSH is down, pituitary disease…There’s none of the considered opinion so characteristic of other specialties… No "....rheumatoid factor is positive, but I think this is a red herring as clinically you've got..." or "your angio is normal but your chest pain does sound ischaemic, so what might be happening is...". No. The tests define the disease. So any specialty opinion is superfluous".
Larkin then proceed to give the example of tiredness, "when they mention tiredness, just do all the endocrine tests; there's no point in enquiring after polydipsia, poluria, 'what sort of weather do you prefer (I ask you, what sort of weather do you prefer), when everybody knows you're just going to do the tests. If their T4 is up, it doesn't matter what weather they prefer, you'll decide they're thyrotoxic. If their plasma glucose is 45, then the absence of a family history of diabetes isn't going to change your diagnosis". Perhaps endocrinology is the most mathematical of specialties, and hence its great success.