The best way to begin this article is for me to pose the Hambly’s paradox (Hambly, 1985). If one, who is weighing 600N, sits ideally at the centre of gravity of a stool and if the stool has three legs, the reaction force at each legs would be of course 200N (note that this is a paradox so all the ideal state of symmetricality and so on has been taken for granted). Now, if the stool has four legs, instead of three and by taking into account the reality that there could never be a smooth surface beneath the legs, at least one of the legs must be tilting (not in contact with the ground). This leaves three legs supporting the stool. However, since one of the leg must be diagonal to the tilted leg, satisfying the static moment equilibrium requires that the force in this leg is zero. What this amounts to is that there are actually only two legs supporting the stool thus the reactions in each legs must be 300N instead of the intuitive 200N. A good engineering sense would have realized all these and assist the engineer to decide on the construction of three-legs stool instead of the four-legs stool. He or she should have realized that the construction of the four-legs stool not only uneconomic but also lacks of safety. But, the bigger picture to this is the appreciation of the fact that what seems trivial is not sufficiently necessary – ‘intuition’ is sufficient but ‘technical intuition’ is what sufficient and necessary. Thus, from this paradox, I propose the following conjecture:
i) To be an engineer, one must have a good engineering sense
ii) A good engineering sense, on the other hand, requires a good understanding of scientific notions
iii) But, science is understood and delivered through mathematics
iv) So, engineers must know, at least, their so called engineering mathematics
Those who shy away from mathematics run the very real risk of becoming functionally illiterate……..Indeed, it is this mastery of mathematics and science that distinguishes the engineer from the engineering technologist or technician - (Duderstadt, et. al., 1982, p. 136)
Applying the above quotation to the Humbly’s paradox, we can address those who end up with the four-legs stool as mere technicians instead of real engineers as ‘intuition’ belongs to the technicians but ‘technical intuition’ belongs to the true engineers. Let’s quote further from the father of engineering mechanics:
“Wanting to do work on strength of materials, I had to broaden my knowledge in this area,…I needed to go on elasticity theory. I decided to tackle the most thorough courses in this field, the book by A.E.H Love…in addition I read Lame’s book and several chapters from a book by Saint-Venant…but my mathematical knowledge was insufficient. I read Riemann’s book Partial Differential Equations, in the Hattendorf edition. I learned something about Fourier series… ” –(Timoshenko, 1968, p.83-84).
Also, Karl Terzaghi, another great figure of engineering and widely known as the father of soil mechanics is quoted as:
“ To predict the effective stress at any time was the problem Terzaghi had set out to solve. He understood the physics well……He had been unable to make headway on formulating this physical behaviour until he thought to study the books on the mathematics of heat conduction. This suggested to him simplifications that led to the derivation of a differential equation completely analogous to the well-known diffusion equation governing the time dependent flow of heat in solids”- (Goodman, 1999, p.83).
Had not Timoshenko and Terzaghi, the two greatest figures of engineering turned to mathematics, both of them would ended up as mere technicians according to both Hambly’s paradox and my conjecture. Have they not, engineers of today would still be in the ‘dark ages’, rhetorically speaking.
But, what if one argues “we, engineers never despise our mathematics, we have been learning and applying mathematics our whole life”. To such an argument, I would like to response, time changes everything thus it can change the status of the three-legs engineer to a four-legs engineer simply if the former does not keep up with the ‘wind of change’”. In regard to this, it is again very tempting to tell another story of Timoshenko.
“There I presented my paper on stress concentration….an evaluation of my paper, prepared by the well-known Harvard Professor G. F. Swain, was read. The reviewer obviously had extremely limited knowledge of strength of materials and had never even heard of the high stresses at the edges of round holes. This did not prevent him, however, from vigorously attacking my paper, from branding my theoretical research on stress concentration as a useless fantasy of theoreticians, divorced from any practical application.” –(Timoshenko, 1968, p.255).
Based on the above story, the change of the status of the Harvard Professor G.F. Swain i.e. from a three-legs to the four-legs is made obvious by his ignorance about the stress concentration. But such a change of status is not exclusive, anybody even Timoshenko himself (if he still alive today) would become a four-legs if he fail to keep up with the ‘wind of change’.
But there is another important point that deserves a note. Based on the above, we should learn the danger of ignorance. The four-legs can not only ‘endanger’ himself but most likely the three-legs as well especially if the former has the authority (in this case, it was Professor Swain) . Due to the ignorance, there is a possibility for the former retarding any novelty of the latter. Therefore, this calls for open-mindedness from everybody.