PIERRE DE FERMAT (1601- 65) ANDREW WILES (b.1953)

1637 – France; 1993 – USA

Portrait of PIERRE DE FERMAT

PIERRE DE FERMAT

Fermat’s theorem proves that there are no whole-number solutions of the equation x n + y n = z n for n greater than 2

The problem is based on Pythagoras’ Theorem; in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares on the other two sides; that is x 2 + y 2 = z 2

If x and y are whole numbers then z can also be a whole number: for example 52+ 122 = 132
If the same equation is taken to a higher power than 2, such as x 3 + y 3 = z 3 then z cannot ever be a whole number.

In about 1637, Fermat wrote an equation in the margin of a book and added ‘I have discovered a truly marvelous proof, which this margin is too small to contain’. The problem now called Fermat’s Last Theorem baffled mathematicians for 356 years.

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ANDREW WILES

In 1993, Wiles, a professor of mathematics at Princeton University, finally proved the theorem.

Wiles, born in England, dreamed of proving the theorem ever since he read it at the age of ten in his local library. It took him years of dedicated work to prove it and the 130-page proof was published in the journal ‘Annals of Mathematics‘ in May 1995.

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DAVID HILBERT (1862-1943)

1900 – France

‘Mathematics is concerned with formal symbolic systems. It is an activity that uses a series of symbols and rearranges them according to various formal rules. This separates it from any concrete reality and consequently there is nothing external to its workings that can be used to validate it, so all of its arguments must be capable of justifying themselves’

The pursuit of formalistic ideas led to many developments within mathematics. Hilbert introduced an original approach to ways of considering mathematical invariants. An invariant is something that is left unchanged by some class of functions. In terms of a geometrical transformation, an invariant would be an object that does not alter its shape or size while it is being moved. He proved that all invariants could be expressed in terms of a finite number – a number that can actually be counted.

Hilbert spent the first two decades of the 20th century struggling to construct a self-justifying system of arguments that would prove that a finite number of steps of reasoning could not lead to a contradiction. This work was itself contradicted in 1931 when Czech born KURT GODEL published his incompleteness theorem, showing that every consistent theory must contain propositions that are undecidable. Godel pointed out that when proving statements about a mathematical system at least some of the rules and axioms must derive from outside that system. By doing this you create a larger system that will contain its own unprovable statements. The implication is that all logical systems of any complexity are, by definition, incomplete. In doing this Godel showed that truth is more important than provability.

Amongst the list of 23 problems presented to the Second International Congress of Mathematicians in Paris in 1900, was the question of decidability, the ENTSCHEIDUNGSPROBLEM: is it possible to find a definite method for deciding whether any given mathematical assertion was provable?
image depicting a 3-dimensional Hilbert curve used as link to http://mathworld.wolfram.com/
In modern terms this ‘definite method’ would be called an algorithm. ALAN TURING answered the question in the negative in April 1936, but may have been anticipated by the American logician Alonso Church. While Church appealed to contemporary mathematics to make his point, Turing had introduced a theoretical machine that could perform certain precisely defined elementary operations. By doing this, Turing created a foundation for modern theories of computation.

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MAX PLANCK (1856-1947)

1900 – Germany

‘Energy is not a continuous quantity but it is quantised; it flows in discrete packets or quanta. When particles emit energy they do so only in quanta’

According to Quantum theory, the energy (E) of one quantum (photon) is given by E = hf where f is the frequency of radiation and h is Planck’s constant.
Its value is 6.63 x 10-34 joules per second

h is a tiny number, close to zero, but it is has a finite value. This implies energy is released in discrete chunks, a revolutionary notion.

photo portrait of MAX PLANCK ©

MAX PLANCK

By the late 1800s the science of thermodynamics was developing to the point that people were beginning to understand the nature of energy.
The traditional view was that energy was released in a continuous stream and that any amount of energy could be indefinitely divided into smaller and smaller ‘lumps’. Planck’s work on the laws of thermodynamics and black body radiation led him to abandon this classical notion of the dynamic principles of energy and formulate the quantum theory, which assumes that energy changes take place in distinct packages, or quanta, that cannot be subdivided. This successfully accounted for certain phenomena that Newtonian theory could not explain.

The basic laws of thermodynamics recognised that energy could not be created or destroyed, but was always conserved. The second law was drawn from an understanding that heat would not pass from a colder body to a hotter body.
The study of thermodynamics was based on the assumption that matter was ultimately composed of particles. LUDWIG BOLTZMAN had proposed an explanation of thermodynamics, saying the energy contained in a system is the collective result of the movements of many tiny particles rattling around. He believed the second law was only valid in a statistical sense; it only worked if you added up all the bits of energy in all the little particles.
Among his detractors was Max Karl Ernst Ludwig Planck.

Planck began his work on the second law of thermodynamics and the concept of entropy. He investigated how materials transform between solid, liquid and gaseous states. In doing so he found explanations for the laws governing the differing freezing and boiling points of various substances.
He also looked at the conduction of electricity through liquid solutions (electrolysis).

In the mid 1890s Planck turned his attention to the question of how heated substances radiate energy. Physicists were aware that all bodies radiate heat at all frequencies – although maximum radiation is emitted only at a certain frequency, which depends on the temperature of the body. The hotter the body, the higher the frequency for maximum radiation. (Frequency is the rate per second of a wave of any form of radiation).

Planck had been considering formulae for the radiation released by a body at high temperature. Using ideas developed by ROBERT KIRCHOFF, he knew it should be expressible as a combination of wavelength frequency and temperature. For a theoretical ‘black body’, physicists could not predict expressions that were in line with the behaviour of hot bodies at high frequencies and were in agreement with other equations showing their nature at low frequencies. Thus no law could be found which fitted all frequencies and obeyed the laws of classical physics simultaneously.
Plank resolved to find a theoretical formula that would work mathematically, even if it did not reflect known physical laws. His first attempts were partially successful, but did not take into account any notion of particles or quanta of energy, as he was certain of the continuous nature of energy. In an ‘act of despair’ he renounced classical physics and embraced quanta.

The final straw had been a concept developed by John Rayleigh and James Jeans that became known as the ‘ultraviolet catastrophe’ theory. They had developed a formula that predicted values for radiation distribution and worked at low frequencies, but not at high frequencies. It was at odds with Planck’s formula, which worked for high frequencies but broke down at low frequencies. In June 1900 Rayleigh had pointed out that classical mechanics, when applied to the oscillators of a black-body, leads to an energy distribution that increases in proportion to the square of the frequency. This conflicted with all known data.

Planck’s answer was to introduce what he called ‘energy elements’ or quanta and to express the energy emitted as a straightforward multiplication of frequency by a constant, which became known as ‘Planck’s constant’ (6.6256 x 10-34 Jsec-1). This only works with whole number multiples which means for the formula to have any practical use one must accept the radical theory that energy is only released in distinct, non-divisible chunks, known as ‘quanta’, or for a single chunk of energy, a ‘quantum’. This completely contradicts classical physics, which assumed that energy is emitted in a continuous stream. The individual quanta of energy were so small that when emitted at the everyday large levels observed, it appears that energy could seem to be flowing in a continuous stream.
Thus classical physics was cast into doubt and quantum theory was born.

Planck announced his theory on December 14 1900 in his paper ‘On the Theory of the Law of Energy Distribution in the Continuous Spectrum’. Planck said ‘energy is made up of a completely determinate number of finite equal parts, and for this purpose I use the constant of nature h = 6.55 x 10-27(erg sec)’

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When ALBERT EINSTEIN was able to explain the ‘photoelectric’ effect in 1905, suggesting that light is emitted in quanta called ‘photons’, by applying Planck’s theory – and likewise NIELS BOHR in his explanation of atomic theory in 1913 – the abstract idea was shown to explain physical phenomena.

Planck was awarded the Nobel Prize for Physics in 1918.

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ALFRED NOBEL (1833- 96)

1901 – Sweden

  • 1866 – Invents dynamite

  • 1876 – Invents blasting gelatin

  • 1886 – Invents ballistite

  • 1896 – Nobel Foundation set-up to comply with the terms of Nobel’s will

  • 1901 – First Nobel Prizes awarded

A Swede educated in Russia, France and the United States, Alfred Nobel was a chemist who set up a factory to manufacture the relatively unstable nitro-glycerine to serve the civil-engineering market. After a disaster in 1864, Nobel found a way to stabilise the liquid explosive with kieselguhr, which he called dynamite. He went on to develop blasting gelatin, ballistite and a series of detonators.

The success of Nobel’s dynamites was compounded by his holdings in oil, leading to vast personal wealth. He left much of his fortune to funding the establishment of a series of awards, one of which included an accolade for peace.
There was another dedicated to literature, with the remaining three presented for achievements in the sciences. The first Nobel prizes for medicine (or physiology), physics and chemistry were awarded in 1901.

The prizes are awarded annually, according to the terms of Nobel’s will; ‘to those who, during the preceding year, shall have conferred the greatest benefit on mankind.’

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IVAN PAVLOV (1849 – 1936)

1903 – Russia

‘A conditioned reflex is a learnt response to an environmental stimulus’

The process of learning to connect a stimulus to a reflex is called conditioning.

An innate or built-in reflex is something we do automatically without thinking (such as moving our hand away from a flame).

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Ivan Pavlov and his staff


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ALBERT EINSTEIN (1879-1955)

1905 – Switzerland

  1. ‘the relativity principle: All laws of science are the same in all frames of reference.
  2. constancy of the speed of light: The speed of light in a vacuüm is constant and is independent of the speed of the observer’
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EINSTEIN

The laws of physics are identical to different spectators, regardless of their position, as long as they are moving at a constant speed in relation to each other. Above all the speed of light is constant. Classical laws of mechanics seem to be obeyed in our normal lives because the speeds involved are insignificant.

Newton’s recipe for measuring the speed of a body moving through space involved simply timing it as it passed between two fixed points. This is based on the assumptions that time is flowing at the same rate for everyone – that there is such a thing as ‘absolute’ time, and that two observers would always agree on the distance between any two points in space.
The implications of this principle if the observers are moving at different speeds are bizarre and normal indicators of velocity such as distance and time become warped. Absolute space and time do not exist. The faster an object is moving the slower time moves. Objects appear to become shorter in the direction of travel. Mass increases as the speed of an object increases. Ultimately nothing may move faster than or equal to the speed of light because at that point it would have infinite mass, no length and time would stand still.

‘The energy (E) of a body equals its mass (m) times the speed of light (c) squared’

This equation shows that mass and energy are mutually convertible under certain conditions.

The mass-energy equation is a consequence of Einstein’s theory of special relativity and declares that only a small amount of atomic mass could unleash huge amounts of energy.

Two of his early papers described Brownian motion and the ‘photoelectric’ effect (employing PLANCK’s quantum theory and helping to confirm Planck’s ideas in the process).

1915 – Germany

‘Objects do not attract each other by exerting pull, but the presence of matter in space causes space to curve in such a manner that a gravitational field is set up. Gravity is the property of space itself’

From 1907 to 1915 Einstein developed his special theory into a general theory that included equating accelerating forces and gravitational forces. This implies light rays would be bent by gravitational attraction and electromagnetic radiation wavelengths would be increased under gravity. Moreover, mass and the resultant gravity, warps space and time, which would otherwise be ‘flat’, into curved paths that other masses (e.g. the moons of planets) caught within the field of the distortion follow. The predictions from special and general relativity were gradually proven by experimental evidence.

Einstein spent much of the rest of his life trying to devise a unified theory of electromagnetic, gravitational and nuclear fields.

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ROBERT MILLIKAN (1868-1953)

1909 – USA

The charge on the electron’

Photograph of ROBERT ANDREWS MILLIKAN ©

ROBERT ANDREWS MILLIKAN

Millikan measured the charge on the electron.

His experiment showed that the electron is the fundamental unit of electricity; that is, electricity is the flow of electrons.
From his experiment Millikan calculated the basic charge on an electron to be 1.6 × 10-19 coulomb.
This charge cannot be subdivided – by convention this charge is called unit negative, -1, charge.

Millikan also determined that the electron has only about 1/1837 the mass of a proton, or 9.1 × 10-31 kilogram.

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