Paul Dirac was one of the most proficient theoretical physicists of the 20th century. In this interview, Dirac himself talks about the existence of dimensionless, fundamental constants in physics. These constants are numbers, usually ratios, which contain no units and are therefore universal. No matter what units you use, a dimensionless number will always be measured to be the same. The most widely known example, from mathematics, is Pi which is the ratio of the circumference of a circle to the diameter of the circle. No matter in what units you measure both the circumference and diameter of a circle, be it feet, meters or stadia, Pi will always be 3.14159265.....
The fact that fundamental physics constants exist as dimensionless constants means that they can act as landmarks in a physical theory; if predictions are made by a theory that create a dimensionless constant of a given value and if that value is then confirmed by experiment then the theory is deemed acceptable in physics. An example of this is the lepton g-factor in physics, by which it is possible using Quantum Electrodynamics, to predict its value theoretically which matches experimental measurements to an accuracy of 1 part in a billion. This is the most accurate prediction in the history of mankind and demonstrates the predictive power of quantum mechanics, no matter how strange it is.
Paul Dirac also seems to make reference to his famous and controversial "Large Number Hypothesis" here, which is not accepted by many theoretical physicists, stating that the similar sizes seen in some of fundamental physics constants is coincidental. It seems a crucial part in Dirac's thinking what there were no coincidences in physics and that theory should be able to explain why certain numbers are so large, independent of units, compared to others.
In a way Dirac could be correct in certain aspects of this, as the observed range of the fundamental forces of nature, independent of units, are a result of the nature of the forces themselves are are not entirely coincidental. For example, the weak nuclear force is short ranged due to the Higg's mechanism and the Strong nuclear force is short ranged due to the property of Asymptotic Freedom in QCD, where the strong force screens itself at short distances, making it easy for the force to bind quarks and gluons, and strong at large distances making quarks difficult to break apart thus keeping them confined in protons and neutrons. The strength of the electromagnetic force is also dependent on the distance, as screening of the vacuum is diminished at extremely short distances making the electric charge infinitly large at singular points. These infinities led to the development of renormalization being developed to perform calculations in quantum field theory which was a procedure which Dirac disliked, possibly because it was a way to eliminate large numbers rather than explain their existence. Nevertheless, this method does work so there may in fact be no reason for why large numbers exist in physics, anymore than the reason for the infinite regress of decimal points seen in many mathematical constants, such as Pi or Euler's Constant.
His dislike of renormalization, developed by Feynman and others to reconcile theory and experiment, and his pursuit of the Large Number Hypothesis meant that Dirac was also a controversial figure in the physics community, sometimes viewing pure theory as being superior to experimental verification.
Paul Adrian Maurice Dirac was one of the founders of quantum theory and reconciled quantum mechanics with Einstein's theory of special relativity in the equation which bears his name, the Dirac Equation.
Using this equation, Dirac predicted the existence of antimatter by finding that for his equation to work, charged quanta as described by relativity must be roots of the square root of the normalized charge of a particle, hence forming positive and negative terms.
From the relativistic terms in his equation Dirac also described the existence of quantum spin as a relativistic phenomenon and described the existence of fermions, by a reinterpretation of Dirac's equation as a "classical" field equation for any point particle of spin ħ/2, itself subject to quantisation conditions involving anti-commutators.
Werner Heisenberg later interpreted the equation as a quantum field equation accurately describing all elementary matter particles -- today quarks and leptons -- this Dirac field equation is as central to theoretical physics as the Maxwell, Yang--Mills and Einstein field equations. Dirac is regarded as the founder of quantum electrodynamics, being the first to use that term. He also introduced the idea of vacuum polarisation in the early 1930s. This work was key to the development of quantum mechanics by the next generation of theorists, and in particular Schwinger, Feynman, Sin-Itiro Tomonaga and Freeman Dyson in their formulation of quantum electrodynamics.