One of
the most elegant aspects of science is how, with mathematical modelling, we can
infer universal truths from limited data. Often direct measurement isn't
possible, but mathematics provides a reliable and consistent framework for
exploring and understanding complex systems. Some things are obviously objective.
Take, for example, Newton's F = ma - it is a fact (in macroscopic systems) whether you're a man in Nepal
or a woman in Sweden. The reason being, it has nothing to do with subjective
opinion, because force is equal to the time derivative of momentum, so the
relationship between an object's mass m, its acceleration a, and the applied
force F is F = ma wherever you are in the world (assuming Euclidian space).
Suppose a crank in Chile decided to opine something different about the quantitative calculations of dynamics, and he came up with a different, but provably wrong, idea about how velocities change when forces are applied. It's an opinion to which he can claim no justifiable entitlement, because objective facts about reality transcend the culture and geography in which they are discovered. The theory of evolution by natural selection is always based on factual accounts of billions of years of biochemical history, irrespective of whether you live in England, India or Brazil.
We know that the nuts and bolts of creation is assessed objectively because the language of mathematics reveals objective truths about physical reality. For example, in the early 1800s, astronomers set out to improve the tables of predictions for planetary positions they had created from Newtonian mechanics by undertaking calculations of the orbit of planets in relation to their neighbouring planets. At the time, Uranus was the farthest planet, but its calculations were proving to be inconsistent with the rest of the planets in our solar system. One suggestion made by several astronomers was that perhaps Uranus's behaviour proved that Newtonian mechanics is not universal. But with further mathematical calculations, a better proposition was offered; one that demonstrated the predictive power of mathematics.
By taking the discrepancies in the orbit of Uranus, astronomers were able to speculate about the possibility of a further planet, and use Newtonian predictions to calculate the possible size and location of a possible adjacent planet that would explain the anomalies with Uranus's behaviour. Using Newtonian mechanics, we could predict the potential whereabouts of a further planet (which we would later call 'Neptune') simply by assessing the behaviour of Uranus, and that is what we did.
For another way to capture the essence of how mathematics allows us to extrapolate from what we can measure to what we cannot directly observe, let’s return to Newton’s second law of motion:
“A body experiencing a force F experiences an acceleration a related to F by F = ma, where m is the mass of the body. “
Force is equal to the time derivative of momentum, so the relationship between an object's mass m, its acceleration a, and the applied force F is F = ma, where acceleration and force are vectors, and the direction of the force vector is the same as the direction of the acceleration vector. This enables us to make quantitative calculations of dynamics, and measure how velocities change when forces are applied.
Newton’s laws were formulated from observations that were made on local objects; for example - dropping objects from high places, calibrating acceleration of gravity for a falling object, observing motional trajectories, and looking at planetary positions. Although Newton's laws are formulated as universal statements, we can infer universality from what we observe locally (although this isn’t an irrefutable claim).
When Newton gave the formula for gravitational force, he claimed the law to be true for any two masses in the universe. But what warrants that leap of induction, and how would one develop certainty about the universality of it? For example, one doesn't directly observe the force of gravity between the earth and the moon - it is evidenced from things like tidal effects, lunar orbit and satellite measurements. Yet we gain evidence for scientific statements that are universal and cannot be measured directly. Mathematical models rely on established principles and constants (like the gravitational constant) that have been empirically derived.
What is required must be described in terms of mass and distance – this gives us the force. However, we cannot measure the force between the earth and any other object that we cannot weigh on a scale. We can weigh any easy to handle localised object (a football, or a snooker ball, or a cannonball, etc) and determine the attractive force of gravity between the earth and the localised object, but we cannot do this with the moon. However, what we can do in the absence of being able to hold the moon in our hand is work it out with simple mathematics, where we can infer this force through indirect measurements, and applying Newton's laws to predict their trajectories.
F = G*M(moon)*M(earth)/distance^2 where G is the gravitational constant, 6.67x10^-11 m^3/kg/s^2, the mass of the earth is 6x10^24 kg, the mass of the moon is 7.3x10^22 kg, and the distance between them is about 3.84x10^8 m.
The force is measured because gravitational force decreases inversely by the square of the distance, so by measuring the distance between the earth and the moon (it varies but its average distance is approximately 239,000 miles), and then the earth’s radius, followed by dividing the earth’s radius into the distance between the two objects, one gets the square result. Using mathematics we have accomplished something that we couldn’t achieve with physical testing.
Not only does a scientific theory work best when it is formulated such that in the Popperian sense it produces highly falsifiable implications, one must also distil from a theory a vast nexus of predictability – in the case of Newton’s laws - a web of implications on the behaviour of all masses under forces including gravity*.
Given that we can distil from this theory a vast nexus of predictability, we can infer this web of implications from the mathematics underpinning the law – we do not have to put a body in various regions of space and repeat-test this theory. We cannot, of course, put the moon on a set of scales, but we do not have to, there are easier methods. We know that the only possible orbit under Newton's Laws is an elliptical one, and we also know that the stronger the gravity of a planet, the farther an object can orbit. By sending a satellite to orbit the moon, we can measure its mass quite accurately – something Newton couldn’t have done in his day, of course.
Nowadays, we can even work out the effect the moon has on our seas and calculate its mass, but there would be greater margins of error if this was the only method we had. In the past, the moon and the earth were closer together (they are moving further apart each year at a rate of about 3cm per year) so the gravitational force would have once been much stronger. Now of course simple calculations would tell us that if that were the case the tides would have been higher than they are now. Once again, levels of consistency are found in such theorising; geologists frequently find fossilised tidemarks that demonstrate tides were higher in the past – and of course, subject to other earthly consistencies, future tides should become lower as the earth and moon separate further.
Given the position of an orbiting body at two points in time, Newton's laws will also tell us where that object will be at any point in the future. The better a theory, the greater its predictive value, in so far as it produces accurate and useful forecasts that one can anticipate, test and then verify or falsify. With theories such as motion, gravity and evolution, our predictions are always confirmed with localised evidence and simple mathematical equations. In the case of Newton, all orbits for anything we observe are forbidden to act in a way that departs from the predictions and implications of his own laws.
However, Newton's laws did run into trouble in the late 1800s, as Maxwell’s theory of electromagnetism was propounded describing all electromagnetic phenomenon and predicting the presence of electromagnetic waves. The electromagnetic field is a field that exerts a force on charged particles. Naturally, the presence and motion of such particles affects the outcome. Once it was discovered that a changing magnetic field produces an electric field, and that a changing electric field generates a magnetic field, we were able to discover electromagnetic induction - the discovery of which laid down the foundations for the vast array of electronic innovations (generators, motors and transformers) that followed.
Again, the predictive value here is essential - there must be uniformity and regularity for such endeavours to occur. The theoretical implications of electromagnetism brought about the development of Einsteinian relativity, from which it was evident that magnetic fields and electric fields are convertible with relative motion – that is, the perception of these fields changes depending on the observer's frame of reference, particularly how electric fields can transform into magnetic fields and vice versa depending on the relative motion of the observer and the source - allowing us to (among other things) correctly predict how forces increase exponentially for particles approaching the speed of light (this led us further to knowledge of how Euclidian geometry is challenged with the knowledge that space-time does not quite correspond to our own Euclidian intuitions, nor our intuitive view of past, present and future). This (the electromagnetic force) is one of the four fundamental forces of nature - it shows that the electromagnetic field exerts a fundamental force on electrically charged particles. Add to this the other fundamental forces; the strong and weak nuclear forces (the former is what holds atomic nuclei together), and the aforementioned gravitational force, and we have the four fundamental forces of nature, from which all other correlative forces (friction, tension, elasticity, pressure, Hooke's law of springs, etc) are ultimately derived. Aside from gravity, the electromagnetic force affects all the phenomena encountered in daily life - that is, all the objects we see in our day-to-day life consist of atoms which contain protons and electrons on which the electromagnetic force is acting. The forces involved in interactions between atoms can be traced to electric charges occurring inside the atoms.
But even though Newton’s laws were improved upon, they were still good approximations to reality. Newton's universe was a mechanical universe which has been supplemented by the likes of Maxwell, Einstein, Schrödinger, and Heisenberg, who themselves laid down the foundations for all the 20th century physics and cosmology that was to come. Newton appeared to be right for over three hundred years, but 19th discoveries caused us to reassess his theories and, in this case, augment them.
The main two measures we have of a theory’s veracity is the ability to make accurate predictions from it, and the localised evidences for it. As Newton has shown us, all scientific theories are only approximations of what is really at the heart of a complex nature. Approximations are not necessarily inaccurate, but are instead simplified models that apply under certain conditions. Newton's laws still work in situations that are non-relativistic (that is, at speeds much less than the speed of light), but Einstein’s theories work for both non-relativistic and relativistic situations. Einstein, Maxwell, Schrödinger, Heisenberg and any subsequent physicist and cosmologist all owe Newton a great debt – we are observing that science is progressive and that theories are there to be developed upon.
Once a theory is reached that reconciles quantum mechanics and general relativity, we may see that Einsteinian relativity in its current form will be viewed as inadequate. Just as special relativity the provided a framework that included both Newtonian mechanics (as an approximation at low speeds) and Maxwell's equations, demonstrating how they coexist in a broader relativistic context, so too a future theory (perhaps even a theory of multi-dimensions) will almost certainly resolve the current tension between quantum mechanics and general relativity. And any such theory that unifies the two would have to be consistent with separation of scales between high and low ends of the complexity spectrum – that is, the quantum effects which mostly deal with the "very small (that is, for objects no larger than ordinary molecules) and the relativistic effects that deal mostly with the "very large". Aspects of string theory, superstring theory and quantum gravity suggest progress, but the grand theory that unifies quantum mechanics with general relativity eludes us at present.
Applying this to epistemology, we can see that all these theories have provided provisional approximations of nature that produce highly accurate and useful predictions, but that by themselves they do not encapsulate a self-consistent whole. Newton’s approximations are better at slow speeds, but once we approach the speed of light the discrepancies start to show, and we require what is called a "relativistic correction" to Newton's predictions. But although we do not accept scientific theories as if they are the final word on a cosmic universal truth, we know that because we can test their implications with degrees to which certainty prevails at the greater universal levels than at the local levels (we rely on mathematics to prove this) we actually possess greater degrees of certainty about the universal levels than we do the local levels.
If we were merely non-mathematical creatures relying on local evidential observations the best we could do is postulate simple deduction with some further intrepid attempts at induction. But with laws, axioms and the vast nexus of contingency that is woven into the mathematical fabric we can make grand theories at the universal level that we know will apply at the local level too. Because of our mathematical fecundity we have made predictions about, and found consistency in, masses, motion, and forces, and we are as certain about these as we are about most localised discoveries.
There is an important balance to be stuck between the broad applicability and predictive power of universal laws and the localised contexts. For example, Newton's the law of universal gravitation applies to all objects - it applies equally to planets, apples, and snooker balls. But when observing a snooker ball on a table, local factors such as friction, air resistance, and imperfections in the surface of the baize can introduce uncertainties. These local conditions can make precise predictions about the ball's motion more challenging than applying the general laws to predict gravitational interactions between celestial bodies. The cosmological model of the Big Bang provides a framework for understanding the cosmic narrative of the universe as a whole, predicting phenomena like cosmic microwave background radiation and the large-scale structure of the universe. But at the local level, when trying to model the formation of a specific star or planet, numerous local variables (such as the presence of nearby stars, gas density, and magnetic fields) can introduce complexities and uncertainties that unsettle the potential accuracy of predictions. Or the laws governing radioactive decay are consistent and can be predicted with high accuracy over large populations of atoms (for example, calculating the half-life of a particular isotope). But predicting the exact moment when a specific atom will decay is inherently uncertain due to quantum mechanics. These local uncertainties does not detract from the reliability of the universal law; they simply illustrates how local predictions can be less reliable despite the overall framework being robust.
Science may not provide us with all the answers, but its own rewards are evident by the human progress it has ushered in; science by its very definition should always lead to progression, and every Kuhnian paradigm shift ought to qualitatively supersede the last. It is easy to look back into history and be under the illusion that many of these advancements were quick and easy, but they were not. Einstein’s relativistic standpoint didn’t swiftly refine the framework of classical mechanics to accommodate Maxwell’s electromagnetic standpoint, yet the retrospective viewpoint may give us the illusion that these transitions were smooth. When one thinks of the many other transitions; not just from Newton, Maxwell and Faraday to Einstein, Schrödinger and Heisenberg, but from the Ptolemaic cosmological view to the Copernican view; from classical mechanics to quantum mechanics; from Becher’s phlogiston theory to Lavoisier's caloric theory of combustion, right through to the science of thermodynamics; from Lamarckian inheritance to Darwinian natural selection and the reconsideration of Lamarck’s ideas with ‘epigenetics’ which identify possible inheritances of acquired traits - what these shifts (and many others) ought to tell us is that we are always in transition and ought to be prepared for black swans and new knowledge that will augment our present foundations.
