We have two possibilities open to us. The first option is that perfect shapes do exist, in some transcendent non-physical form. This belief is based on that posited by Plato, the ancient Greek philosopher. He held that objects in the material world are ‘shadows’ of an immaterial realm known as the realm of Forms. We recognise circles because of the Form of Circle, and our mathematical knowledge is knowledge about the relevant Forms. However, this view seems somewhat unpalatable. It is unclear how Plato’s Forms can interact with the real world, and provide knowledge to it. Plato himself suggests that our souls were in the realm of Forms before we were born, so our recognition of things such as circles is a result of our pre-birth exposure to the Form of Circle. We have gone from seeking an answer to the question of what mathematics is about to positing an entire metaphysical realm and accompanying dualist beliefs. It seems a great deal to suggest with little justification.
The second option is that perfect objects do not exist, but are instead products of our mind – inventions. Under this view the subject of mathematics is a fiction; circles, numbers and sets do not exist except as concepts in our minds. Mathematics is the study of these entities, and mathematical knowledge is therefore invented rather than discovered. To begin with this seems unsatisfactory; how can a statement such as ‘2 + 2 = 4’ be anything other than true? It is surely not an invention; at the very least we can demonstrate it empirically. However, we must consider the difference between ‘two apples and two apples makes four apples’ and ‘2 + 2 = 4’. I would argue that the two statements are in fact qualitatively different. While the first is an empirical observation (of the type that I shall deal with below), the second is a general, abstracted statement that, as with the circles above, is about nothing in particular. We cannot observe ‘2’ itself, even if we can observe items with ‘2’ as a property (pairs of shoes, for example). The statement that ‘2 + 2 = 4’ deals with abstract entities, and unless we are willing to return to Plato’s Forms, we must accept that the knowledge that ‘2 + 2 = 4’ is an invention.
This point is even clearer if we look further into mathematics. At a basic level we may accept proofs by intuition – it is ‘obvious’ that two parallel lines will never meet. This emotional way of gaining knowledge is straightforwardly a product of the human mind, and qualifies as invented knowledge. But in the deeper reaches of pure mathematics, proofs proceed through application of rigorous symbolic logic, and no step can take place which does not follow the rules for symbol manipulation. It might seem that this is the purest form of reason as a means for discovering knowledge. However, if we take a closer look at formal mathematics, we still find axioms at the base of it which by their nature can be justified, if at all, by intuition only. Regarding them as arbitrary does not change mathematics at all, because mathematics is the study of the consequences of those axioms. For examples, non-Euclidean geometry is a self-consistent field in which parallel lines can meet. Because the axioms are produced by us, we must regard mathematics as a whole as a human product, and so any kind of mathematical knowledge is necessarily invented.
Interestingly, this analysis hints at a way that mathematical knowledge could be made into discovery rather than invention. If instead of saying ‘2 + 2 = 4’ we said ‘accepting certain axioms implies that 2 + 2 = 4’, the resulting set of statements could be regarded as discovered rather than invented. It is only when we claim ‘2 + 2 = 4’ as a true piece of knowledge that it must be regarded as an invention. It is the latter, however, which more accurately describes what we regard as mathematical knowledge, and although the justification is intuitive rather than observational, I think that we are still justified in making mathematical statements.
What other possible area of knowledge could then truly be described as discovered rather than invented? The obvious candidate seems to be the natural sciences. Unlike mathematics, intuition does not play the key role in developing scientific knowledge, but instead sense perception is given pride of place. Knowledge that arises from an observation of the world seems to me certainly to qualify as discovered knowledge. If I take a set of measurements of the heights of students in my class, there is no invention in the data that I get. However, once we get beyond the territory of direct observation the case is not so strong. Areas of modern physics in particular make claims about the world which in some cases are not observable at all. We cannot ‘see’ electrons, only their effects. A distinction must be made between the aspects of science that consist of direct observations (which are undeniably discovered knowledge) and the aspects of science which consist of theories, and conjectures which help explain those observations.
The status of the theory itself is more variable. A theory may propose mechanisms as well as formulae for making predictions. In some cases the mechanism can be directly observed, but in others the mechanism may be less concrete. Einstein proposed that gravity is due to the curvature of spacetime; the mechanism here is not something that we can observe. We may, however, observe its consequences and determine that it gives credible results. I would argue that this is not enough to qualify the knowledge of mechanism as discovered. Although we work through reason (albeit perhaps with intuition giving us the initial inspiration), the mechanism is something that we posit without directly observing it, so it is invented.
The final area of knowledge I shall briefly examine is ethics. There are many approaches to finding ethical knowledge. Historically theists have taken their morals from God, and this method of finding knowledge qualifies as discovery, because the individual takes commands directly from God. However, this is the only type of ethical knowledge which can justifiably be labelled as discovery. There are several secular systems of ethics, such as utilitarianism and Kantianism, which seek to provide ethical guidance without recourse to God. However, these inevitably rely on a fundamental value at the core – pleasure in utilitarianism, a ‘good will’ in Kantianism – and this value is not justified by reason or observation. Similarly to mathematics, secular ethics must be classed as invented because of the invention at the core.
Examining the nature of our knowledge in its various areas has revealed that our initial assumptions about them may be false. Although we have an experience that feels like discovery, the actual processes that take place to give us knowledge in mathematics, ethics and (parts of) the natural sciences are more accurately classified as invention. The area of knowledge which genuinely qualifies as discovery is much smaller, and encompasses only direct observations of the world around us, possibly with the addition of moral commandments that come from God, for a theist. The majority of knowledge is therefore invented. Does this mean that we should not trust it? I believe that the fact that knowledge is invented does not necessarily weaken it beyond the point of usefulness. Mathematical knowledge is fundamentally based on invented assumptions, but this does not mean that the knowledge is not knowledge, or that we cannot make mathematical statements confidently. Any argument based on reason is in a sense invented (including this essay), but that does not mean that such an argument has no power.
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