Logical statements may refer to external things, but the principles that they are based on are not external things themselves. Rather, they are properties of thought, or – if you – reject an anthropocentric viewpoint – the properties of language.

The 18th century philosopher Immanuel Kant discusses this in his work, Critique of Pure Reason, where he argues that logic is a priori (preceding our experience of reality) rather than a posteriori (based on experience), and is found in the part of the mind that deals with concepts. Because it transcends (or more precisely, precedes) experience of phenomena, he calls it ‘transcendental logic’. He contrasts this with ‘general logic’, which involves making abstractions from experiences of sensed phenomena.

Ludwig Wittgenstein disputed Kant’s view that aspects such as logic exist a priori, instead arguing that it resides in language. Because logic – as a language – is a structure for conceptual organization, it makes no sense to say that logic would exist without human minds, or that it exists abstractly in the external world, any more than it does in relation to language. The reason I would argue that some claim that logic has an independent existence is because of the fallacy of misplaced concreteness.

Take a rock for example. We can label a thing with a certain shape, constitution and behaviour as a rock, and understand that it is distinct from a giraffe. But this is a categorical identity that is defined through the concepts of language. Some languages do not identify giraffes as distinctive animals, so to say that a rock is not a giraffe would make no sense.

When pushed a little further, the objectivist interpretation of category descriptions begins to break down. We can accept that a rock is distinct from a giraffe. But what happens when we replace that rock with another similar rock? Logically, A still equals A, but empirically it is a different rock – it has a different ontological identity (although the same categorical identity). It is not possible to say, in this instance, that a rock is a rock and not not a rock. Rather, it is a rock, but not another rock!

The rock differs from the giraffe, not because they are separate entities in external reality, but because they are different categories within language. Suppose the giraffe eats the rock. They are no longer distinct identities in an empirical sense, because they now occupy the same space and time. It is only in categorical terms that we can argue that the rock does not properly belong in the giraffe, and if we were to x-ray the giraffe, we would say that a rock has lodged itself in the giraffe which does not properly belong there (if all goes well with the digestion process, the rock will soon find its way back to the surface where it naturally belongs).

What happens if the giraffe dies on the spot (maybe because the rock gets lodged in its respiratory tract), and it becomes fossilised? Is it then a giraffe or a rock? In category terms we might say now that it is neither A (a rock) or B (a giraffe), but C (a fossil). But then again, it would not be wrong to say that it is A, B or C – take your pick!

However, in Kant’s defence, when we introduce the concept of space, time and causality, we can start to draw distinctions between things. With the concept of time, for example, we can determine that the giraffe, once its remains are fossilized, is not really a giraffe anymore if understood as a living, flesh-based animal. It is no longer living, and its flesh has long decayed, leaving just an impression in the rock. As such, a fossil of a giraffe is not equivalent to a giraffe itself. We simply have a rock that has the impression of a giraffe, and causally, we know this to be likely because it is where a giraffe once died (spatially speaking). Nevertheless, these kinds of physical explanations (which are only relatively recent in the history of humankind) are not as important as the basic linguistic difference, which is based on distinctions identified through the senses (which views a rock and a giraffe as different) in combination with certain socially agreed concepts (e.g., a rock is not alive whereas a giraffe is).

The same point can be applied to explaining mathematical rules. Actually logical realism is usually seen as the weaker debate compared to the one over mathematical realism, which is the question of whether mathematical properties have an objective existence. Mathematical realism is a stronger claim than logical realism, because it is generally accepted that mathematics is more integral for explaining the behavior of physical objects, as evidenced by the centrality of mathematics in physics.

It is clear that mathematical operations are not found in the physical world. Objects simply move about (or not move about) and follow their own nature. They do not behave in any way dependent on mathematical properties, but we can use mathematics to represent their behaviour. How is it possible that mathematical principles do not exist in the physical world, yet can predict the behaviour of physical objects in universal ways? The physical world has an order, which acts in accordance with the laws of physics (another aspect that is debated in terms of its externality). Mathematics is a human tool for measuring things within this order, just as logic is a human tool for thinking about this order (as well as other things). Because real world objects have order, it is possible to measure their behaviour and predict how they will respond to different stimuli in different environments using mathematics. It is the order, not mathematics, that is independent of the human mind. Mathematics is a human representational system – a notational language if you will – for precisely mapping time and distance in ways that accord with the universe. Time and distance (or space), as Kant points out, are a priori to human experience.

The philosopher Gottlob Frege argues that mathematics is derived from logic, and hence comes from the same place as logic does. While there have been attempts to reduce mathematics to formal logic, the complexity of the logical sequences are such that it is difficult to view these as intuitive in any sense. A similar difficulty lies with attempts to reduce mathematics to more simplistic representations such as those of set theory, which utilize sets of real world objects as the basis for mathematics. Beyond the basic natural numbers, it becomes very complex to view real-world objects according to sets that might be treated mathematically.

The philosopher Edmund Husserl disagreed with Frege, and argued that mathematics – like logic – are correlates in subjective categorisation. Whereas logic is directed towards judgement, Husserl argued that mathematics is directed towards ontology. Like Frege, Husserl did not believe that either of these categories originated in psychological processes. Instead, he subscribed to the Platonic notion that logic and mathematics are ideal forms independent of subjectivity and intuitively grasped through the subjective mind. As such, Husserl postulated a transcendental ‘third’ realm (in addition to subjectivity and objectivity) where abstract ideals reside.

It might be argued that Platonism has similarities to TAG postulates of a spiritual realm, but neo-Platonists do not necessarily accept that such a realm (to the extent that it is even a ‘realm’ as such) requires a form of intelligence or consciousness. Nevertheless, the postulation of a third realm is problematic, and others, such as the psychologist Jean Piaget, have suggested the psychological explanation is more persuasive.

To suggest that mathematical principles are mental abstractions that are correlated to the physical universe – like a mechanical clock is a correlate to the movement of the sun – enables us to see these aspects as a formalised language, albeit one with real-life application. Like any language, mathematics involves a capacity for abstraction that has been made possible due to human evolution. As a language, mathematics is not dependent on the subjectivity of any particular individual, but has become formalised within inscribed rules and symbols that are independent of us in the same way that a book is independent from human subjectivity, even though it is a product of human subjectivity. If we lost our knowledge of mathematics, we would have no means of reacquiring our mathematical understanding in any easy way, but we would have to go through the long process of rediscovering mathematical properties.

The more difficult question is how to treat the laws of physics, including cause and effect, which more directly relate to external phenomena. As ‘laws’, they are rules that account for the order of the Universe. The reference to them as ‘laws’ can be misleading, because we tend to think of laws as rules imposed on us externally by society, and so it is tempting to view the laws of physics as external to humans. But the laws are predictive theories that are produced through scientific thought, and as such, they are susceptible to change as our theories about the universe undergo change.

To the extent that certain physical laws remain empirically supported, they mostly go unchallenged. But science has encountered certain revolutions (the most radical one being Einstein’s theory of relativity) that radically alter the way we understand the Universe, in what philosopher Thomas Kuhn refers to as a paradigm change. Paradigm changes particularly occur when certain problems in ‘normal science’ are encountered that require a new paradigm in order to explain them. Given the amount of problems that are now being seen in terms of scientific explanation of quantum behavior, it is possible that we are on the verge of another major paradigm change – time will tell.

So the laws of physics are human-made predictive theories that achieve longevity and the veneer of concreteness (and hence become recognized as ‘laws’) by virtue of their ongoing value in explaining physical behavior. Do objects themselves observe the laws of physics? The answer is no. Their behavior accords with the laws of physics (as best approximations), but this is different from saying that they observe the laws of physics. Instead, they are bound to an order that physicists endeavor to account for through theoretical models and principles. Objects do not adhere to those laws – they do what they do as part of the natural order, and we apply those laws to understand their manner of behaving.

Physical laws, mathematics and logical principles are employed to describe and account for the natural order – they do not form part of the order itself. This is not to imply, however, that these conceptual elements are divorced from external reality (although some neo-Kantians make such a claim). On the contrary, I would argue that they have a close sympathy with external reality (which is not to necessarily say they resemble reality, although that might be so). These elements are the means by which our understandings can align our interactions with external reality – they reflect the alignment of human subjectivity with the objective order of the universe.

What these rules do is constrain the excesses of human thought so that we do not impute false qualities to this order. In the case of logic, its rules prevent muddled thinking by means of precise categorization and formal reasoning using categorical statements. In the case of mathematics, its rules assist human consciousness to perceive distance in its correct proportion. In the case of physical laws, its rules enable humans to predict physical processes in accord with the nature of those objects.

These laws, rules and principles help us to obtain conceptual clarity about how the universe operates, with scope for further clarity as our categories, logic, measurement and theories improve. Without the human mind, these laws, rules and principles would not exist. Their universality is not a reflection of any independent origin (as Platonist argue), but a reflection of the uniformity in the things – bound by time and space as relative constants – that they seek to classify and measure.

If there is one place to look for the possibility of a divine Being, it is not in the rules that humans use to understand the physical order that surrounds them (an epistemological matter), but how that order came to be (an ontological matter). This is the subject of the final article in this series.