Abstractions

In my last post, I began the next major part of my blog by illustrating a tripartite view of reality, dividing what we seem to experience into three realms: the physical, the mental, and the abstract. Now I am going to start exploring the question of which of these three realms actually exist.

I was going to begin by looking at whether the physical realm exists, since it seems like the answer to that question, at least, is fairly obvious. Instead, for some reason I have decided to reverse course completely and talk about the realm of abstractions. It is a little more philosophical of a subject, but I think it can help to clarify things.

Abstract objects are somewhat hard to define, but I can give some examples to illuminate the concept:

  • Propositions are abstractions of meaning. As an abstract object, a proposition is usually taken to exist independently of any sentence that has that proposition as its meaning, and two different sentences can express the same proposition.
  • States of affairs are abstractions of reality, descriptions of ways that reality can be. States of affairs are usually taken to exist whether or not reality actually is the way that they describe.
  • Things that exist have properties, such as being red, or being made of wood. Two different things can have the same property, and this is often said to explain how things can resemble each other.
  • Similarly, things that exist can stand in relations to each other, such as the relation of one thing being taller than another.
  • Mathematical objects such as numbers or sets are paradigmatic examples of abstract objects.

If they exist, abstract objects are thought to lack any kind of causal powers. Numbers, for example, can’t cause any kind of change in reality. This is commonly taken to be the defining feature of being abstract, next to being neither a physical nor a mental entity. Abstract objects are contrasted with concrete objects, which are physical or mental entities and can affect reality in some way.

Many of the above examples are also thought to exist necessarily, outside of space and time. But there are things that are reasonably construed as abstract that lack these features. For example, the equator can be thought of as an abstract object with a definite location that came into existence at a certain time (when the Earth formed into its approximately spherical shape).

So much for what abstract objects are. The interesting question now is, do they exist?

Platonism is the philosophical position, named after the ancient Greek philosopher Plato, which holds that abstract objects are real things that exist, and that they exist just as fully as concrete objects. If platonism is true, numbers, for example, exist independently of any minds, outside of space and time. Nominalism, on the other hand, is the belief that platonism is false, and that abstract objects do not exist. But if nominalism is true, we have to find some way to explain how we can talk about things like numbers.

NOTE: Since writing this post, I have come to realize that the question of abstract objects is more nuanced that I presented it here – in particular, there are more positions than just platonism and nominalism, and when I conclude in the next post that “abstract objects such as properties are just a means of speaking about the way that things exist,” what that ends up looking like when formulated more precisely is something more like platonism and less like nominalism than I originally thought. More to come.

An Argument for Platonism

One of the oldest arguments in favour of platonism is this: consider a red apple, and a little red wagon. How is it that both of these things can be red? How can they resemble each other, seeming to have the very same redness? Platonism explains this by saying that the apple and the wagon exemplify a property, the property of being red, and it is the very same property that they both exemplify. Properties are universals, things that can be exemplified in more than one place at once (so that more than one thing can be red), and that exist changelessly across time (so that redness today is the same as redness tomorrow).

Philosophers nowadays have generally recognized that the issue of resemblance in this argument is a red herring. If you can explain how the apple is red, presumably you can explain how the wagon is red in the same way. And then nothing more is needed to explain the fact that they are both red. So the real challenge is to explain how something can be red.

Nominalism responds by saying it can do this just as well as platonism can. Things are red because (i) they reflect certain wavelengths of light, and (ii) we perceive those wavelengths as red. In principle, we can continue offering explanations for facts (i) and (ii) until we arrive at a number of statements that will presumably be about the basic constituents of reality. All of these statements can be put into a form like this: Xs are Y. (For example, electrons are negatively-charged.)

The platonist explains facts like these by saying that Xs have the property of Yness. But the nominalist says that this is not an explanation – saying that Xs are Y because they have Yness is no more informative than saying that Xs are Y because they are Y. We might as well just accept that when we reach the fact that Xs are Y, we have reached an irreducible fact about reality. (This is especially plausible if all of the fundamental facts we have reached in this way are necessary facts, facts that could not possibly have been false, since necessary facts are often thought of as being explained by their own necessity.) There is no explanatory gain in accepting the existence of properties like Yness.

NOTE: In fact, I have realized that what I have said in the last paragraph may very well be false. This is in part because Platonism is not trying to explain statements like “Xs are Y” so much as it is trying to understand what such statements actually mean – what it actually means for Xs to exist in such a way as to be Y. The nominalist response that I gave, in that case, is missing the point.

So the nominalist thinks we can talk about properties as a convenient way of expressing the ways that things can exist and resemble each other, but that we don’t need to actually believe they exist, at least not for any explanatory power that they offer.

Another Argument for Platonism

Here is another argument in favour of platonism. Even if we can sometimes paraphrase statements to avoid talking about abstract objects (instead of saying that Xs have the property of Yness, just say that Xs are Y), we can’t always do this. And often when we talk about abstract objects, we don’t have any paraphrase in mind at all – we just mean what we are saying.

So if we say something like there are infinitely many prime numbers, and we think this is true, we are committed to believing that the things we are talking about exist. Prime numbers have to really exist, the platonist argues, in order for it to be true that there are infinitely many of them. And some statements like these have to be true – they are indispensable for our understanding of reality. Mathematics, for instance, is a necessary foundation of our scientific theories. Arguments like this can be made for the existence of many different kinds of abstract objects.

Can we mean what we are saying without committing ourselves to the existence of what we are referring to? I want to suggest that we can. In fact, we do it quite often. You might say something like “she has a heart of gold,” and mean it, without believing that anything medically odd is going on. Or you might say “I’m overflowing with joy,” without believing that an emotional state has somehow liquefied and started pouring from you into the surrounding environment. Or you can say “the price of apples is skyrocketing” without believing that anything is being rapidly propelled into the air.

In other words, we can speak figuratively, and we can even do this without being aware of it. There is a theory in the field of cognitive linguistics, the idea of conceptual metaphors, that metaphors are a fundamental aspect not just of our language, but of our whole way of thinking, and that we use them often without realizing it. (A great book to read on this subject is Metaphors We Live By, by George Lakoff and Mark Johnson.) Completely literal language is the exception, not the norm.

“Cognitive discourse at its most dryly literal is largely a refinement. … It is an open space in a tropical jungle, created by clearing tropes away.” – W. O. V. Quine

If we have reason to think we are speaking figuratively, and not literally, when we are talking about abstract objects, then this argument for platonism falls flat – we don’t need to believe that abstract objects exist in order to meaningfully speak as if we did.

Even more than that, abstract objects do not need to exist for our figurative statements about them to be objectively true or false. Using the example of numbers: if our concept of numbers is sufficiently determinate – if the laws of logic determine what numbers have to be like, given our conception of them – then there are objectively right or wrong answers to questions we ask about them when we are speaking as if they exist, whether or not they actually exist.

And if our concept of numbers is not sufficiently determinate, their real existence is no help. Since in that case the laws of logic leave some things about numbers undetermined, the truth of a statement like there are infinitely many twin primes could either be true or false, and it would just be a coincidence if our belief about that statement lined up with reality. (Compare this with what is usually thought to be true, which is that our concept of numbers is determinate, and that the above statement is either necessarily true or necessarily false – whichever way it is, it could not possibly be any other way.)

In my next post, I’m going to look further at the idea that our abstract object talk is figurative language, or something similar to it. I think this theory spells trouble for belief in the existence of the abstract realm.

Realms of Reality

In my exploration of my whole belief system, I have finally come to the point where I can start talking about the nature of reality, and what it contains. Up to this point, I have merely been laying the foundation by examining the concepts of truth and knowledge. But now I can really get going.

In his book The Road to Reality, mathematical physicist Roger Penrose describes three different realms of reality that seem, at least in some way, to exist. First, there is the physical realm; the external realm of earth, sea, sky, and space that we observe through sensory experience, and study through the physical sciences. Second, there is the mental realm; the realm of consciousness where we have the introspective experience of our internal mental lives. And third, there is the abstract realm; the realm of abstract objects including mathematical entities like numbers, sets, and functions.

On Penrose’s view, each of these three worlds are interconnected. We perceive the physical world to be connected to the mental world through the experience of our embodied existence; we seem to be both mental and physical beings. The mental world is connected to the abstract world through our ability to perceive logical and mathematical truths; many people feel when they do mathematics that they are discovering something that is already there, rather than inventing it. And the abstract world is connected to the physical world by the astonishing fact that the laws of physics conform to precise mathematical descriptions.

I think Penrose’s view is useful as an illustration, though I do not necessarily believe that it is correct. In order to form a comprehensive worldview, we eventually have to answer the question of which of these three apparent realms of reality actually exist.

Does the physical world exist? The common-sense answer is “yes, of course it does.” But there are some people who would disagree, or who at least think that we cannot know that it exists, and that we should have greater confidence in the existence of one of the other two worlds instead.

Does the mental world exist? Can our conscious experience be explained entirely by the physical, or does it require something more? Is there any truth to the idea that we have minds distinct from our bodies? The answers to these questions have huge implications for our self-understanding.

Does the abstract world exist? Does the objective truth of mathematics, and its indispensability in characterizing the laws of physics, imply that these kind of abstract objects really exist independently of the other two worlds? Or are abstract objects simply concepts that we use to describe reality?

These, and more, are the questions I will be turning to in the following weeks.

Science, Philosophy, and Religion

What do you think of when you think about how we, as a society, learn and acquire new knowledge? It seems like the dominant mindset in our culture finds the pursuit of knowledge to be closely associated with the activity of science. Science is upheld by many as the epitome of rationality, or even as the only valid way of justifying our beliefs and gaining knowledge.

In contrast, the field of philosophy is now widely held to be useless, having been eclipsed by science as a means of finding out the truth. This attitude is expressed by none other than physicist Stephen Hawking in his book, The Grand Design (co-authored with Leonard Mlodinow):

“Traditionally these are questions for philosophy, but philosophy is dead. … Scientists have become the bearers of the torch of discovery in our quest for knowledge.” – Stephen Hawking and Leonard Mlodinow

And close on the heels of these kind of statements, there is the popular belief today that religion is the exact opposite of science, the epitome of anti-rationality. Religious faith is thought of as being, by definition, belief without evidence, or even belief in spite of the evidence. Religion and science are believed to be incompatible at a fundamental level.

These ideas – that science is the only source of knowledge or that it has eclipsed all others, that science and religion are incompatible, and that religious faith is opposed to reason – are, to put it bluntly, completely wrong.

Science and Knowledge

“You are right in speaking about the moral foundations of science, but you cannot turn round and speak about the scientific foundations of morality.” – Albert Einstein

It seems to me that the most appropriate way to define science is as the rational and empirical study of nature (including humans and their activity within nature). This definition encompasses the physical sciences, such as physics, chemistry, biology, geosciences, and astronomy; as well as the social sciences, such as history, economics, political sciences, psychology, and sociology.

Note that this definition does not include the formal sciences, logic and mathematics, as these lack the empirical methodology that characterizes science proper. Nor does it include the practice of the applied sciences, such as engineering and medicine, as they are aimed at applying knowledge rather than discovering it. (However, research activities in the applied sciences do properly fall under science as I have defined it here. Which is why you can get a M.Sc. in engineering.)

What all of the disciplines within science proper have in common is the scientific method. This is not really a single method, but a group of interrelated methodologies, including observation, hypothesis, prediction, confirmation or disconfirmation of theory through evidence and experiment, and peer-review to eliminate error and bias as much as possible. Scientific theories are prototypically falsifiable and open to discussion and criticism.

And science works. It has shown itself to be incredibly successful at discovering the workings of the natural world. It works well precisely because it embodies the virtue of critical thinking, by demanding justification for its beliefs in the form of evidence. But the fact that science is successful does not imply that it is the only way to acquire knowledge about reality, or even that we should hold it up as a model that all other ways of acquiring knowledge ought to resemble.

The simple reason for this is that science rests on philosophical foundations that cannot be justified by science itself. Let’s say you want to use science to study the electromagnetic interaction between electrons and protons. Before you can do that, you need:

  • epistemological beliefs about what is rational and what is not;
  • further epistemological beliefs about the validity of deductive, inductive, and abductive reasoning;
  • metaphysical beliefs about the existence of the physical world and its relation to our perceptions of it;
  • further metaphysical beliefs about the uniformity and predictability of the physical world, for example, that electrons and protons in the same conditions will always behave in the same way;
  • even further metaphysical beliefs about the nature of causation, if you want to move from merely describing what happens to electrons and protons (the particles moved in a certain way) to explaining why it happens (because the electromagnetic field exerted a force on them);
  • ethical beliefs about the right way to conduct science, for example, that you must report your research truthfully and that you must not plagiarise.

The scientific method presupposes all sorts of philosophical beliefs like these, and none of them can be justified by the scientific method alone. But if these beliefs that underlie science are not justified, then science itself cannot justify any other beliefs.

Scientism, which is the belief that science is the only valid source of knowledge, is itself a philosophical belief rather than a scientific one. And scientism is self-defeating. If it were true, the philosophical presuppositions of science, and therefore science itself, would be unjustified. So science would not be a valid source of knowledge after all – if scientism is true, then it is false. It’s unfortunate that even incredibly intelligent scientists like Hawking can apparently fail to recognize this.

So, science is not and cannot be the only source of knowledge. Good thing, too – so far in this blog, all I have been doing is philosophy, and it would be a shame if all of that effort had gone to waste. Now, I don’t want to give the impression that philosophy is the foundation of all knowledge, either. Really, what I have argued is that the foundation of our knowledge is common sense.

(That being said, philosophy is an invaluable resource in clarifying the foundations of knowledge and the implications and interrelations of knowledge from all sources. You can’t really avoid doing it if you undertake to build a coherent and comprehensive belief system, as I am doing.)

Science and Religion

“There is superficial conflict but deep concord between science and theistic religion; and superficial concord but deep conflict between science and naturalism.” – Alvin Plantinga

To give an overly simple definition, religion is a set of beliefs about the supernatural (and more broadly, about all of reality as it relates to the supernatural). Religious faith is trust or confidence in those beliefs. If science is a way of getting knowledge from the natural realm, religion could be considered a way of getting knowledge from the supernatural realm. This kind of knowledge comes about in much the same way as other knowledge: people have religious experiences and intuitions, they draw conclusions about the supernatural from other beliefs through reason, and they pass on their beliefs to others through testimony.

If religion refers to a set of beliefs, or even if it refers to the practices and rituals associated with those beliefs, it should be evident that religious faith is not necessarily opposed to reason, or to science. The concept of faith that seems to be widely held outside of religious circles (and unfortunately, by some within them) is that it is irrational belief without justification, even going so far as celebrating ignorance as a virtue. But this is a caricature of the concept of faith as it is understood by many believers of different religions.

I am most familiar with the Judeo-Christian tradition, so that is where I will speak from. The kind of faith in God that is commended in the Bible is not irrational confidence in belief without evidence. Rather, it is confidence in one’s beliefs specifically because of the evidence that supports them. In the Old Testament, the nation of Israel is commanded to remember how God brought them out of Egypt, as evidence of his power and faithfulness. In the New Testament, the writers of the Gospels say they wrote what they did specifically to give evidence for belief, through testimony. And the apostle Peter writes that Christians should be ready to give reasons for their faith.

So the biblical concept of religious faith is complementary to critical thinking, not opposed to it. Which is one of the reasons I think that, if they are understood properly, science and religion are united by a common purpose: to conform your beliefs to the truth. Science is about doing this for truth we can discover in the natural world, religion for truth we can discover in the supernatural. Of course, there can be overlap between these two pursuits, for example, if you believe that God created the world and has revealed some things to us about it. But because truth cannot contradict truth, true science and true religion cannot contradict each other.

Of course, religion can find truth in the supernatural realm only if the supernatural exists. Whether or not it does is a question for a later time. But for now, I just want to recognize that there will be a conflict between science and religion only to the extent that science is seen to be inextricably linked to metaphysical naturalism, the belief that there is nothing supernatural in reality.

But I don’t think you can show that doing science requires you to accept naturalism: many scientists are also adherents to one religion or another, without feeling that science compels them to abandon religion, or vice versa. And naturalism, as a fundamentally philosophical belief, is not one that science can justify: merely studying nature cannot prove that nothing outside of nature exists.

Nothing about studying the natural world requires that we deny the existence of the supernatural. Certainly, beliefs about the supernatural can affect how we do science. If you have a pagan conception of the universe that deifies the forces of nature and hosts a pantheon of capricious, conflicting gods, who were ultimately born from a primeval chaos – well, then, science might be a non-starter, since everything is effectively supernatural. Conversely, though, a belief in a rational and transcendent God, separate from the universe He created, can produce or strengthen a scientific belief in the order and predictability of nature.

Arguably, it has done just that, for at least one scientist:

“The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.” – Johannes Kepler

So I think we can move on from the ideas that science and religious faith are fundamentally opposed to each other, or that science is the only fount of knowledge that we can drink from. Instead, let’s search for truth, with critical thinking and epistemic faith, wherever it is to be found.

 

A little post-script: there are a lot of authors who have explored, much better than I have, the idea that science and religion are not really in conflict the way that they are often proclaimed to be. Here are some good books to read about this, if you’re interested:

  • Galileo Goes to Jail and Other Myths about Science and Religion, by Ronald Numbers
  • God’s Undertaker, by John Lennox
  • Where the Conflict Really Lies, by Alvin Plantinga