Why Philosophy Cannot Progress like Mathematics and Science

8 min read

Progress in Mathematics and Science has been the result of a shared language — something Philosophy lacks

Philosophy wants to build off certainty. Thus, it wants to build off the a priori truths in experience to achieve knowledge, but is unable to create the shared language needed to describe them

[a priori = knowledge gained from your existing. The truth of it is part of your existing. For instance, the colour ‘red’ is there in your vision. You cannot deny it being there, although you might dispute where the ‘redness’ came from]

Language barrier

Let us keep in mind that language originated in survival.

A priori truths refer to fundamental facts of existence. When referring to fundamental emotions such as ‘pain’, language does a pretty good job. This is hardly surprising, as these concepts are very important to communicate — although even here pain can refer to many things and language fails to fully capture what we mean.

Concepts such as truth remain much more elusive. Again, within a social group, there is normally a broadly similar idea of what truth is. For instance, when you talk to friends you don’t get stuck defining truth when they ask you what you had for dinner last night. Across social groups, this can vary: a group of Christians and a group of Buddhists might have a different conception of what constitutes truth in a spiritual/religious context.

When I read the word truth I understand it, even if I cannot put my finger down on what it is. Here lies the issue. There are certain concepts which I cannot explain to others whatsoever without assuming they already understand it. Try explaining colour to a blind man who has never seen — I cannot ‘explain’ the colour red in any meaningful way to him. Similarly, could you explain ‘truth’ without merely using synonyms?

So, these basic concepts give us trouble because we cannot express what they are to others. If the other person lacks these experiences we are stuck.

As we think and construct arguments using words, our inability to encapsulate these a priori concepts in a single word causes trouble.

Meanwhile in Mathematics and Science…

Mathematics and Science have progressed through shared languages. Einstein’s astounding breakthrough — that the speed of light was constant despite the implications — nevertheless relied on an enormous inherited infrastructure ranging from mathematical tools to his education, measuring equipment and data. Any new theory needs to be informed by an understanding of the data so it isn’t a wild guess, and a whole network of other scientists and their results were needed for this. Good mathematical notation took millennia to develop and is often taken for granted.

With a shared language, knowledge can be passed down and accumulated.

Shared language in mathematics and science (not essential)

(Skip to next section if you want)

Mathematics first developed through arithmetic. We developed integers (for counting), and we could construct other numbers from geometry. These are founded in a shared image in front of us (geometry) and understandings about basic concepts such as ‘amount’ or ‘length’. For example, you can build all rational numbers and many irrational numbers using geometry — in fact you can visually find the solutions to a quadratic and some cubic equations using geometry. (See below if interested)

I admit it’s not immediately obvious!


Crucially, it is clearer in Mathematics what the concepts are (and we assume we share them). Mathematics nowadays is not built off physical intuition (try imagining infinite dimensional Hilbert space) but instead using a set of axioms and deductions. It may take some time to get accustomed to it, but mathematicians all understand the axioms they are using. For instance, each proposition in classical logic is assumed to be either true or false

In Science, we have an outside world to test things against. As a result, Any misunderstanding in concepts is shown up in the data (or the misunderstanding doesn’t matter because both provide accurate models of the world).

Arguably, in Mathematics the same is true — that we can test ‘axioms’ by seeing if they lead to contradictions or not. (I wrote an article on this a while back, see below)


Unmixing a cocktail

The issue with Philosophy is the greater amount of certitude demanded. A Philosopher isn’t happy with a vague understanding of truth, nor a vague understanding of her experiences. After all, doesn’t uncertainty about the axioms leads to uncertainty about the conclusions?

Unfortunately, this requires the Philosopher to un-mix her cocktail of emotions and experiences. Then it requires that to be expressed. In contrast, a Mathematician is content to accept the classical logic assumption each proposition is either true or it is false without getting too bogged down in definitions of truth.

When we say a word such as ‘truth’ we reference a concept which we simply assume the other person has a similar understanding of. We might realise they are using the word in a different way and guess how they use it by thinking what you would do in a situation (and thus assume similar experiences). Nevertheless, you lack the ability to directly access what they understand by the concept.

This causes difficulty. Our emotions and experiences are complicated cocktails, and when we reference part of them with a word we want the other person to know what we mean.

Yet with the most fundamental concepts and experiences, the words ‘splurge’. They may have subtly different connections and meanings to individual people, which are hard to correct. Providing definitions for fundamental concepts fails to help much as they end up being self-referential. (e.g. Q: what is right to do? A: What is moral)

This means that arguments end up meaning different things to different people, and persuasion and verification is difficult.

An example

Let’s look at the ‘Kalam Cosmological Argument’ for the existence of God. (There are later steps which look at the nature of whatever was the cause of the Universe but I won’t look at them now)

  1. Whatever begins to exist has a cause.
  2. The Universe began to exist.
  3. Therefore, the Universe had a cause.

Why are some people persuaded by this and others not?

First, ‘whatever begins to exist has a cause’ is very abstract. It references these concepts of existence, beginning and cause. How do you ensure you understand the same thing by ‘cause’, ‘exist’ and ‘begin’? In an everyday situation, you have fairly easy ways of making sure you mean the word in the same way, but not in this context. In these contexts you relate the concepts directly to some things happening in the world, which removes ambiguity. Yet, could you even define a ‘cause’?

Also, ‘begin to exist’ makes sense in my everyday context. But how the hell are we going to make sure we mean the same thing when talking about the beginnings of universes?

‘a priori’ confusion

Most of these Philosophical arguments just leave you with confusion.

The concepts accessed are connected in complicated ways. Yet we would struggle to be entirely sure what the linkages and concepts even meant, let alone be sure that others understood them in the same way.

Thus, if a Philosopher wanted to write down a set of a priori truths in words, she would face the difficulty of not being sure what concepts to express and explain what she meant to others.

It also means there is too much work for the individual to do. Humans rely on the insights of millions of others to do things. Even me writing this relied on the mathematical and scientific innovations and implementations to make it possible for me to type and create words on a screen to send around the world. Heck, it required language itself to have been created and taught to me.

Thinking of suitable distinctions and definitions for deductions is hard, but for this type of reasoning, in Philosophy, it is upto each individual to work it out for themselves, as we cannot express it to others. We have no shared reference point in the outside world for the a priori truths within us.

Imagine you had to do everything yourself in mathematics — you’d struggle to create addition, let alone set theory, differential geometry, group theory, number theory, Hilbert space… Creating the right definitions is one of the most crucial part of mathematics and is hard. Likewise here we struggle to think of the right definitions to help us make deductions.

📷Photo by Nick Hillier on Unsplash. You’d struggle to invent counting by yourself, let alone the differential calculus below
Photo by Roman Mager on Unsplash

An Empirical Question?

The Greek’s referred to Scientia, a broad term which covered Science, Geometry, Theology and more. Reading Plato’s Dialogues I am left with the impression that he viewed deductions of Philosophy as valid as those of Mathematics.

This makes it seem like an empirical question. The guarantor we have of Mathematics and Science is merely that we test our ‘shared framework’ against the world. It’s not a satisfying answer, but at the end of the day they work. When they didn’t work (such as when Mathematics led to contradictions near the end of the 19th century) we changed how Mathematics was built.

Abstract arguments about the existence of God or what precisely a priori truths we have don’t have a way of being tested — in part because Philosophy doesn’t assume an ‘external world’ to test things in!

A Collective Endeavour

Humans learn collectively. Ever single insight rests on millions of others. To solve one difficult problem in mathematics, I rely on the insights, language and teaching developed over centuries. Yet collaboration requires a common language, whether it is French, English, Python, Mathematical notation, art, music…

Philosophy has a barrier to collective learning and advancement through the lack of an adequate language. As an individual effort it will never succeed.

The author is ‘The Sociable Solipsist’, who studies at Cambridge University, UK. In his spare time he does Mathematics, Philosophy and writes flattering bios of himself.