Does mathematics contain fundamental flaws, and are mathematicians and teachers peddling a pseudo-religious falsehood designed to corrupt and indoctrinate young minds that all starts with the argument that 0.999… = 1? I recently published a video on the Mathsaurus YouTube channel discussing the statement that 0.999… = 1 and the ways it can be interpreted and justified and received a very long and detailed comment arguing exactly this case.
I really hope that I’ve been the unwitting victim of an educator trying to provoke philosophical discussion, but the sad truth is that we live in a climate where the public confidence in experts has been eroded by clear falsehoods in political narratives. These sorts of arguments do persist and are increasingly being used by people who want to exploit public misunderstandings of scientific truths for political or financial gain and so I wanted to take a closer look at the arguments raised, which I have phrased in the form of a letter to this viewer.
Dear viewer,
If you are a troll, as indeed I can only hope, you seem to be a highly educated one. You have clearly understood all of the arguments against your own position but somehow you don’t accept their consequences. I am afraid I can’t support your conclusion that teaching about recurring decimals is some sort of indoctrination. Indeed, in the current political climate of increasing disbelief in science it seems quite dangerous to phrase your objections in the way you do to an audience containing many students less informed than you on these matters.
On the existence of the infinite, and mathematical objects in general
Your objections appear to centre around the concept of infinity, and a rejection of any form of mathematics that you cannot experience directly in what you call the ‘real’ world. I would like to try to address a few related but very distinct questions:
(a) Should we reject infinity as a ‘real world’ concept?
(b) Should we reject infinity as a mathematical concept?
The philosophical position that infinity does not exist in the ‘real world’ seems to me to be a very valid one – I do not believe that anyone can truly say whether time or space is fundamentally ‘discrete’ and not divisible into chunks smaller than a certain tiny size, or that it is in some sense continuous. Or equally whether it is possible to say that the universe is takes up a finite or infinite amount of space. But I think we could argue in a similar way that no mathematical objects really ‘exist’ in this sense. Before we argue about infinity, I would like you to show me the number ‘2’ in the real world, for example. We could assert that for mathematics to have meaning that things made of the same matter in the ‘real’ world must be represented equally in mathematics. Then I can take 1 piece of paper, tear it into 2 pieces and tell you that therefore 1=2. But this does not mean that I have broken mathematics, just that I need to be careful in my definitions. Indeed, so much of school mathematics, and algebra, is about trying to understand what the equals sign really means. The questions of mathematics are about understanding definitions as much as they are about solving equations.
When I make the argument that 0.999… = 1, because we can establish by long division that 0.111… = 1/9, and that 0.999… = 0.111… x 9 = 1/9 x 9 =1 I am not in fact claiming to prove that 0.999… = 1. Nonetheless it is a very useful argument to show teenagers, that if we are to establish an equivalence between 0.999… and a number, that it could not be any number other than 1. We might be able to say that we have proved 0.999… = 1, but only once we’ve established what we mean by 0.999… and by ‘=’. In this case, we interpret 0.999… as a limit, and ‘equals’ as an appropriate statement about this limit. It turns out recurring decimals are quite a nice way to express and do algebra with these limits, but that is another matter.
Now your objection seems to be that this limit does not actually ‘exist’ because you cannot find it in the ‘real’ world. But when mathematicians assert the existence of a mathematical object, the ascribe this a very particular meaning. We might later be able to use the idea in the ‘real’ world, but the existence only truly occurs within an abstract mathematical world. Perhaps at an early stage of your education you have ascribed the term ‘real’ number to mean something to do with your everyday conception of the term ‘real’. But this is the perhaps the most misleading name in all of mathematics. Real numbers are no more ‘real’ than complex ones, or perfect lines or circles for that matter, and as stated above, the existence of this limit is no different to the existence of the number 2 or any other mathematical object.
Another possibly unhelpful distinction, is that which is often made between ‘pure’ and ‘applied’ mathematics. We tend to think of pure mathematics as a discipline that constructs hypothetical universes containing abstract and often beautiful theorems. Pure mathematics may be viewed as a subject that, despite the wide uses it has found, needs no justification. It can be seen as self- contained, perhaps even as a form of art. In fact, most of applied mathematics also sits within a theoretical universe, just one where the abstract objects like differential equations are imagined by their creators to have the potential to be useful for applications in the real world.
Actual applications of mathematics sit outside of the discipline of mathematics itself, and the question of whether a particular mathematical structure is a good model for a physical situation is a question for other disciplines, for example in the applied sciences or engineering. Although mathematics does not need such applications to be valid, there is no doubt that it has found a great many such uses and celebrated success. Would you discredit the use of circles in design because they are impossible to construct perfectly, or in using lines in geometry because they are meant to have zero width? The creator of a wheel may never be able to create a perfect circle, but the fact that such an object has been imagined is of immense benefit. If you are going to reject the notion of a limit of an infinite sequence, then you may as well reject all mathematical concepts – as none exist in the firm form you seem to want them to – that of being able to actually experience or directly perceive them.
Mathematics as religion
I could stop here, but I would also like to address your description of mathematics educators as something akin to religious evangelists. As a mathematician, much of this discussion will centre around a definition of terms, but I also believe that this framework will unveil some fundamental meaning.
I agree with you that there is a sense in which mathematics can be seen as a belief system – many enjoy disputing whether mathematics is invented or discovered, but it is certainly propagated by and popularised by humans, and in so in that sense viewing it sociologically and comparing its information and communities to religious ones seems to be an interesting angle. I would argue that mathematics is quite different in character to most religions, and that it may not fit the way most people would interpret the ordinary meaning of the term ‘religion’. Of course, if you wish to not believe in some or all of the axioms of mathematics and their consequences and to live your life accordingly, then that quite fundamentally is your choice, one which we may or not usefully characterise as ‘freedom of religion’. Personally, I enjoy the internal beauty of the pure forms discrete and continuous mathematics, and I have seen them model a great deal of truth and meaning in the world. So, in the sense that its a question of belief or faith, then you can count me firmly as a believer.
One fundamental distinction between mathematics and ‘other’ religions is that mathematics has sought to establish itself around a core set of irrefutable axioms and a system of logic. Other religions have guiding principles and reference points, or perhaps point to a notion of God or godliness or another set of rules as a system for making decisions, but none have quite the level of rigorous definition as the axioms of mathematics. The axioms of mathematics were formed later than many of its fundamental results were discovered, but this analysis of the foundations of the subject led to some results being discarded. On this point there are many choices of the axioms of mathematics, and we can choose to some extent which to accept or reject, but we must generally agree on their consequences.
Now the axioms are abstract and so do not exist in the sense you might want them to. Perhaps there is a parallel here to the existence of a God in a religion. Whilst we can never prove God’s existence or non-existence in our human mind, many do see evidence of God’s existence in the world around them, in the beauty of the natural world or in the actions of the people they love. Many also find evidence of God’s non-existence in the outbreaks of deadly diseases, or in human actions like wars, whilst others may simply infer that God is working in mysterious ways.
Not all religions place deities at the centre, and even where they do many followers may still not pay much thought to questions of existence. Many instead find stories about God or religious teaching a useful way to navigate a path through life without having to ascribe them literal meaning. In that sense, many who consider themselves non-believers might still gain useful practical or moral meaning from the stories contained in religion or other systems of belief. In dialogue between religions, and between believers and non-believers, we can learn to understand each other’s perspectives better. Frustrations can also develop between a mathematician who wants to centre discussion around logical concepts, whilst a religious person may wish to take God as a starting point, or other social or emotional considerations. Similar difficulties can occur between people of different religious faiths or groups where they fail to find common starting points.
Teaching in mathematics and other religions
One significant difference between mathematics and ‘other’ religions is the level of agreement within the mathematical community about the definitions and rules of the system. Within a system that takes the existence of God or the contents of a text as a key axiom of a guiding belief system, there is a great deal of interpretation possible of the details of what God intended when they appointed some humans to write down their will in the form of a book. Religions tend either to try to resolve their disagreements either by delegating authority to certain individuals or groups of people, or otherwise they allow ambiguity to persist, but either way the interpretation of God’s word is an ever-evolving narrative.
If you believe that mathematics is created by humans, then it also evolves over time, or if you believe it is simply being discovered, then perhaps it exists in some perfect form but is gradually being discovered over time. Either way, whilst advancements in our understanding of mathematics may be made by individuals, disagreements about it are resolved by reference to a robust system of logic that has pervaded over centuries. There may be human and political discussions about how mathematics is funded, and for which purposes it is applied, but questions about the discipline itself are resolved only by the watertight rules of the system.
However, to an outsider, it may seem that there is little difference to a religion – professors are the clergy, and disputes about mathematical misunderstandings may be referred to them for clarification. From the student’s perspective of relative ignorance, they may not understand either the content of or the nature the advice they are being given by a teacher. Indeed, bad teachers of both mathematics and religion might abuse their power by placing themselves at the centre of the experience, seeking to establish positions of power as guardians of truths not discoverable by students. They encourage adherence to rules, but not a critical appreciation of them, and they do not pave a path to either mathematical or spiritual nourishment. Good teaching guides students to search for meaning themselves, guides them around common pitfalls and encourages them to persevere in the face of challenges.
Bad teaching may have tarnished the reputation of mathematics in the eyes of many generations of students, though they do not discredit to the discipline of mathematics itself. Neither does the existence of abusive religious leaders have any bearing on the arguments for or against the existence of God. In the same way, religious leaders who design systems to cement their own power rather than to nurture the communities they serve may be considered selfish or abusive, but they do not necessarily undermine the philosophical case for religion itself. A great many people may legitimately continue to adhere to a religious belief that they view as being misinterpreted by its current administrators. Similarly, most students of mathematics have had to persevere in a pursuit of knowledge despite at least one bad teacher.
The case neither for nor against mathematics
I set out to show that religion and mathematics are somehow very different, yet my arguments seem to have found many things in common. Most fundamentally, despite mathematics educators’ claims of mathematics fundamental truths, there is no way that I can prove to you that its system of mathematics is correct in the ordinary sense of the word. Indeed, Godel’s famous incompleteness theorem proves the impossibility of establishing any such truth from inside the discipline. Even as the most abstract of its theorems are often perceived as more beautiful for their abstract nature, many believe even more so for belonging to an abstract world, unperturbed by the messy imperfections of human existence. But even these statements are self-referential, for it is only human existence that is leading to the experience of appreciating their beauty.
To understand this point further let us consider two related statements:
Statement 1: ‘This statement is false’
If Statement 1 is true, then it must be false. However, if it is false, then it must also be true. Statement 1 is often referred to as a paradoxical statement, though I shall simply state that it is undecidable. Statement 1 is not to be confused with a contradiction, which would assert something like ‘The sky is both blue and not blue’. This is a contradiction – it is a statement that is very much decidable, and is always false.
Statement 2: ‘This statement is true’
The problem with this statement is much less obvious and it would be easy to glance over it without further thought. Indeed, if Statement 2 is true, then t is true, as it states, so surely there’s no problem. But, and here’s the catch, if Statement 2 is false, then since it states that it is true, it is false! So it is also an undecidable statement. It could be either true or false, and either way is logically consistent. Again, Statement 2 should not be confused with a tautology, like the statement ‘Either the sky is blue or the sky is not blue’, which is always true by its definition.
So, whilst a well-meaning teacher may have convinced you that mathematics is somehow ‘true’ and you may have enjoyed the security that mathematical problems seem to always have clear-cut right and wrong answers, be aware that the discipline itself makes no such assertion. Indeed, when asked the question ‘are you true?’ the discipline of mathematics replies unequivocally with the response ‘it’s impossible for me to say’.
The case for belief in mathematics
So, despite this, why do I still state so happily not only that I am a firm believer in mathematics, but also that whilst I will accept whatever choice you make on a personal level, I do think that you are fundamentally wrong to reject mathematics, including the mathematics of the infinite?
Firstly, it is because of the sheer weight of evidence in favour of mathematics that I have seen. I do believe personally that I have had mathematical experiences in the real world – when I look to the stars in the sky, or to the fractal patterns on the leaf of a tree, I believe that what I am seeing is mathematics. I know that tis is subjective, and that many others might say that they are experiencing God in these same moments so however convinced I am I know I will need to present more evidence. Perhaps you will be convinced by its use?
When I see people using reasoned, mathematical logic to help settle debates I see its power to bring people together, to provide a neutral language in which to explain the world, and to systematically understand different perspectives. When I see science help limit the spread of a deadly pandemic, or cure my family and friends of otherwise deadly diseases, underpinned by mathematics, I witness its value, its power and its worth and I can only believe my life is far better with mathematics than without it. And when I construct the limit of a geometric series represented by a recurring decimal, or some other abstract mathematical object, I may in that moment be engrossed in the beauty and consistency of an abstract world, but what I am experiencing is also profoundly linked to these concrete realities of my existence.
I think it is probably misleading to compare mathematics to religion. When the word ‘religion’ is used in everyday language It generally refers to not just to a doctrine or a system of beliefs, but also about groups of people and about organisations. At their best, these groups point towards universal truths, create nurturing communities, and nourish the souls of their adherents. At their worst, they enshrine power and corruption for the benefit of their leaders at the expense of human wellbeing. But I could say the same thing about the schools and universities that teach mathematics, or the banks and other businesses that profit from it – they can do so in good and bad ways and the reality is usually a messy middle ground. But to the extent that any of these organisations, mathematical or religious, are truly effective in finding truth, I believe that they will also eventually find mathematics in some form. So perhaps mathematics is not the religion, but rather the God. One who challenges us with difficult problems, but who also rewards our perseverance with beautiful insights, who teaches us to think for ourselves and to be robust in the face of those who seek to push us down unfruitful paths.