Critical Thinking Toolkit: Inductive Reasoning

Deductive reasoning is a very useful way to gain knowledge, but it is also very limited. Deduction is a type of reasoning that intellectually compels you to believe something if you accept some number of other things. But very few aspects of reality are like this. In most situations, there are a range of available options for explaining certain aspects of the way the world is. This is the case in science, history (especially ancient history), psychology, and so many other areas of our daily lives. It is therefore important to discuss methods of reasoning that help us deal with situations where a variety of options are available to us within the bounds of deductive reasoning.

In this post, I want to discuss one of the two most important methods of reasoning that are not deductive in nature. This method is called inductive reasoning or induction.

What is Induction?

Although most of us have a familiarity with induction in one particular context, I don’t want to define induction using a specific kind of induction, but by using the broadest form I possibly can. Inductive reasoning, in the most general sense, draws from specific examples some kind of general conclusion. (Notice that specific examples do not necessarily lead you to some conclusion, but nonetheless it is often quite reasonable to use specific examples to justify a conclusion).

I think probably the most helpful general example I can use for induction is the way that a toddler learns about the world around them. It is probably quite obvious to every one of us that when we are born, we don’t really know anything about the world – at least not consciously. Perhaps we have certain inherent instincts when we are born – an example of such an instance might be something like when I experience such-and-such a sensation, that means I need urinate. Beliefs about hunger and pain are probably quite similar. But consider a different belief – that liquids are wet. You certainly are not born with this belief. But over time, as you gain experience with more and more things that look liquid, you will discover that each of them feel wet. As a toddler, you learn that liquids feel wet.

Is learning that liquids feel wet deductive? Not at all. For a toddler, liquid is basically something visual – you would identify liquid in terms of what a thing looks like (namely, fills the space it occupies, sloshes around). For the same toddler, wet is a tactile experience – you feel wetness on your skin. Since the senses are distinct, you won’t be able to use anything deductive to connect them (at least not for a toddler – perhaps there are such arguments that I am unaware of). So if the toddler isn’t using some kind of deductive reasoning in their subconscious, what is happening here? The answer is that the toddler is learning about their world inductively. They have enough examples in their lives where such-and-such a visual experience is related to such-and-such a tactile experience that they come to conclude that the two are closely related to one another.

Could they be wrong? Well, yes. There are non-Newtonian liquids (so-called oobleck is an example) that sometimes become hard when you press on them – this doesn’t seem wet. And yet when left to themselves, non-Newtonian fluids certainly look liquid. So this would be a counterexample to the toddler’s experience. So deductively, “wet if and only if liquid” is not really true. But nonetheless, we all know that it is quite true that wetness and liquidity are quite related, even if not identical. This is the kind of knowledge that we can gain by induction. The toddler experiences enough examples of wetness and of liquids, and they come to infer – in an inductive way, not a deductive way – that wetness and liquidity are related.

This is the sort of learning that induction is all about. But… how do we actually use induction?

The Scientific Method

This is the example I was referring to vaguely earlier. All of modern science is at its core an inductive enterprise. The scientific method involves conducting experiments over and over again, trying to figure out how any slight variations in how you do your experiment affect the result of that experiment. When you do this, what you are really doing is trying to build up examples of physical experiments in an organized way. The organization and scientific control here is supposed to help you do induction faster than would otherwise be reasonable. It might take a baby a year to realize that liquids are wet – a scientist could arrive at a conclusion like that much, much faster.

There is such a thing as theoretical science, that uses theoretical principles to derive theoretical conclusions – but at the end of the day, these results are only meaningful when verified experimentally, which is where the inductive method comes in. If theoretical science turns out to be false, then it still has value as a piece of theoretical mathematics, but not as genuine physics that applies in the real world.

Daily Life

We often use induction in daily life as well. Muscle memory is sort of like induction – although it is not cognitive, muscle memory is acquired in a way that resembles in a lot of ways the way that we obtain knowledge of the world through induction. This is also probably similar to the way we learn language and skills like driving, cooking, a sport, or even child-rearing. Anything that you learn “by experience” usually has a lot to do with induction.


Induction is an important way that we come to learn information. Although deduction does give us more certainty than induction does, induction is more widely applicable than deduction. Both are useful and they have their contexts, but it is important to recognize both.

As a mathematician, I am all too familiar with the failures of induction. There is a reason that mathematicians use deduction and not induction. For the mathematician, Skewes’s number is a key example of why we don’t place our absolute trust induction. It would take a while to explain all of the details that go into Skewes’ number, so I will be brief. Prime numbers are an important kind of whole number, and in mathematics we often write \pi(x) to represents the amount of prime numbers between 0 and x. Mathematicians now have an easier to compute function, which we will call Li(x), that very closely approximates the much harder to compute \pi(x). For a long time, it was believed that Li(x) was always an underestimate for \pi(x). But the South African mathematician Stanley Skewes has shown that eventually this is no longer true. But all we know today is that the failure happens when x is less than 10^{10^{10^{964}}}. The number 10^{964} already has essentially no meaning in the physical world we live in – the number of subatomic particles in all of the world is so much smaller than 10^{964} that it is funny. The number 10^{10^{10^{964}}} is insanely larger than this. What is our point here? The point is that you can have millions of millions of millions of examples of a true claim in mathematics, only for that claim to turn out to be false later. This ought to give us pause about any absolute trust in induction. Induction gives us a huge degree of trust when done correctly – as something like science usually is – but it is still not infallible.

“The Unreasonable Effectiveness of Mathematics in the Natural Sciences” by Eugene Wigner (Summary & Comments)

It is only relatively recently in human history that the deep connection between math and physics was realized. Early civilizations knew of some simple connections regarding a pretty intuitive level of mathematics. Counting has an obvious connection to our world, as does much of geometry. And yet, that isn’t really what science is any more. It is no longer intuitive kinds of mathematics. Scientific theories today use extremely abstract ideas – including manifolds and tensors, just to name a few – to explain their theories about how our physical world works. Why would something as abstract as a tensor product have anything at all to do with our world?

This is the topic of physicist Eugene Wigner’s famous 1960 essay, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” published in Communications on Pure and Applied Mathematics – an academic journal that primarily publishes papers relating to mathematical science, applied mathematics, and physics. Wigner, who won the Nobel Prize in physics in 1963 for some of his groundbreaking work in elementary particle physics, had a lot to say about the relationship between mathematics and physics. Despite being a convinced atheist himself, he concludes this essay with the following remarkable and surprising line:

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.” [1]

We ought to be taken aback by an atheistic physicist calling something deep at the heart of modern physics itself a miracle that we don’t understand or deserve. What is going on here? What would lead a Nobel Prize winning physicist to make such a strange remark? For the remainder of this article, I’d like to give a brief summary of Wigner’s deeply interesting article and to expound upon some of his observations.

  • As a note, the square brackets [ ] with numbers inside of them are used for citations, and parentheses ( ) with numbers inside within my summary of Wigner’s article at points that I later want to reference and make commentary on.

Summary of Wigner’s Paper

Introduction to the Idea

“There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. “How can you know that?” was his query. “And what is this symbol here?” “Oh,” said the statistician, “this is pi.” “What is that?” “The ratio of the circumference of the circle to its diameter.” “Well, now you are pushing your joke too far,” said the classmate, “surely the population has nothing to do with the circumference of the circle.”

This is how Wigner’s article opens. Of course, the approach of the classmate it a bit naïve. Yet, there is something eerie about it. Why exactly should the circumference of a circle have anything to do with populations? The classmate goes a bit too far in calling the claim a joke, after all sometimes bizarre things do happen, yet surely there is something unusual here. Wigner points out two things [1]:

  • “The first point is that mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections.”
  • “Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate.”

Again in Wigner’s words, “We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.” In other words, is it really true that mathematics is a good way to talk about physics, or are we just getting lucky?

What is Mathematics?

What is it that mathematicians do? Well, to oversimplify, mathematics is very much the invention of concepts that we can exercise our logical skills on in a particularly ‘mathematical’ sort of way (1). Why then would mathematics have anything to do with the real world? Well, from a mathematical perspective, you could argue that the axioms (which are the most basic definitions and assumptions of the mathematician) are themselves grounded in things that we have real-world experience with – like the basics of counting and addition – and so we ought not be surprised that mathematics and the real world line up. This is true, for elementary mathematics. So fair enough, we ought not be surprised that elementary mathematics has anything at all to do with the real world.

But some mathematics isn’t like that. Actually, some isn’t the right word… more accurately, almost all mathematics is not like that. Very, very little interesting mathematics can be done without adding in either additional definitions that are not intuitively grounded in the real world, or additional axioms that are not intuitively grounded in the real world (2). Surely this kind of mathematics can’t have anything to do with the real world?

We will get to that later. For now, the point is that mathematicians, in Wigner’s own words, “ruthlessly exploit the domain of the permissible” with abstract concepts like Borel sets, complex numbers, algebras, and linear operators (particularly on infinite-dimensional spaces). Mathematicians are not even trying to remain within concepts that we see in the physical world – they are basically trying to come up with concepts entirely foreign to the physical world. On this recklessness, Wigner adds the following [1]:

“That his recklessness does not lead him into a morass of contradictions is a mircle in itself: certainly it is hard to believe that our reasoning power was brought, by Darwin’s process of natural selection, to the perfection which it seems to possess.” (3)

So, not only is this process of mathematics entirely unrelated to the physical world, it is not at all clear on a totally physical view of the world how we are even able to do this kind of reasoning without falling into “a morass of contradictions.” But what exactly are we talking about here?

The complex numbers are a helpful example. As a reminder, the foundation of all complex numbers is defined by i = \sqrt{-1}. We must immediately remember that even attempting to define such a number radically defies everything we know about multiplication. This number is supposed to satisfy the equation i^2 = -1, but for real numbers – the only kind of numbers that physical reality suggests to us – this is utterly impossible. If you ask the mathematician, they will tell you that the study of complex numbers was not at all motivated by physics when they were first studied – rather, complex numbers arose in the theory of equations, in efforts of describing solutions to cubic equations (and eventually, equations of higher degree).

What is Physics?

We’ve now discussed what mathematics is about. But what is physics about? In order to understand physics, it is critical to look at the concept of a law of nature.

Wigner’s first observation is summarized by a quote from Schrodinger – that it is “a miracle that in spite of the baffling complexity of the world, certain regularities in the events could be discovered” (emphasis mine). This is quite right. None of us can just look around at the world and predict what is going to happen. For instance, try predicting the weather tomorrow without looking at a weather app. You just can’t do this sort of thing. The world seems in our daily lives to be incredibly unpredictable. And yet, the amazing observation is, we actually can predict some things. Finding this predictability in a sea of unpredictability is surprising.

But what are these regularities? A prototypical example is Galileo’s observation that if you drop two rocks from a tower at the same time and same height, they will always hit the ground at the same time. We find two sorts of regularity in the laws – their constancy across various circumstances, and that only an extremely limited set of factors have any influence at all on the fall. Given the same rock and the same height, it will fall in the same amount of time every time. If you hadn’t grown up being taught this, it wouldn’t be at all clear that this should be the case. But even more surprising is the shockingly few factors that play a role. Things like size, shape, color, location, wind, the time of day and even the weight of the rock have nothing whatsoever to do with the fall. It is not at all obvious why some factors determine how the rock falls and others have nothing at all to do with the fall.

Without both of the criteria just described, there could be no physics – or at least no physics that human beings could actually do or understand. Without the regularity, you cannot predict anything with any certainty. Without the extremely limited number of conditions that are relevant to a given event, problems would be too complicated for a human being to even possibly understand. It is not at all clear that ‘laws of nature’ like these should exist [and Schrodinger himself says that this miracle may be beyond human understanding] [2]. It is not at all obvious why there should be laws of nature that enable us to predict future events given present states of affairs, and why human beings should be able to understand them at all compounds the mystery.

These ‘laws of nature’ are conditional statements and they only relate to a very small slice of the world in which we live. And these laws give surprising information. Why ought we be able to always know the acceleration of any falling body anywhere on earth while being completely ignorant of their position and velocity (4)? And, with respect to actually predicting future events, the laws of nature only give us probabilities – we cannot predict future events with perfect precision even given the laws of nature – although the laws get us very close.

How Math and Physics Interact

When we do physics, we use mathematics to make the calculations. We normally don’t think much of it. But, there is an underlying background assumption here. This whole approach assumes that mathematics is the right language for expressing the laws of physics, as the famous quote of Galileo goes. There are some elementary aspects of the physical world – like counting and a large part of geometry – that were obviously developed for the purpose of looking at the physical world, and so we ought not be too surprised when we find that addition, multiplication, and elementary geometry help us do physics.

But most of physics isn’t like this. Take for example quantum mechanics. There, the two basic concepts (states and observables) are vectors in a Hilbert space and self-adjoint operators on those vectors, respectively. Possible observational values are the characteristic values of the operators. Unless you have some sort of degree in mathematics or physics, these terms from the advanced theory of linear operators will probably be meaningless to you – as well the term linear operator. Which is rather the point we want to make. Even once you learn what these things mean – it is not at all clear that such concepts should make an appearance in physics. You can make a similarly daunting list with general relativity and particle physics.

This is unlike the previous situation, because now the concepts are not ‘so obvious that they were sure to arise in a physical theory.’ Take as an example the complex numbers, which we mentioned earlier. One would at first guess that \sqrt{-1} can’t possibly have anything to do with the physical world – after all, it defies the laws of multiplication that are familiar to us. And yet, they are absolutely crucial in essentially all of modern physics. Complex numbers are indispensable. The application of so-called analytic functions to quantum mechanics is similar – there is no reason whatsoever to expect the mathematical theory of analytic functions to play the huge role in physics that it in fact does play.

The so-called “empirical law of epistemology” – the assumption that the laws of nature shall play out as mathematical constructions that human beings are capable of understanding – is “an article of faith of the theoretical physicist.”

Why the Miracle?

In light of all this, Wigner says the following: “It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that a human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them.” The closest explanation we have is Einstein’s statement that we will only ever accept beautiful physical theories – but that is a statement of what kinds of theories we want to look at, not about what theories might be true about the world.

One possible explanation would be to accuse the physicists of irresponsibly assuming there is a deep mathematical connection going on every time he sees something he recognizes as looking like some other thing he learned in math class. Perhaps this is so. But then how in the world does such crude experience lead to extremely precise predictions? And it is even more surprising, because the predictions made upon a certain evidence base are able to make discoveries that have no relation to the dataset used to construct the theory. Take for example Newton’s gravitational laws. There was very little empirical evidence at the time to back up his theory, and the fact that the second derivative played such a massively important role was repugnant to some, or perhaps many, at his time. Why the second derivative and not the first derivative, or just the underlying function itself? And yet this crudely developed theory has now been shown to have an accuracy of less than one ten thousandth of one percent, and Newton’s law of gravity was used to discover a planet never before seen.

Elementary quantum mechanics has similar features. After it was initially developed, some physicists noticed a parallel between the mathematics of matrix algebra and of this rudimentary quantum mechanics. So, at the time there was no evidence whatsoever that their matrix mechanics would actually work in real-world scenarios. The move to use matrices was purely based on a cursory mathematical similarity with little to no empirical evidence, and yet the matrix mechanics worked, and solved extremely difficult problems about hydrogen and helium atoms to a degree of accuracy of less than one part in one million. In Wigner’s words: “Surely in this case we “got something out” of the equations that we did not put in.

We reemphasize our point with the example of Newtonian gravity again. Why should we expect that, even though there is no simple equation that tells us the velocity location of an object at any moment in time, there is a simple equation that tells us the acceleration of any object whatsoever at any moment. Why should there be an equation like that? It is patently absurd to think that this is self-evidently true (i.e. obviously true). Some may argue that things will be much clearer once we have a unified view of all of physics – but it is not even clear that physics can be unified. The two most successful scientific theories ever – general relativity and quantum mechanics – make logically contradictory assumptions about the way the universe is structured (5). One operates in an infinite dimensional Hilbert space, and the other in a four-dimensional Riemann space. These are very, very different sorts of objects. This ought to at least give us serious doubts about whether unifying physics is even possible. And even if we did unify physics, it is very likely that the non-obviousness of these laws of nature will be equally present in a unified theory, if not more so. Even though essentially all physicists believe a unifying theory is possible, this is by no stretch a foregone conclusion.

There is yet more surprise here. Because of the contradictory nature of general relativity and quantum mechanics, we know that at least one of them has a foundational assumption that is false. But then why does a theory with such a deep flaw at its very center make such incredibly accurate predictions about the world? This is not the dream scenario of the “article of faith” mentioned before – we are now entering the nightmare of a totally false theory producing amazingly accurate results. And this nightmares, for all we know, may actually be real.

In conclusion, Wigner writes the following:

“Let me end on a more cheerful note. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.”

Additional Discussion

I’ll start of by saying that I find Wigner’s actual article quite readable and not all that long. I’d highly recommend it to anyone interested in the discussion here. I have provided a citation at the end.

As a mathematician myself (or a mathematician-in-training, depending on your definitions), I agree with what Wigner says about mathematics. It is very similar to the opinion defended by most – or more likely all – professional mathematicians that I know, summarized best in G.H. Hardy’s A Mathematician’s Apology. The point here is that mathematics – as practiced by the mathematician – is comparable to poetry, painting, or theatre. It is an art form, seeking to shed light on beautiful patterns and to weave logical ideas into interesting stories. Although we use a different medium from the painter and poet, mathematics is a form of art. I think that probably the closest direct comparisons would be poetry and music.

The point Wigner makes that I agree so firmly with is that, by and large, all but the most absolutely basic of mathematical concepts are motivated by aesthetic reasons rather than practical ones. This is why Wigner sees such surprise in advanced mathematics applying to physics – it would be like discovering an actual poem found uniformly across nature. It is not obvious why such abstract mathematics should be relevant in the world of physics.

I am no physicist, but it seems that again Wigner is perfectly correct in his analysis of physics and the relationship between mathematics and physics. We find that framing laws of nature as mathematical formalisms has been extremely useful and accurate, and yet there really is not any a priori reason to think that laws of nature ought to be mathematical. For if such a thing were a priori obvious, why didn’t more ancient cultures have this idea? Ancient Greeks had engineering and some pretty advanced mathematics, but never applied the same degree of mathematical precision to study nature itself. It took monumental figures like Galileo and Newton to get the “physics is written in the language of mathematics” theme to actually catch on.

The mystery here is quite real. On the one hand, the study of, say, Riemann surfaces or Hilbert spaces, was entirely motivated by a desire to study a rich mathematical beauty and had nothing to do with physics. And yet, these beauty-motivated studies pop up in physics all over the place. And it isn’t just a few concepts. You find the same pattern all over physics. What is going on here? Is this actually true? And if so, how do we explain this incredibly non-obvious fact about the world?

I know of only one way. If the universe were created by an intelligent being capable of recognizing mathematical beauty and structuring the physical world according to beautiful ideas, then this apparently miraculous fact would make a lot of sense. This same hypothesis can also explain why human beings are capable of observing this structure – because the intelligent being that created us purposefully gave humanity the ability to see both beauty and logical structure in the world around us. This sounds quite a lot like theism – the view that God exists (polytheism would be shaved away by Ockham’s razor unless some evidence were be provided that more than one intelligent being is required).

Wigner himself was an atheist, but he never provided any explanation for how or why the world has this mathematical structure. So you can maintain that there is no God and yet recognize the facts that Wigner presents. But I’m not aware of any account other than theism that makes sense of why the world has this surprising feature.

My purpose here is not to lay out the full argument for the existence of God, so I won’t do so. But I have provided resource links to helpful videos in which the points Wigner brings up are synthesized into a more rigorous presentation of the idea I have just put forward. I also have in the video resources a rebuttal by a popular online atheist and a response to that rebuttal in order to give both perspectives.

I hope anyone who has read this far finds Wigner’s observations in this famous essay as interesting and thought-provoking as I have.


(1) This includes operations with ideas like quantity, size, shape, and most importantly, studying pattern, symmetry, and structure.

(2) The axiom of choice would be a good example of an axiom that cannot be grounded in our physical experience of the world.

(3) The problem of arriving at a system capable of abstract logic via natural selection aimed only at survival is quite a serious problem apart from the difficulties it brings here. See Christian philosopher Alvin Plantinga’s book Where the Conflict Really Lies or atheist philosopher Thomas Nagel’s book Mind and Cosmos: Why the Materialist Neo-Darwinian Conception of Nature is Almost Certainly False for in-depth discussion of this conflict.

(4) To know the position and velocity of a body in the future, you must first know its position and velocity in the present. This is not so with acceleration. Acceleration (due to gravity) is constant, and so you do not need present acceleration to know future acceleration. But, since acceleration is defined as the first derivative of velocity and the second derivative of position, it is surprising that such closely related concepts do not have similar levels of predictability.

(5) The way general relativity and quantum mechanics define space is contradictory. In quantum mechanics, space is quantized (built out of discrete, indivisible bits) and in general relativity, space is a manifold (completely smooth, not made out of discrete, indivisible bits). Both of these conditions cannot be simultaneously true.

Video Resources


[1] Wigner, E.P. (1960). “The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959”. Communications on Pure and Applied Mathematics. 13: 1-14. (Link)

[2] Schrodinger, E. What is Life? (Cambridge: Cambridge University Press, 1945), p. 31.

What Is a Limit? (Explaining Calculus #2)

Sometimes, it doesn’t matter as much where things are right now, but where they are going. For example, I am writing this roughly six months into the COVID-19 quarantine in the United States, with the 2020 presidential election on the horizon. With an election nearing, there are a lot of polls out there about who is ahead of who. Although those could be interesting, what I think all of us care most about is how those polling results change over time. For instance, will the recent death of Supreme Court justice Ruth Bader Ginsberg alter public opinion on the election towards Republicans, towards Democrats, or neither? I imagine most people who read this will know the answer to this question – but as I write this we don’t know yet, and in any case that isn’t my point. My point is that we don’t merely case about what polls say right now – we also care about where they are going. To use fewer words, we care about the trend over time of the polls.

Polls, of course, use percentages. This makes polls very mathematical in nature. It is quite reasonable, therefore, to ask what mathematics has to say about polls. As has just been described, we as humans tend to care quite a bit not only about how things are at the moment, but how things change over time. There are numerous examples of our tendency to care about change over time, including climate change, political elections and our physical health, just to name a few. Mathematics has always been used to count things and quantify various values, but can we use mathematics to quantify how things change over time?

Thanks to calculus, the answer is yes. In fact, you could actually define calculus as the mathematical study of how things change over time. I of course am leaving out a lot of detail here, but for now this definition will suffice. When you hear calculus, think about the various problems we face in trying to understand how the world around us changes over time. How do the planets move around us? How do we move around the Sun? How does my bank statement change as I acquire interest? How does the location of a plane change as that plane flies? All of these questions – and so much more – lie in the realm of calculus.

A Foundational Concept: “Zooming In”

How do we study change over time? One way is through averages. For instance, if one year ago I had $1000 dollars in my bank account, and without ever depositing any extra money I ended up with $1010 in my account – then how much did I earn per day in interest? If I use averages, then I arrive at

\dfrac{\$ 1010 - \$ 1000}{12 \text{ months}} = \dfrac{\$10}{12 \text{ months}} \approx \$0.83\text{/month}.

This would tell me that, on average, my bank account gained 83.3 cents per month. But is that actually true? Well, not necessarily. Because of how interest works, it is actually more likely that I earned something a little bit closer to 83 cents in the early months and something a little closer to 84 cents in the later months. In this situation, the ‘error’ is pretty small. But if I tried to do the same averaging over much longer periods of time – say 10 or 20 years – then the average would become further and further from actually true.

Since averages tend to get less and less accurate as you expand the timeline, wouldn’t it make sense that narrower timelines are always better? Certainly that must be true. And in mathematics, it is true. The concept of “narrowing the timeline” is at the heart of what calculus is. In more visual terms, we can view this process as “zooming in.” The concept of “zooming in” is at the absolute heart of what calculus is. Our goal is to use the “smallest possible timeframe” for zooming in. Ideally, we’d like to use a “single instant” of time as our timeframe… but what does that mean?

The Concept of the Limit

The limit is the mathematician’s way of making sense of ‘taking an average’ at just a single point in time. This actually took a long time to figure out. For aficionados of mathematical history, you might know that Archimedes came very close to coming up with this idea in ancient Greece, about two thousand years ago. But he didn’t quite get all the way there. Nonetheless, the basic idea Archimedes used is the same core concept used in limits. The idea is to look at the trend of a process over time. Let’s use an example.

Consider the list of numbers: 0.3, 0.33, 0.333, 0.3333, 0.33333, 0.333333, \dots, a list that keeps going and going. There is a pattern in this list – each number has an extra 3 added on to the end of it. One natural question to ask about a thing like this is “does this pattern take us anywhere?” In others words: Are we moving towards something, ever closer without ever quite getting there? The answer here is yes – intuitively you’d want to say 0.3333333... with 3’s going on forever. This actually is a number – this turns out to be exactly the value of \dfrac{1}{3}.

Continuing a process like this all the way to the ‘end’ of a process is basically what we mean by a limit. As we get closer and closer to some ‘final point’ – is there a pattern? If so, what is the pattern?

Framing the Limit Quantitatively

This concept of a trend towards some ‘final result’ is a fairly intuitive idea. But how can we frame this into actual mathematics? To show how we do this, let me define how I would write this down as a mathematician, and then explain how a mathematician can make more exact how to calculate these things (although a bit abstractly).

The easiest way to go about this is to use function notation. The role of the ‘pattern’ is played by a function f(x). If we want to, we could be more specific about what sort of pattern f(x) is, but we don’t have to. Imagine now that instead of wanting to what value f(2) represents, we instead want to know what f(x) is close to when x is very close to 2. You might think of this as the ‘trend’ of f(x) as x approaches 2. If there turns out to be an actual answer to this question, we call that answer

\lim\limits_{x \to 2} f(x),

and we read this out loud as “the limit as x approaches 2 of f(x)“. Now, what would it actually mean for there to be an answer to the question? To explain how this works, let me use a more explicit example. Let us say that the ‘pattern’ we actually care about happens to be described by the function f(x) = 2x + 3. What would it mean for a limit, say \lim\limits_{x \to 2} f(x), to be equal to some number, say L? One way to think about it is you can “make f(x) zoom in on L” by “making x zoom in on 2″ close enough. In more visual terms, you can zoom in on the height of f(x) by zooming in on 2 on the x-axis.

A Numerical Example

For a more ‘numerical’ approach, I’ll briefly explain that \lim\limits_{x \to 2} f(x) = 7. Let’s say that we are in a scenario where we are building something that will collapse if we don’t get our measurement close enough to 7 – to be more specific, we have to be within \pm 0.01 of 7. We know that if we were to use an exact value of 2 for x, we would get exactly 7. But this is the real world – measuring exactly isn’t possible. So we want to make sure that we are using a measurement device that gets ‘close enough’ to 2 for our building to be safe. How close will that need to be?

Well, it would help to have an equation that represents, given the measurement x we end up taking, how far off we are the value we need it to be – 7. We can express this equation as

Error = f(x) - 7.

We have stated earlier that our error has to be within \pm 0.01 in order for the building to be safe. Using the equation for error we just came up with, we can now say that we need

-0.01 < f(x) - 7 < 0.01.

We can add 7 to everything, since this is an equation. When we do this, and remember that f(x) is really just 2x + 3, we end up with the equation

6.99 < 2x + 3 < 7.01.

As with all equation-solving, our goal is now to get x on its own. We can subtract the 3 from both sides to get closer to that goal. Once we do this, our equation looks like

3.99 < 2x < 4.01.

Finally, we can divide both sides by 2. If we do this, we conclude that

1.995 < x < 2.005.

So we now know the actual values of x we require. If we instead what to know how far away from two x needs to be, we can subtract 2 from both sides and conclude that -0.005 < x < 0.005. This means that in order to get our measurement close enough to 7 for our purposes, we need our input measurement x to be $\pm 0.005$ away from what we want it to be.

There wasn’t really anything special here about \pm 0.01. We could have made that number much smaller, but as long as x was within plus-or-minus half of the value, we would have ended up close enough to 7. This is what we mean by a limit.

A “Fancy” Definition of the Limit

For the sake of completeness, I should define the limit in the exactly language that a mathematician would use. If your only goal is to learn what calculus is about conceptually, you probably won’t actually need this part. The intuitive descriptions given previously will serve you well in understanding what we mean by a “limit”. But if you want to know more about how calculus is “done”, then this may well be of use.

In the previous discussion, we have made frequent use of very small numbers. In order to make math easier to read, mathematicians try to use certain variables only for these very small numbers. The two most popular choices are the Greek letters epsilon and delta – which are written \varepsilon and \delta. In the definition you are about to see, you should imagine both \varepsilon and \delta to be some very small numbers.

Definition: \lim\limits_{x \to 2} f(x) = L is considered to be true if, for whatever choice of small number \varepsilon > 0 we want, we could choose another small number \delta > 0 so that whenever x is within \pm \delta of 2 (also written |x - 2| < \delta), the value of f(x) is within \pm \varepsilon of L (also written |f(x) - L| < \varepsilon).

There is of course nothing special about the 2. We could have put any number there. In fact, there is also a way to define limits “going to infinity”. If we replace being “close to 2” with “being extremely large”, then you basically end up with what a limit going to infinity represents. For a formal definition,

Definition: \lim\limits_{x \to \infty} f(x) = L is true provided that, if we choose any number \varepsilon > 0 we want, we can choose a large number N so that whenever x is chosen to be larger than N, it will always be true that |f(x) - L| < \varepsilon.

We are now done discussing the big-picture concept of what a limit is all about, worked out a numerical example to build on that intuition, and even given a “fancy” definition of what we actually mean by limits. These examples should be very helpful in building up our intuition for understanding more topics in calculus. Next time, I will walk through some important examples of how to actually calculate limits, examples that should point out some of the special things you can calculate using limits that you can’t do any other way.

Critical Thinking Toolkit: Rebuttal versus Refutation

When you disagree with someone, there are a variety of ways to approach that disagreement. The most obvious kind of approach would be to ask someone why they disagree with you and what evidence they have that underlies the disagreement. After all, in order to have a productive conversation you need to be talking about the same thing! But even if everyone involved in a discussion is ‘on the same page’ in this way, there are still a variety of ways that disagreement might happen.

In order to best express these distinctions, it is helpful to give a simple definition of what we mean when we say we know something. There is debate in philosophy about what knowledge actually is and what qualifies as knowledge, but for my purpose here we will use the most ‘basic’ idea of what knowledge is. This will be called the JTB model for short – which stands for justified true belief. Here is the basic idea. When we talk about knowing something, we of course mean at least that we believe it. However, knowing and believing are not the same thing. After all, the way we normally use the word know is underpinned by the idea that what we know actually is the case – that it is true. So knowing something, at least, must mean having a true belief. But this is also not quite enough. You might have a true belief that your team will win the baseball game tomorrow, but if your team is a massive underdog, most people would say you are being irrational. This example shows that you need some kind of reason for knowing something, some kind of justification. Using the JTB model, once we have a justified true belief, we can now say that we know something.

The reason I bring up this model is because knowledge is a helpful framework within which to analyze a disagreement. Say you are arguing with your friend Sarah about X. Sarah says that she knows X, but you don’t believe X – that is, you think Sarah is in some form mistaken. Using the JTB way of thinking about Sarah’s claim to know X, we might ask what went wrong. Since Sarah is claiming to know X, surely Sarah believes X. There isn’t much reason (at least in normal situations) to question whether your friend is being honest about what they believe. So how are you going to go about your disagreement with Sarah?

When you bring reasons forward in disagreement with a person, these reasons are often called defeaters. Put another way, a defeater would be a new piece of information that is meant to convince you to change your mind about something that you know (or think you know). Defeaters are whatever someone might bring to you as counterevidence to what you already believe/know. In the disagreement with our friend, we are wanting to give Sarah some sort of defeater for something she believes and claims to know.

Option 1: Refutation (Attacking the T of JTB)

One straightforward way to disagree would be simply claiming that she is wrong. You could say that Sarah doesn’t actually know X because X isn’t even true. In this approach, you would try to bring forward reasons to convince Sarah of the falsehood of X. These approaches are collectively called refutations, or refuting defeaters. If Sarah becomes convinced of your refuting defeater, she will come to believe that X is false, and because your friend is an honest person, will stop believing X.

Option 2: Rebutting (Attacking the J of JTB)

There are other ways a person can be flawed in their thinking that are not necessarily about matters of truth or falsehood. You might just think your friend is being hasty or biased. This is the case with the example of the baseball game I used earlier. It could very well be true that the team will win the game, but it might not be a very reasonable thing to believe until it actually happens if the team is much worse than their opponent. This kind of situation gives rise to a very different conversation.

If your friend Sarah was endorsing something along these lines – say that her favorite team will win the championship this year – perhaps you might point out to her that most experts think her team won’t even make the playoff. Notice the difference – you aren’t actually saying that they won’t win. Strange things happen sometimes. What you have done here is to point out that Sarah doesn’t have any good reasons for believing what she does about her team. Instead of undermining the T in JTB, this kind of defeater attempts to undermine the J in TJB. These are called rebuttals or rebutting defeaters in philosophy. Whereas a refutation tells someone they are incorrect, a rebuttal is more like telling someone that they are being unreasonable/irrational/biased.

Why Does All of This Matter?

These distinctions might be philosophically interesting, but does it really matter? Why would thinking about the differences between rebuttal and refutation impact anything in day to day life?

The difference actually matters a good deal. In fairly insignificant situations – like that of the upcoming baseball game – it doesn’t matter too much. But if the disagreement is over something much more consequential – say matters of politics, religion, ethics, or health – avoiding erroneous thinking becomes all the more important. In these areas, accepting something false or rejecting something true can cause harm.

The actual impact of rebuttals and refutations are different. To see why, suppose I am a Democrat for reasons A, B, and C. Then, imagine I get into a conversation with a conservative friend, who makes some really good rebuttals, and in the end convinces me that A actually isn’t a very sensible thing to believe. Does this now mean that I should cease to be a Democrat? Well, not necessarily. It might be that B and C are strong enough reasons to remain justified in your overall political stance and you only need to make some small changes. Or perhaps B and C are fairly weak reasons and A was your central reason, in which case some kind of bigger picture change in your beliefs would be justified and perhaps even necessary. But even in that case, these rebuttals don’t force you to become a conservative.

Now, imagine a different situation. Maybe you are an atheist (i.e. you believe that God does not exist). Maybe you have some reasons X, Y, and Z why you think there is no God. Imagine then that your friend, who thinks there is a God, gives you his reasons A, B, and C for thinking that God does exist. This ought to give you pause, since this is a refutation, not a rebuttal. Your friend is not claiming that your reasons are bad, he is claiming he has even better reasons that prove you wrong. In this case, what matters is the relative strengths of the reasons. You ought to base how you go forward based on the relative strength of your reasons versus his. And if you find his reasons better than yours, then you have an obligation to admit you were wrong and to change your mind.

The point is that since rebuttals and refutations take their aim at different parts of human thinking, we should process them accordingly. Rebuttals can be perfectly valid but not touch the truth or falsehood of the belief in question, whereas refutations always deal with truth or falsehood. But this distinction does not mean rebuttals can be ignored – it depends on what they target. If someone gives you a rebuttal against your belief that your brain functions properly, you better take that seriously! The nature of these questions is complicated and depends on context. This is why, when we engage in conversation with others, we should be careful to understand which situation we are in, what the consequences of people’s rebuttals/refutations actually are, and to hold ourselves to a high intellectual standard – being neither too hasty nor too hesitant in admitting when we are wrong.

Topics from “Pre-Calculus” (Explaining Calculus #1)

Before we can go into a discussion of calculus itself, it is important to set up some of the underlying concepts that calculus uses. You might view this as something analogous to learning the letters of the alphabet before you can start learning to read words and sentences. Without letters, you aren’t going to be able to grasp words. But once you start getting the words down, you don’t have to think about the letters very often any more. In the same way, there are certain mathematical ideas that are important for understanding how calculus works and what it can do, and you have to understand those first. But as you grow in understanding of calculus, you will have had enough practice with these ideas that you don’t think much about them anymore because they begin to come naturally.

My goal here is to lay out some of the key concepts that will come up at various times in the discussion of what calculus is, what it means, and what it does. You don’t have to be fluent in these ideas to understand the underlying concepts, but it certainly doesn’t hurt.


One of the most important uses of mathematics is in tracking how things are related to each other in the world around us. As an example, you might ask “if I throw a ball this hard, how far will it go?” The underlying relationship is between the force you put into your throw and how far the ball travels. In any situation like this one, you’re going through in your mind a mathematical concept called a function.

In mathematics, a function is a rule for transforming one kind of object into some other kind of object. Usually, a function turns one number into another number. The central idea of a function is that if you know the input, then there is only one way to get an output out of that. As in our example, if you know how hard you threw your ball, the the laws of physics determine exactly how far that ball will go (of course there are other factors like wind and angles, but you get the point).

Certain shorthand notations are helpful in talking about functions. Very often, we use the letter f to symbolize a function. If we need to use a few different functions at the same time, we often use the letters g and h next (just because those are next in the alphabet). You can use any symbol you want to represent your function, the choice of f, g, and h is merely a matter of convenience. We will also often write f(x) to denote the function f with the input value x. We then write f(x) = ... and in the realm of the dots, we write the rule we are supposed to use. For example, the function f(x) = 2x + 3 is a rule according to which you take your number x, multiply by that 2, then add 3. If you want to use a specific number in your function, you replace all the x‘s with the number you intend to use. For example, f(2) = 2*2 + 3 = 7 and f(-1) = 2*(-1) + 3 = 1.

There is one other slightly less common but equally important notation we should introduce. Sometimes, it is important to know the specific input values we are about. For example, you can’t throw a ball with negative force, so if you write a function describing the relationship in our example, you won’t care about whether or not you “can” plug in negative numbers. You might also care what sorts of numbers your output looks like – you can’t throw the ball a negative distance. In these situations, we use the notation f : A \to B (read “f is a function from A to B). When we write f : A \to B, what we mean is that A symbolizes all the inputs allowed for the function f, and all of the outputs of f will be somewhere in B.

When we express A and B, we usually use what is called interval notation. Interval notation is a way to express a range of values by writing down the endpoints of the ranges. When we want to write all numbers, we write (-\infty, \infty). You can basically just read this as “all numbers.” We can also use interval notation to express a limited range of numbers. For example, the interval (1,3) represents all numbers between 1 and 3, not including 1 and 3 themselves. The interval [1,3] represents all numbers between 1 and 3, including 1 and 3. In these examples, the round/open brackets ( ) translate to “not including” and the square/closed brackets [ ] translate to “including.” As another example, the interval (1,3] represents all numbers between 1 and 3, including 3 but not including 1. In intervals, \infty is used to mean that the interval never ends one some side. For example, (1,\infty) represents all numbers larger than 1. Finally, you can combine intervals using the symbol \cup (read this as “union”). So, (-3, -1) \cup (1,3) represents all numbers that are either between -3 and -1 or between 1 and 3.

Types of Functions

Functions can be just about anything you want them to be, but there are some special and helpful examples that are often used as examples in calculus and other areas of math. So, we will go through a few of those examples here.


A polynomial is any function built by adding/multiplying variables with numbers or other variables. Normally this takes the form of multiplying x by itself a certain number of times, multiplying the result by some constant number, and adding together other terms constructed in the same way. Here are a few examples of polynomials:

x^2 + 2x + 1, \ \ \ x^7 - 6x^3 + 3, \ \ \ x + 2, \ \ \ x^{101} + 53x^2 - 56x.


An exponential is formed by multiplying a constant to itself a variable number of times. One example of an exponential function is f(x) = 2^x. To calculate a few values of this exponential function, f(3) = 2^3 = 2 * 2 * 2 = 8 and f(5) = 2^5 = 32. An exponential function can use any number as its input, although the way to compute an exponential with a fraction (like 8^{1/3}) or with even weirded exponents uses some different methods… and very often you can’t really “simplify” an exponential expression.

If you use the definition of an exponential as repeated multiplication, you can deduce a few rules about how exponentials work. For example, 2^x * 2^y = 2^{x+y} and (2^x)^y = 2^{xy} will always be true. These rules form the basics of exponentials. Using these, we can also figure out a few other rules. What if, for instance, we want to know 2^0? Well, the first of our two rules would tell us that 2^2 * 2^0 = 2^{2+0} = 2^2. This quick equation leads us to conclude that 2^0 = 1. We can do something similar with negative exponents to see that a negative exponent is really just division. That is, 2^{-x} = \dfrac{1}{2^x} (see if you can see why – as a hint, you can use the first of the two rules along with the fact that 2^0 = 1). We can use the second rule to decipher the meaning of 8^{1/3}. Whatever this means, it should satisfy (8^{1/3})^3 = 8^{3 * 1/3} = 8^1 = 8. Since 2^3 = 8, this leads us to conclude that 8^{1/3} = 2. Similar ideas can be used for other fractional exponents.

The need to do calculations with exponentials is actually relatively rare, but it is important to know that there are these rules that can help us simplify expressions that have exponentials in various ways.

Graphs of Functions

Graphs are universally taught in school because they are a good way of visualizing information about mathematics. The idea is to use a two-dimensional grid to show how a function relates numbers to other numbers. This two-dimensional plane is often called the xy-plane. The reason is that the “horizontal” dimension is usually represented by an x variable, and the “vertical” dimension is usually represented by a y variable. Points on a graph are normally written as (x,y), which means “x units horizontally and y units vertically from the starting point.” This “starting point” is often called the origin (as it is the point from which the others ‘originate’ in a sense).

All equations, and in particular functions can be represented in the form of a graph. The way to do this is to plot all the points with coordinates (x,f(x)). In other words, the x-coordinate is the input, and the y-coordinate is the output. Below is an example, a graph of the function f(x) = x^2:

Circles and Distance

The concepts of distance and circles are tightly intertwined, and so we discuss them at the same time. Before doing so, it is useful to discuss a third concept – the Pythagorean Theorem – that is fundamental to understanding each. The Pythagorean Theorem tells us how to calculate the side lengths of a right-angled triangle. More specifically, if a,b,c are the side lengths of a right-angled triangle with c the longest side, then the formula a^2 + b^2 = c^2 will always be true.

How does this apply to the notion of distances? Imagine drawing two points in the plane (i.e. the xy-plane). How far are these from each other? The answer is the length of a line segment that connects them. If you’re having trouble visualizing this, draw two points on a piece of paper and connect them with a straight line. Now, draw a line in the east-west direction starting at one point and ending “beneath” or “above” the other point. Then connect this new line to the second point with a line running in the north-south direction. If the north-south line has length a, the east-west line has length b, and the distance between the two points is d, then the Pythagorean Theorem tells us that a^2 + b^2 = d^2. By taking square roots, we obtain

d = \sqrt{a^2 + b^2}.

If the two points have coordinates (x_1, y_1) and (x_2, y_2) in the xy-plane, then this distance formula turns into

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.

This is because a = y_2 - y_1 or y_1 - y_2, depending on which is positive, and similarly b = x_2 - x_1 or x_1 - x_2.

But how exactly does all of this relate to a circle? Well, the definition of “circle” involves the notion of distance. A shape is a circle if it is built out of all points that are the same distance from a central point. To visualize why this works, imagine taking a piece of string. Attach one end of the string to a fixed point on a piece of paper. Put a pencil in the other string, and move the pencil around the paper, keeping the string tightly pulled the whole way. If you do this correctly, you will draw a circle. The length of that piece of string is the “distance” involved in the definition, which we call the radius of the circle. The center of the circle is point that was pinned down to the paper.

Let’s use this to make an equation for circles. We want to know whether a point (x,y) is on a circle centered as (a,b) with radius r. That is, we need to know whether (a,b) and (x,y) are distance r apart. Using the distance formula we previously came up with, this becomes

r = \sqrt{(x-a)^2 + (y-b)^2}.

If we square both sides (and swap the right and left-hand sides), we come to the easiest-to-look-at version of the circle equation:

(x-a)^2 + (y-b)^2 = r^2.


This completes the discussion of “pre-calculus” topics. You don’t have to know how to do detailed work with these ideas in order to understand what calculus is about. But I’ve found through teaching that these concepts are quite helpful and are worth thinking about.

Database: What Do History’s Great Mathematicians Think About God?

There is very often debate about whether science and religion are in conflict. I know from my own studies and will continue to affirm that these are not at all in conflict. There are quite a few ways to go about this, and I plan to take on every angle that I can. Since I am studying to be a mathematician, I thought I’d start there. For if Christianity is anti-science, then you wouldn’t expect to see very many Christians among the greatest mathematicians. Therefore, I thought it would be informative to do a sort of statistical study of the greatest mathematicians of history.

My aim here is to take a subset of the greatest who have ever lived in my professional field of mathematics – where I have the background knowledge to confirm the greatness of these mathematicians. I have used a website that has an all-time top 200 list of mathematicians (for this list of mathematicians and their accomplishments, see This list is obviously not sectioned to bias any particular religious worldview, and so I figured I’d do a skim through it and do some considerations of what these great men and women believe about God.

To be properly balanced, I ought to say plainly that nothing can be totally proved from such a list, one way or another. This is purely statistical, and so nothing can be demonstrated beyond doubt. However, this can serve as evidence that can be evaluated in a ‘scientific’ way. Because the two opinions involved – the opinion that Christianity is anti-science at its core and that Christianity is compatible with science at its core – make different predictions about how many scientists and mathematicians embrace Christianity. Thus, while not conclusive, it ought to provide at least some insight into the questions at hand.

Over time, I hope to fill in more of the listings that I have put as ‘unclear’ or as ‘likely’ something-or-other. Turns out, this kind of biographical research isn’t always easy, as many people keep their beliefs about religion/God private. There are times that I have made educated guesses, and in those situations I have listed as ‘likely’ the conclusion. If I didn’t put ‘likely,’ this means that I found something that I found convincing (although I may well still be wrong). In any case, errors and inability to find information should approximately balance out.

The Top 200 Ranking, along with Views on God

  1. Isaac Newton – Christian
  2. Archimedes of Syracuse – Likely polytheist
  3. Carl F. Gauss – Christian
  4. Leonhard Euler – Christian (Calvinist)
  5. Bernhard Riemann – Christian
  6. Henri Poincaré – Atheist
  7. Joseph-Louis Lagrange – Agnostic
  8. Euclid of Alexandria – Likely polytheist
  9. David Hilbert – Agnostic
  10. Gottfried Leibniz – Christian
  11. Alexandre Grothendieck – Theist
  12. Pierre de Fermat – Christian (Likely Catholic)
  13. Évariste Galois – Unclear
  14. John von Neumann – Likely Agnostic
  15. René Descartes – Christian (Catholic)
  16. Karl Weierstrass – Likely Theist
  17. Srinivasa Ramanujan – Hindu
  18. Hermann Weyl – Deist
  19. Peter Dirichlet – Christian (Catholic)
  20. Niels Abel – Likely Christian (Lutheran)
  21. Georg Cantor – Christian (Lutheran)
  22. Carl Jacobi – Christian (from Judaism)
  23. Brahmagupta – Hindu
  24. Augustine Cauchy – Christian (Catholic)
  25. Arthur Cayley – Unclear
  26. Emmy Noether – Likely Jewish
  27. Pythagoras of Samos – ‘Pythagoreanism’
  28. Aryabhata – Likely Hindu
  29. Leonardo ‘Fibonacci’ Pisano – Christian (Catholic)
  30. William Hamilton – Unclear
  31. Appolonius of Perga – Unclear
  32. Charles Hermite – Christian (Catholic)
  33. Pierre-Simon Laplace – Agnostic
  34. Carl Siegel – Likely Theist
  35. Johannes Kepler – Christian
  36. Diophantus of Alexandria – Unclear
  37. Muhammad ibn al-Khwārizmī – Muslim
  38. Kurt Gödel – Likely Christian (Lutheran)
  39. Richard Dedekind – Unclear
  40. Felix Christian Klein – Unclear
  41. Bháscara (II) Áchárya – Likely Hindu
  42. Blaise Pascal – Christian (Catholic)
  43. Élie Cartan – Unclear
  44. Archytas of Tarentum – ‘Pythagoreanism’
  45. G.H. Hardy – Atheist
  46. Alhazen ibn al-Haytham – Muslim
  47. Jean le Rond d’Alembert – Atheist
  48. F.E.J Émile Borel – Unclear
  49. Julius Plücker – Unclear
  50. Hipparchus of Nicaea – Unclear
  51. Andrey Kolmogorov – Unclear
  52. Joseph Liouville – Unclear
  53. Eudoxus of Cnidus – Unclear (student of Plato)
  54. F. Gotthold Eisenstein – Christian (Protestant)
  55. Jacob Bernoulli – Christian (at least some ministry)
  56. Stefan Banach – Likely agnostic (possibly Catholic)
  57. Jacques Hadamard – Atheist
  58. Giuseppe Peano – Unclear
  59. Panini of Shalatula – Hindu
  60. André Weil – Agnostic (with heavy Hindu influence)
  61. Jean-Pierre Serre – Unclear
  62. Jakob Steiner – Unclear
  63. Marius Sophus Lie – Unclear
  64. Gottlob Frege – Christian (liberal Lutheran)
  65. François Viète – Christian (probably Protestant Huguenot)
  66. Christiaan Huygens – Christian (Protestant)
  67. M.E Camille Jordan – Unclear
  68. Joseph Fourier – Christian (Likely Catholic, considered becoming a monk)
  69. Bonaventura Cavalieri – Christian (Catholic, Jesuate monk)
  70. Hermann Grassman – Unclear (but I think Christian)
  71. Albert Einstein – Agnostic (maybe pantheist)
  72. James Clerk Maxwell – Christian (Evangelical or Presbyterian)
  73. Girolamo Cardano – Christian (Likely Catholic)
  74. Aristotle – Theist
  75. Galileo Galilei – Christian (Catholic)
  76. Michael Atiyah – Unclear
  77. Atle Selberg – Unclear
  78. Alfred Tarski – Atheist (yet also formally a member of the Catholic church)
  79. Gaspard Monge – Atheist
  80. L.E.J Brouwer – Unclear (I’d guess atheist, but am not sure)
  81. Liu Hui – Likely Taoist or Buddhist
  82. Alan Turing – Likely Theist (maybe atheist – a bit hard to tell)
  83. Jean-Victor Poncelet – Unclear
  84. John Littlewood – Unclear
  85. Shiing-Shen Chern – Unclear
  86. James Sylvester – Jewish
  87. Henri Lebesgue – Unclear
  88. Johann Bernoulli – Likely Christian (Protestant)
  89. Ernst Kummer – Unclear
  90. Johann Lambert – Christian (Likely Protestant)
  91. George Pólya – Agnostic
  92. Felix Hausdorff – Likely Atheist
  93. George Birkhoff – Unclear
  94. Pafnuti Chebyshev – Unclear
  95. Adrien Legendre – Unclear
  96. John Nash – Atheist
  97. Siméon-Denis Poisson – Unclear
  98. John Wallis – Christian (Clergyman)
  99. Omar al-Khayyam – Agnostic/Atheist
  100. Thales of Miletus – Likely Theist
  101. Hermann Minkowski – Jewish
  102. Simon Stevin – Likely Christian (Calvinist)
  103. Nicolai Lobachevsky – Atheist
  104. Andrei Markov – Atheist
  105. Daniel Bernoulli – Christian (Protestant)
  106. Mikhail Gromov – Unclear
  107. Paul Cohen – Likely Jewish
  108. John Milnor – Unclear
  109. Robert Langlands – Unclear
  110. John Conway – Likely Agnostic (Maybe Christian?)
  111. Pierre Deligne – Unclear
  112. William Thurston – Unclear
  113. Edward Witten – Likely Jewish
  114. Saharon Shelah – Likely Jewish
  115. Terrence Tao – Unclear
  116. John Thompson – Unclear
  117. Simon Donaldson – Unclear
  118. Vladimir Arnold – Unclear
  119. Stephen Smale – Unclear
  120. Timothy Gowers – Unclear
  121. Pappus of Alexandria – Unclear
  122. Sofia Kovalevskaya – Likely Atheist
  123. Leopold Kronecker – Christian (late-life convert from Judaism)
  124. Thabit ibn Qurra – Muslim
  125. Siméon Denis Poisson – Unclear
  126. Paul Erdős – Agnostic (maybe atheist)
  127. Jean Gaston Darboux – Unclear
  128. Nasir al-Din al-Tusi – Muslim (Shia)
  129. Ferdinand Georg Frobenius – Likely Christian (Protestant)
  130. George Boole – Christian (Unitarian)
  131. Hippocrates of Chios – Unclear
  132. James Gregory – Likely Christian (Protestant)
  133. John Napier – Likely Christian (Protestant)
  134. Norbert Wiener – Likely Jewish
  135. Lennart Carleson – Unclear
  136. Emil Artin – Unclear
  137. Ptolemy of Alexandria – Unclear
  138. Tullio Levi-Civita – Likely Jewish
  139. J. Müller ‘Regimontanus’ – Christian (Catholic)
  140. Abu Rayhan al-Biruni – Muslim
  141. Girald Desargues – Likely Christian (Catholic)
  142. John Tate – Unclear
  143. Alfred Clebsch – Unclear
  144. Oliver Heaviside – Christian (non-practicing Unitarian)
  145. Alexis Clairaut – Unclear
  146. Oswald Veblin – Unclear (maybe Christian, Lutheran)
  147. Colin Maclaurin – Christian
  148. Qin Jiushao – Unclear
  149. Henri Cartan – Unclear
  150. Henry J.S. Smith – Unclear
  151. Lars Valerian Ahlfors – Unclear
  152. Rafael Bombelli – Christian
  153. Michel Chasles – Likely Christian (Catholic)
  154. William Clifford – Likely Theist (?)
  155. Samuel Eilenberg – Likely Jewish
  156. Maurice René Fréchet – Unclear
  157. Lars Hörmander – Unclear
  158. Kunihiko Kodaira – Unclear
  159. Edmund Landau – Likely Jewish
  160. Peter Lax – Likely Jewish
  161. Leonardo da Vinci – Christian (Catholic)
  162. Augustus Möbius – Unclear
  163. Nicole Oresme – Christian (Catholic, Bishop of Lisieux)
  164. Roger Penrose – Agnostic
  165. Grigori Perelman – Unclear
  166. Plato of Athens – Theist (or at least close)
  167. Lev Pontryagin – Unclear (maybe atheist)
  168. Waclaw Sierpinski – Unclear
  169. Yakov Sinai – Likely Jewish
  170. Isadore Singer – Unclear
  171. Thoralf Skolem – Unclear
  172. Evangelista Torricelli – Christian (Catholic)
  173. S.G. Vito Volterra – Likely Jewish
  174. Zhu Shiejie – Likely Shamanism
  175. Jamshid Al-Kashi – Muslim
  176. Eugenio Beltrami – Unclear
  177. Bernard Bolzano – Catholic (priest)
  178. Raol Bott – Likely Catholic
  179. Luigi Cremona – Unclear
  180. Max Dehn – Likely Jewish
  181. Paul Dirac – Atheist
  182. Eratosthenes of Cyrene – Unclear
  183. Gerd Faltings – Unclear
  184. Michael Freedman – Unclear
  185. Marie-Sophie Germain – Unclear
  186. Vaughn F.R. Jones – Unclear
  187. Kazimierz Kuratowski – Unclear
  188. Jean Leray – Unclear
  189. Benoit Mandelbrot – Jewish
  190. Yuri Matiyasevich – Unclear
  191. Eliakim Moore – Likely Christian (Methodist)
  192. David Mumford – Unclear
  193. G. Personne de Roberval – Unclear
  194. Claude Shannon – Atheist
  195. Goro Shimura – Unclear
  196. Endre Szemerédi – Unclear
  197. Brook Taylor – Unclear
  198. Jacques Tits – Unclear
  199. Karen Uhlenbeck – Unclear
  200. Oscar Zariski – Atheist

Given this list, it would be interesting in light of the question of a “science/religion conflict” what the overall statistic of this group are. If, say, Christianity is anti-science at its core, we should expect to see very few Christians on this list. The totals will only be accumulated for those figures I feel confident that I have reliable information on.

Statistics on Beliefs

Total Counted: 116 (Uncounted: 84)

Theist / Atheist + Agnostic / Other: 79 / 26 / 11

Christian: 49

Atheist/Agnostic: 26

Jewish (religiously): 15

Uncommitted Theist/Deist: 9

Muslim: 6

Hindu: 5

Other: 6

Notes for the Reader

  • As in all of my database articles, this article may be updated from time to time. I will update this database both in accordance with new information that I learn as well as with any information that seems reliable provided to me by any of my readers.

Series: Databases of Resources

I am publishing this post as a preface to a sequence of posts that I will continue to update as long as I run this blog, and hope to continue updating for as long as I live – compilations of references on what I have studied during my life. I’ve decided to call these databases of resources.

Why do this? Well, whether my reader is Christian, Buddhist, Muslim, non-religious, or whatever else, I want to be intellectually responsible. If people have questions about what I believe, I want to provide responsible answers. As I see it, there are at least two things that I must do in order to do that well.

First, I should do my best to explain my reasoning in detail in important matters. Some of what I write on this blog is not quite like that. Sometimes, what I write is related to emotional issues in such a way that although I want to write clearly, I do not feel a need to address those topics in an academic writing style. However, many of the fundamentals of Christianity are not at their core emotional (though they have important emotional implications) but are based on historical and/or philosophical realities – both of which are based on objective truth rather than human perception and emotion. In objective matters, one ought to be careful to verify what you are saying. I may get things wrong from time to time, but if I make public my reasons for what I believe, then it will be easier for me to figure out where I am wrong. And if I am wrong or unclear, I can be corrected. This I welcome, that way I can correct my beliefs if I am mistaken or become more sophisticated and nuanced in my beliefs if I am merely unclear.

Secondly, I feel I should do my best to provide a way for people to double-check what I say. These databases hopefully will play that role. These are largely for people who want to learn more about a topic themselves, people who want to do their own research and come to their own conclusions. I encourage that kind of critical thinking. I believe strongly that Jesus would join me in this encouragement. I believe that the central claims of Christianity – which are sometimes called Mere Christianity in reference to CS Lewis’ foundational book by that name – are objectively true. Since objective truths that impact all of humanity must be based on some kind of evidence “out there” accessible to all of us, it is only fair that I try to provide sound reasoning based on principles that others can see – of which some examples are science, historical studies, philosophy, and ethics.

Databases may include posts like lists of useful resources, good books and authors, academic resources, or passages of Scripture that are relevant to a certain issue. I hope that as I continue to compile these kinds of database posts, they can serve as useful references for people who are interested in learning about areas that I am also interested in.

Explaining Calculus (Intro)

In our day and age, this word strikes fear into many hearts. Much like ‘rocket science,’ it is used in our culture to represent anything that is extremely difficult or even something incomprehensible. I am talking about calculus. Those who have taken a course in this topic will have a good idea of what kinds of problem calculus can answer – although in my experience those who have not taken a course in calculus don’t really know what it means. Or perhaps they do know what it is about, but are too intimidated by the word ‘calculus’ to trust themselves. Or – and I suspect most people are in this position – perhaps they actually do understand the core concepts behind calculus, but just don’t realize that they do.

Because I am convinced that most people understand at least parts of all the basic concepts that go into calculus, I am convinced that it shouldn’t be so scary. In fairness, mastering all of the computational details can be quite difficult. Just like mastering a sport, this requires a lot of practice and repetition. But, this doesn’t mean learning about what calculus is has to be difficult. Just like a person can understand the strategy and details of a sport without being a professional athlete, I also believe that we can all understand what is ‘going on’ with calculus without mastering the more difficult calculations.

I write this post at the beginning of the Fall 2020 academic semester. This semester, I am teaching a course called “Survey of Calculus 1” – which is the first calculus course a student takes. To be clear, this isn’t some sort of ‘easy’ version of calculus. The fact of the matter is that there is a natural way to split up the subject of calculus into two segments. For now, let’s just call them (A) and (B). The two segments are very closely related. In fact, if you already know (A), you can use (A) to teach people (B). And if you already know (B), you can use (B) to teach people (A). Even more, (A) and (B) both arise out of a common origin using the concept of limits. For those who already know this, I am talking about differential calculus and integral calculus. If you don’t know what these are, that doesn’t matter. When we talk about “Calculus 1” courses, we basically mean differential calculus, and “Calculus 2” is integral calculus (and a related third topic called infinite series). So, I will be teaching a semester-long course focusing on differential calculus.

Ever since taking my first course in calculus, I have found it fascinating and have been saddened by the fear others experience towards calculus. I don’t think we should have that fear. Because I have this conviction, I want to write about the course as I teach it. My goal right now is to write about this topic using the same line of thoughts and topics that I’ll be using in my actual course

I know that I have some people reading this who know calculus already, and some who know a little bit about it, and some who have no idea what it is, and I plan to write this series in such a way that I hope it will be interesting and informative for anyone. In order to do this, I will try to place emphasis on both the intuitive framework that gives calculus its intrigue as well as some of the mathematical details and how these details work to tell us not just about the world of numbers, but about the world in which we live every day.

This is my goal. I will be doing lots of writing on this subject, and hopefully it will be interesting!

Critical Thinking Toolkit: A Priori Assumptions

This is one of the most important – perhaps the most important – of the many tools in the “critical thinking toolkit.” I don’t say this because I like this topic most among the topics I want to write about – although I do enjoy this topic a lot. The main reason I think this is so important is because it is always relevant to all discussion involving two people who disagree, and I see this issue underlying almost all of “public ideological battles” today. Perhaps I am exaggerating slight here… but only slightly. The topic I want to discuss here rears its head in pretty much every discussion.

There is a need to clarify some terminology, because the terminology that is normally used here is actually Latin. The term is a priori. I think in order to most clearly define this terminology, it is helpful to introduce the terminology that is usually used as its opposite – a posteriori. For readers who don’t speak Latin (which includes myself), in order to compare the two, the phrases a priori and a posteriori can be translated basically as from the earlier and from the later. In order to understand what we mean, we must answer “Earlier or later than what”?

The answer is this – earlier or later than experience/observation. In light of this, what I mean by a priori means ‘before experience’ and a posteriori means ‘prior to experience.’ To understand what I mean, let’s give an example. Consider the statement, “Because I am 10 years old, I am more than 5 years old.” Now, numbers can be defined without reference to our personal experience of the world, so you can affirm the previous statement without referring to your own experience of the world. It is true that you can experience that 10 is more than 5, but you don’t have to experience it in order to know that it is true. This kind of situation is what is meant by a priori.

On the other hand, consider the statement “I am 10 years old.” In order to know whether this is true or not, you need some experience. You need to know, for example, who is speaking. You also need to know when they were born. Those are not aspects of reality you can understand without drawing from experiential reality. This is what is meant by a posteriori.

This is the sort of thing that is mean by a priori and a posteriori. More specifically, if you are in a debate with a person, an a priori assumption is something that you hold to be true that you hold prior to investigating the evidence from your investigation.

Common A Priori Assumptions

Below, we give some examples of commonly held a priori assumptions that can easily get in the way of having productive, intellectual conversations. I will try to point out some of the flaws involved, and later I will discuss how we can do our part to address these flaws.

Political Alignments

One example that comes to mind in particular is in the abortion debate. To be absolutely clear, I am doing my best to state what I consider to be assumptions – meaning I have rarely or never heard anyone of that position actually say these things out loud. As far as I can tell, most pro-choice supporters carry the subconscious assumption that the only reason for opposition to abortion is sexism of some form (namely, discrimination with respect to reproductive rights). In other words, “My Body, My Choice“. And as far as I can tell, many pro-lifers carry the assumption that everyone consciously believes/knows that everyone agrees that a fetus is a human being with the same rights as all other human beings. In others words, “Abortion is Murder“.

I am not here taking a side on the abortion issue. My point is entirely separate from this. If you look at mainstream dialogue, as far as I can tell, the argument rarely gets down to the important points that lie at the core. If I spent longer thinking about this, I could probably come up with of other neglected but important points of discussion within the abortion debate, and I can come up with similar underlying points within other political debates. For an example of such a position, I believe that the entire gay marriage debate basically boils down to what the word marriage means and what legitimate role the government has within that institution. As far as I can tell, it has almost nothing to do with homophobia. It just doesn’t. You could hate gay people but still support gay marriage, and you could affirm loving homosexual relationships and reject gay marriage. It all depends on what you think these words mean.

My only point here is that in politics, we often don’t every get around to discussing our real differences. We get caught up in catchy and clever-sounding sound-bite comebacks. But every time I’ve thought about a “sound-bite” from either liberals or conservatives, I have found it incredibly lacking in content. We need to do better in political discourse.

Scientific Atheism/Methodological Naturalism

There is another public debate in which I see a priori assumptions playing a significant role, and this is the debate between science and religion. Both sides tend to assume, a priori, that their own perspective is superior to others in obtaining truth about reality. That is, you have many religious people that, although both science and religion can hold truths, say that truths taken from the Bible automatically override truths from science. And many atheists do the opposite – they assume that anything science says automatically overrides anything outside of science.

But, how do we know which of these assumptions is true? And is it possible that science and religion are actually entirely compatible? I believe they are compatible, but that isn’t the point I’m trying to make here, so I will leave that for another time.

People who are very religious (let’s say Christians for the sake of discussion, although a similar idea holds for other religions) may subconsciously assume a doctrine of inerrancy and hold their atheist interlocutor to that standard. Similarly, an atheist may implicitly assume methodological naturalism (a philosophical doctrine that only non-supernatural entities can be posited as explanations) and expect a religious person to also hold to that. But these are fundamentally incompatible – so of course people who hold to these two, respectively, might not agree on something. The real question is, which is a better assumption – methodological naturalism or inerrant Scriptures? That is a question that debates about evolution and the book of Genesis cannot answer – the discussion has to be philosophical

What To Do With Assumptions When Others Disagree?

Very often, our disagreements with people boil down to our assumptions. Therefore, in order to be effective in convincing a person of your beliefs, your approach should take into account the underlying assumptions of both yourself and others.

Figure Out What the Assumptions Are

The first thing we must do is to boil down any debate to its presuppositions is to ask lots of questions. In particular, the beginning of any discussion should consist in lots of “what” questions. These help to clarify the topic of the discussion. After that, we can ask “why” questions in order to boil down the discussion to its foundational pieces. The theme of what/why questions is almost always helpful in any discussion, and is always a great thing to keep in mind.

How will we know when we are at the assumption level? One way to know would be if the “why” question no longer has a helpful answer. For instance, asking a person how they know they exist wouldn’t really have a helpful answer. Alternatively, the level of ‘assumption’ can be located by starting from the obvious claims – things like “we all exist” – and moving gradually upward in complexity/controversy until a point of disagreement arises.

We can view the abortion debate as an example. Although there are multiple ways to approach this debate, and I do not intend here to defend one side or another, I think that since people are fairly familiar with this debate, it will serve as a helpful example. As a jumping-off point, here is a “bottom-up” structure of how a pro-life advocate might arrive at his or her pro-life position.

  • God exists.
  • God is the creator of humanity.
  • God values humanity.
  • God values all human beings individually.
  • All human beings have infinite moral value.
  • An unborn child is a human being. Therefore, an unborn child has infinite moral value.
  • Unjustified killing of a human being is evil. Therefore, killing an unborn child is evil.
  • Abortion kills an unborn child. Therefore, abortion is evil.

The point here is not to agree or disagree with any of these ideas. My point here is that you could disagree with this person’s pro-life position for a variety of reasons. If you don’t think God exists, then you will have a problem with a fair amount of the points involved. If you believe God exists but you are a “deist,” then you will disagree with “God values humanity,” since on deism God is distant and removed from the physical world. You might disagree on the proper way to instill moral value on a human life, or when a human life begins, and so only disagree with this person later in their train of thought. If two people are having a conversation and misunderstand where the disagreements actually lie, then it will be an unhelpful conversation. This is one of many reasons why identifying underlying assumptions is so helpful.

Now, we can discuss some options of how to carry forward in a discussion once our assumptions are identified.

Option 1: Convince them of the assumption: One way to handle a disagreement like this is to stop debating the ‘endpoint’ but to shift the conversation to the real disagreement – the underlying assumptions.

Option 2: Convince them of your point using their assumption: This method won’t always work, but sometimes it does. Sometimes, you might be able to convince a person that they have not carried out their assumptions to their logical conclusions. This is very similar to the mathematical and philosophical method of reductio ad absurdum.

Critical Thinking Toolkit: Ockham’s Razor

When we are in debates, very often there is more than one way to explain something. When we are presented with more than one way of explaining some aspect of reality – be it scientific, historical, religious, or anything else – these situations arise. When they do, we want to be able to differentiate between the various alternatives in a reasonable way. But it is not immediately obvious how to do something like this. If Steve and Carol provide two totally different explanations for the same event, how do we analyze which one of them is more likely right? After all, we cannot evaluate the alternatives based on their conclusions, since the conclusions are identical. What then do we do?

Ockham’s Razor to the Rescue

To give some kind of intuitive explanation of the principle of reasoning that I will define here, let’s use an example. Suppose that we are tasked with explaining why there are Christmas presents under a Christmas tree. There are two possible answers available to us. For us, Explanation 1 is that Santa Claus lives in the North Pole and delivers presents to all children throughout the world on the night before Christmas morning every year. Explanation 2 is that it is actually the parents of individual children that deliver the presents, and that the story of Santa Claus is a fictional tale meant to inspire fun and imagination in children. Notice that Explanations 1 and 2 both completely describe why there are Christmas presents under the tree, so we cannot evaluate between 1 and 2 based upon which one leads to the correct conclusion (of course we could bring in other information that shows us why Santa can’t exist as described here, but for the sake of argument we will pretend we don’t know any of that). Well then, how might we choose which of the two is more likely true?

This where the principle I want to talk about here comes in. It is not meant to be an absolutely foolproof method, but it is quite reliable. The method, called Ockham’s Razor (Ockham can also be spelled Occham or Ocham) is summarized by the statement “entities should not be multiplied without necessity.” To be more specific, Ockham’s Razor tells us that if you have different ways of explaining exactly the same thing, you ought to take the explanation that makes the fewest assumptions. In the previous example, we already know that our parents exist, so the assumptions involved in explaining Christmas presents via our parents are extremely few – the only assumption we have to add is that our parents are lying to us (if we neither believe nor disbelieve in Santa, that’s really all we need). In order to explain Christmas presents by Santa visiting our house, we have to add significantly more assumptions. Therefore, unless we have new evidence that points strongly one way or the other, Ockham’s Razor tells us to accept the alternative with fewer assumptions – namely, that Santa does not exist.

How to (and not to) Use Ockham’s Razor

Ockham’s Razor is a frequently used and very important tool in critical thinking. I must make that very clear – it is a great principle and we really must take advantage of this way of thinking about complicated issues. But, it is also very necessary to emphasize both the powers and limitations of Ockham’s Razor.

Ockham’s Razor is not All-Powerful: One extremely important reality we have to acknowledge is that this principle does not always work. Ockham’s Razor is meant to ‘shave off’ additional assumptions only if those additional assumptions are not helpful in other ways. Ockham’s Razor favors simple explanations over more complicated explanations, but only when the two complicated explanation doesn’t override the simple explanation in other ways. For example, Isaac Newton’s theory of gravity is much simpler than Albert Einstein’s theory of gravity (aka general relativity). Since the two theories analyze the same concept – namely gravity – they can be compared. If that were the only information we knew, we would have to prefer Newtonian gravity to general relativity. But scientists use general relativity today in the most important applications, not Newtonian gravity. Why? Because general relativity is more accurate than the Newtonian theory. The increase in accuracy is more important than the simplicity. So, if we want to use Ockham’s Razor, we should make sure that the two ideas in question are similar in other respects.

When Occham’s Razor Applies, It Almost Always Works: When you look throughout your own life or the history of any discipline of study in human history, I am quite confident that whenever you find a proper situation in which to apply Ockham’s Razor, it will work correctly. In other words, reality tends to favor simple explanations over complicated ones whenever a simple explanation is good enough to explain whatever is going on. Even though you can’t always use Ockham’s Razor, it is very valuable.

It is Almost Always Helpful, Even if it Doesn’t Work: Ockham’s Razor is not meant to be an all-or-nothing principle. But even when you can’t make a full-blown choice of your beliefs based on Ockham’s Razor, it is still helpful. Roughly, this is because simple explanations require fewer contributing factors than complicated ones, and so in most cases simple explanations have much higher probabilities than complicated ones. Although Ockham’s Razor can be viewed as a way to choose between competing ideas that are equally good at explaining the world, it can be used in another way as well. Since simple explanations are favored over complicated ones, you can also apply Ockham’s Razor at the ‘beginning’. But if you apply it at the beginning, it isn’t conclusive. When investigating various ideas, each idea will have an “initial likelihood” in your mind – this is often called the prior probability. Ockham’s Razor has a place in evaluating these prior probabilities. For instance, the prior probability of Santa delivering presents on Christmas is much lower than the probability of everyone’s parents buying parents – but this is because, as adults, we know things that rule out Santa. But, suppose you are 3 or 4 years old. You don’t know enough about the world to decide whether Santa or your parents are better explanations of the presents underneath the tree. But, suppose your parents told you Santa brought the presents. Then since you have nothing else to go on, it is entirely reasonable to believe that Santa brings the presents, and so in this case the prior probability of Santa being the person who brings presents is quite high.

The thing to notice in this example is that the prior probability is always based upon what you already know. If you are 3 years old, the idea that Santa brings presents is the most simple, because all you have to do is believe your parents, whereas denying this requires devising why exactly your parents are lying to you – which is more complicated. But if you start off with the knowledge of an adult, then the entire situation flips on its head – because you know more information. To take this even further, if any of us were to actually live out a Christmas movie where Santa shows up, then probably the situation would change yet again to believing in Santa, because Santa’s actual existence is more straightforward than trying to explain how a bunch of reindeer were flying around. Although if you learn about special relativity theory, then the coin would probably flip again and the unlikelihood of flying reindeer might be balanced out by an outright violation of a law of physics.

Basically, Ockham’s Razor is a useful tool that states that simple ideas have an advantage over complicated ideas, so in order for a complicated idea to win it needs to gain an advantage somewhere else. There are plenty of other ways an idea can get an advantage. These will be discussed elsewhere.