A. our theoretical understanding of what things are “really” like (physics, atoms, etc.)

with

B. our direct, first-hand experiences as humans.

Examples:

(1) Ethics – most people experience a visceral feeling that some things are inherently and universally morally wrong (e.g., murdering children). Yet it’s unclear what, in the universe of atoms (or in physics), could make (or explain) something being “wrong.”

(2) Free will – we feel as though we constantly make choices (e.g., picking options that we didn’t have to pick). Yet the possibility of choices is hard to square with the existence of laws of physics as we know them. Where could a choice possibly fit into those laws?

(3) Consciousness – we each know we are conscious (in the sense of having experiences / there being something it is like to be us) because we directly witness our own experiences. Yet it’s unclear how or why configurations of atoms could ever give rise to internal experiences.

(4) Identity – we feel like we have a unique, persistent, indivisible identity. Yet, if we imagine thought experiments involving splitting, copying, or rebuilding brains in the physical world, it’s hard to see how a unitary identity could be maintained in those circumstances.

(5) Knowledge – there seem to be many things we each intuitively know to be true (our own names, what orange juice tastes like, how to tie our shoelaces), yet it’s hard to explain what the state of “knowing” these things corresponds to in the world, or to define what “knowing” is.

(6) Mathematics – we all know it’s true that 1+1 = 2 and that the number 2 “exists” in some sense. But it’s hard to say in what sense this is true/existent because numbers and addition don’t seem to exist in the physical realm the way that, say, a particular sandwich does.

This piece was first written on December 23, 2020, and first appeared on this site on April 17, 2023.

The Setup

Person X and person Y are friends (or lovers or close work colleagues or whatever). Person X and Person Y happen to both feel the same way about each other (i.e., equal amounts of interest, affection, lust, respect, etc.)
Person X’s ideal is to make plans with person Y every two weeks, whereas person Y (who has a lower amount of free time, or less need for social interaction, or a project they are prioritizing, or whatever) wants to see person X every three weeks. Hence, they differ in their preferred time interval between hangouts.

So what happens? Approximately every two weeks, person X asks person Y to spend time together, which means that person X ends up doing essentially 100% of the invites (since three weeks rarely elapse without Y receiving an invitation from X, so Y almost never asks X to spend time together).†

In other words, Y wants to see X at only a moderately different rate than X wants to see Y (e.g., every three weeks instead of every two weeks) but ends up doing 0% of the invitations.

Person X then assumes that their relationship is imbalanced, and person Y must not feel the same way about the relationship that they do (but happens to be wrong). This can lead to awkwardness and relationship problems.

So how should X handle a situation where X ends up doing all of the inviting without Y reciprocating (but with Y agreeing to see X whenever X does send an invite)? We’ll assume in each of these cases that the people actually do spend time together when an invite is made (i.e., it is not a case of one person purposely ignoring another).

Some would suggest that, if both people simply ask each other their preferences, that can work best for the right types of people. Especially if both are on board with such explicitness of conversations about relationships, know the other person is on board too, and are confident that negative ramifications such as damaging awkwardness won’t result from that explicitness. In most cultures, the explicitness of the form “I’d prefer to see you every three weeks, how often do you want to see me?” is not the norm but certainly is normal in some subcultures.

Always Ask Strategy

The default strategy would be for person X to just persist in making the invitations every two weeks. One drawback is that person X might feel bad about always being the one to make invitations. Another drawback is that person Y might feel frustrated because they end up having to stall regularly on those invites to get their desired rate of spending time together, or else agree to see X on X’s preferred schedule rather than their own. Yet another drawback is that person X might be misreading the signs: maybe person Y just feels bad about saying no and so agrees to see X despite not wanting to? Person X knows that person Y is not reciprocating the invitations by sending invites back to X but doesn’t know the reason Y isn’t reciprocating.

Tit-For-Tat Strategy

Another strategy would be for person X to never make two invitations in a row. That means that person X would make the first invite (after two weeks), and then person Y would make the next invite (after three weeks) and then person X the next and so on. This isn’t a terrible solution since they would see each other every 2.5 weeks, which is a nice compromise. However, it does have a very major drawback, which is that if person Y forgets to make an invite back, then they’ll be stuck not seeing each other. In other words, it’s leaves room for mistakes, and could inadvertently destroy a great relationship. Hence far from ideal! This could be expanded to a “Tit-For-Two-Tats” strategy, where X makes two invites in a row but not more than two. But this actually is not really more robust in this scenario than Tit-For-Tat, since after two invites from X (which will definitely happen in this scenario), if Y then forgets to make the next one, no more invitations will occur.

Exponential Strategy

A third strategy would be for person X to double their invite time interval each time their last invite does not get an invite in return, and then reset back to their original invite time as soon as they get an invite back. (It doesn’t have to be double, of course, it could be X multiplying the invite time by any constant C>1.)

To see what I mean in more detail, consider the situation where X still prefers to see Y every two weeks, but now Y prefers to see X every ten weeks. First, person X makes an invite after two weeks, and they see each other at that time. Then since Y doesn’t reciprocate, X multiplies their time by two and so makes an invite after four weeks. Since Y again doesn’t reciprocate, X multiplies their time by two again and makes an invite after eight weeks. Since person Y’s desired time to see each other is ten weeks, then Y will end up making the next invite (since X wouldn’t make their next invite for 16 weeks). Now since a reciprocation occurs, X resets and so sends the next invite in 2 weeks, then the invite after that in 4 weeks, then eight weeks, etc.

There are some neat things about this strategy. First of all, it’s robust to mistakes since even if person Y accidentally forgets to make a reciprocation, person X will still end up reaching back out with an invite. Second, it does a pretty good job of balancing the desires of both parties by finding an average meeting frequency that is a compromise of both their ideals. The math gets a bit complicated†† but, suffice it to say, for most values of K (person X’s ideal timing between invites), C (the multiplier that X applies to their invite time interval with each unreciprocated invite), and N (person Y’s ideal timing between invites), the average time delay of the two people seeing each other will be fairly close to halfway between X’s and Y’s ideal time delay (to be more precise, it’s usually in the range of 25% of the way from K to N up to about 75% of the way from K to N).

Another neat thing about this strategy is that in the event that X has misjudged this situation, and Y actually doesn’t want to spend time together, Y gets pestered with exponentially decreasing frequency, meaning that the total annoyance Y experiences and the total embarrassment from non-reciprocation that X experiences are both limited.

One final point about the exponential strategy is that it works well if both parties use it even in an environment where forgetting is common (i.e., if 30% of the time people get distracted and so forget to make an invite).

Takeaways

In reality, one would, of course, not do calculations like this formally, and this simple model doesn’t include all relevant factors. But perhaps the Exponential Strategy can give us a decent intuition for how we might handle these sorts of situations in real life. When we want to see someone at regular intervals, and the person does agree to see us when we make invites but doesn’t make invites of their own, increase our invite time interval by some constant factor (say, 2) each time we have a non-reciprocation, but reset back to our most desired invite time interval whenever a reciprocation occurs.

† Note that this math works out even if X has some amount of noise in when they ask Y to spend time together, but as this noise gets large, the math eventually breaks down.

†† The average delay D that occurs between invites when person X is using the Exponential Strategy will be:
D = (N + K * sum_{t=0}^{M-1} C^t ) / (1 + M) where M =Ceiling[Log[N/K]/Log[C]]

But can mathematicians actually prove that math is true? If they can’t, does the fact that math is so useful in solving real-world problems provide evidence of its truth? And, if mathematics is not true, then does that imply that conclusions drawn from it are faulty or suspect? Let’s explore those questions.

The first attempt we might take to prove that math is true is to consider real-world situations where equations seem to appear. Some examples are:

• If I have three red balls in a bag and add two more, the bag will then contain five red balls (3 balls in a bag with 2 balls added to the bag gives 5 balls).

• If I am on a train traveling at three miles per hour and throw a ball at two miles per hour (measured with respect to the train), then the ball will be traveling at five miles per hour with respect to the ground (3 mph sped up by 2 mph gives 5 mph).

• If I had three dollars worth of goods yesterday and borrowed two dollars worth of goods from you today, then I have five dollars worth of goods in my possession (3 dollars of goods with an additional 2 dollars of goods borrowed yields 5 dollars of goods).

Each of these three situations seems to imply the equation 3+2=5. But do they actually PROVE that the equation 3+2=5 is true?

One problem with drawing conclusions about mathematics from these examples is that the number ‘3’ is not the same as ‘3 balls’ or ‘3 hours’ or ‘3 dollars’, and the operator ‘+’ is not the same as grouping balls or combining velocities or aggregating wealth.

While 3+2=5 is typically an excellent model for each of these situations, the equation is not precisely equivalent to these situations. Why not? Well, it’s true that when we group balls (by, in this case, placing them in a bag), the procedure generally behaves as though we are performing addition. But now suppose that the objects we are grouping together are made of packed sand. In this case, when we add new objects to our bag, they will sometimes fracture and split into multiple objects. Or if the balls are made of wet clay, they may fuse into a single object in the bag. The addition operator ‘+’ no longer models this situation well because when we place two new objects in the bag, it does not always increase the number of objects contained in the bag by two. So addition is not a perfect model for grouping physical objects in a confined space.

What about the other examples? Einstein’s theory of relativity tells us (in contradiction to the more intuitive but less accurate equations of Newtonian mechanics) that when a person on a train (which is moving three miles per hour with respect to the ground) throws a ball at two miles per hour (with respect to the train), then the speed of the ball with respect to the ground is actually very slightly less than 5 miles per hour, not equal to 5 miles per hour. So while addition is very accurate for modeling that situation, it’s known that it does not, in fact, give the exact correct answer.

What about the last example? If I had three dollars worth of goods yesterday and then borrowed two dollars worth of goods from you today, the total number of dollars worth of goods that I have possession of will not necessarily be five dollars if the value of my original goods changed between yesterday and today (as can happen in real economic markets).

What these examples show us is that the only reason to say that grouping balls or combining velocities or aggregating wealth encapsulates the idea of mathematical addition is that most of the time, the addition operator ‘+’ provides a good MODEL for these scenarios. We can no more conclude that 3+2=5 is a true statement simply because putting two balls into a bag that already has three balls usually produces a bag with five balls, then we can conclude that 3+2=5 is false, simply because if you’re using balls of packed sand, sometimes the balls will fracture into more balls when you place them in the bag. In other words, while real-world situations can motivate the equations of mathematics and provide justifications for applying them, they cannot prove that those equations are actually true.

We have stared at equations like 3+2=5 so many times in our lives that it can be difficult to consider them with fresh eyes in order to ask ourselves what it really is that they are saying. Clearly, ‘3’, ‘+’, ‘2’, ‘=’, and ‘5’ are not objects in the physical universe. You can go to the zoo and see three bears, or see the numeral ‘3’ printed on a sign, or perform arithmetic on paper using the symbol ‘3’, but nowhere in the universe can you find the actual (metaphysical) number ‘3’. This is hardly surprising, since ‘3’ is a concept or idea, not a physical thing. But this line of thought implies that 3+2=5 is a statement about the relationship among the concepts ‘3’, ‘2’, and ‘5’, and not a statement about physical entities that actually exist. The only time that 3+2=5 is a statement about physical things that actually exist is when we use it as a model for real-world properties that are sufficiently similar to the concepts for the model to be useful.

But how do we define the word “true” when it comes to relations among abstract concepts? One possible approach is to say that statements about abstract concepts are true if they follow as a logical consequence of the definitions of the concepts themselves.

This leads us to ask whether 3+2=5 and all other mathematical statements are simply true by definition as a consequence of our chosen definitions for ‘3’, ‘+’, ‘2’, ‘=’, ‘5’, and the other mathematical objects.

Unfortunately, this question cannot be answered without further qualification. How do we choose to define concepts such as ‘3’? Various authors have attempted to define mathematics by developing lists of axioms (which are simply assumed to be true) and then proving that the basic mathematical objects (e.g., integers) and theorems (e.g., a+b = b+a) follow from these axioms. There are a variety of different ways that math can be axiomatized (i.e., built up from basic axioms). Some approaches use sets as the most basic objects (as is done in what is probably the most popular axiomatization, Zermelo-Fraenkel set theory).

In contrast, others use Category Theory to provide the basic building blocks. Still, other theories attempt to axiomatize only small portions of math, such as Euclid’s Axioms of planar geometry, Hilbert’s axiomatization of Euclidean Geometry, and the Peano axioms for arithmetic.

What is even trickier (when it comes to deciding what is true) than having so many conflicting viewpoints for constructing math is that the axioms of these viewpoints are themselves not provably true. If you are willing to assume the axioms of math are “true”, then all of the resulting theorems that can be derived from those axioms are also true, but the axioms themselves must simply be accepted without proof in order for this process to work. If we could prove that the axioms were true, then they would be called “theorems” and not “axioms”!

Even those mathematicians who agree to rely on a single basic axiomatization (such as Zermelo-Fraenkel set theory) sometimes cannot agree on whether certain extra axioms (such as the continuum hypothesis, which concerns itself with the existence of sets of certain infinite sizes, or the axiom of choice which pertains to being able to select one element from each element of a set of sets) should be added or left out. And to top that off, mathematics (as defined by whichever axiomatization you like) has not even been proven to be consistent, meaning that no one has been able to mathematically demonstrate that the axioms of any single axiomatization do not contradict each other. In fact, Gödel’s 2nd incompleteness theorem shows that if mathematics is in fact consistent, then it will not be possible to use math to prove that no inconsistencies exist!

In conclusion, numbers and other mathematical objects are simply concepts, and not things that are actually observable in the universe, so we cannot say that statements like 3+2=5 are true in the same way that we can say that the statement “massive objects exert forces on other massive objects” is true. We might like to think that mathematical statements are true by definition. Still, this idea is complicated by the fact that there is more than one way to axiomatize mathematics, and therefore more than one definition that we might choose in order to define numbers, operators, and other mathematical objects. But even if there were truly only one way to axiomatize math, the axioms themselves would still not be provably true (they would only be assumed to be true), and hence it would hardly seem fair to then conclude that mathematical theorems are “true” in some objective and universal sense.

In the end, while it hardly seems fair to say that math is “false”, it also does not seem fair to conclude that math is “true” in the usual sense of the word. It’s true, conditional on the axiom that we choose to accept, and only insofar as it is talking about the concepts that it defines (rather than the physical world).

Of course, math undeniably provides extraordinarily useful models for making predictions about what will happen in our physical universe. This will perhaps seem less surprising if we remember that mathematics was not originally developed from the ground up using axioms, but rather piece by piece in order to find solutions to problems that appear in the real world (like those related to calculating the size of plots of land, counting money, measuring roads, tracking the movements of the stars, understanding heat flow in cannons, etc.). Humans chose mathematical definitions to model physical reality so that we could make useful predictions, not to encapsulate metaphysical truth, so should we have expected math to be “true” rather than merely (very) useful?

This piece was first written on January 18, 2009, and first appeared on my website on March 3, 2026.