MATHEMATICS is often thought to be universally and unassailably true. I have even heard it argued that God, omnipotent though He may be, could not make math false even if He was impulsive enough to try it. Can mathematicians actually prove that math is true? If they cannot, 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? These are some of the questions that I will try to address.

The first attempt we might make to prove that mathematics is true is to consider real world situations where mathematical 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.

• 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.

•If I had three dollars worth of goods yesterday and borrow two dollars worth of goods from you today then I have five dollars worth of goods in my possession.

Each of these three situations seem 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 ‘three balls’ or ‘three hours’ or ‘three dollars’, and the operator ‘+’ is not the same as grouping together balls or combining velocities or aggregating wealth. It is true that 3+2=5 is typically an excellent model for each of these situations, but nonetheless the equation is not precisely equivalent to these situations. It is also true that when we group balls together (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, or some other delicate substance. In this case, when we add new objects to our bag they will sometimes fracture and split into multiple objects, and occasionally multiple objects will even fuse together into a single object. The addition operator ‘+’ no longer models this situation well because when we place a new object in the bag it does not always increase the number of objects contained in the bag by one.

It is not terribly difficult to annihilate the relationship between the equation 3+2=5 and the other real world situations given above. For example, 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. What’s more, if I had three dollars worth of goods yesterday and then borrow 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 because 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 generally produces a bag with five balls, then we can conclude that 3+2=5 is false simply because velocities have been proven not to add. 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.

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 follows as 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. To begin with, what do we mean by “mathematical objects”, and 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. Unfortunately, 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), while others use Category Theory to provide the basic building blocks, and 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 worse (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” (whatever that means), 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. As a matter of fact, if we could prove that the axioms were true then they would be called “theorems” and not “axioms”!

As convoluted as this discussion has become, matters get still murkier. 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, but 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. These problems are compounded by the fact that we cannot prove that our commonly accepted mathematical axioms do not contradict each other, leading to a still deeper level on which to question the truth of mathematical statements. In the end, while it hardly seems fair to say that math is false, it also does not not seem fair to conclude that math is true. Math is probably neither “true” nor “false” in the usual sense of those words, though it does undeniably provide 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.). Hence, mathematical definitions were chosen by humans to model physical reality so that we could make useful predictions, not to encapsulate metaphysical truth, so really, why should we expect math to be true?

nice! thanks for this essay…

I agree – I think of mathematics as a language… we would never say “Spanish” is true or false… it is a system people use to describe and model their world…

I’m not sure. It seems mathematical statements, like 2+3=5, are among the closest things to truth we have, beyond bare logical statements like -[A+(-A)]. I’m surprised at your suggestion that the statement “massive objects exert forces on other massive objects” has a better claim to truth than 2+3=5. (Actually, I see you are careful not to say that.) Surely 2+3=5 is thought to be more necessarily true than “massive objects exert forces on other massive objects”.

If mathematical statements aren’t true then it is difficult to see what statements could count as true. No?

John:

That would be correct. Plug in someone like Humes radical skepticism into this and you really don’t end up with a lot left.

Thanks for posting this — In discussions of philosophy, I’ve noticed an unsettling tendency on the part of some people to view mathematics with an almost mystical reverence and then use it to prop up whatever other metaphysical ideas they’re defending. It’s refreshing to see someone talking and thinking clearly about it.

hey, bout your first example on the balls and economy and physics, i disagree that 2+3=5 is wrong, instead i think the values are wrong, for example, in the case of physics, speed plus speed in that case does not give you total speed of the ball, therefore, why does one even use 2+3=5 ? instead of using 2+3=5, one should use a formula that is accurate to find the total speed.

one more point, is your claiming that math has less truth than most claim, than how can we be sure that the theory of relativity is true? since they are indeed caculated by math formulae?

I didn’t intend to imply that “2+3=5” is wrong simply because it doesn’t match certain physical examples, merely that it can’t be proven true merely by referring to physical examples. As I say in the post:

“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 generally produces a bag with five balls, then we can conclude that 3+2=5 is false simply because velocities have been proven not to add. 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.”

As to your second comment, there is a difference between math being “true” and math modeling a particular situation. The equations of relativity model aspects of physical reality, and math is the language in which we make this description. That does not imply that math itself as a system is “true”, merely that we can use it to describe aspects of reality. Of course, we can also use it to describe things that have no analog in physical reality.

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Clockbackward,

You say

“But now suppose that the objects we are grouping together are made of packed sand, or some other delicate substance. In this case, when we add new objects to our bag they will sometimes fracture and split into multiple objects, and occasionally multiple objects will even fuse together into a single object.”

While your argument might at face value make sense, if one were to reason, we could conclude that by simply giving a precise definition, for example, that “1” = a unit of packed sand” at time ‘t’. Where “unit” means for example, 2,300 grains of sand. Then we could conclude that 2 units plus 3 units = 5 units. The laws of mathematics are clear, easy to know, and irrefutable.

Please respond.

Yes, that is true, but it’s just another example where addition gives us a good model for something. There are still many examples where addition models the world poorly. The point here is that you can’t use the fact that addition models certain situation well to prove that 1+1=2 is true, just as you can’t use the fact that it doesn’t model other things well to prove that 1+1=2 is false. The fact that addition represents a good model for something does not imply that addition is “true”.

I started reading your essay with great interest as I was hoping you could prove your point. Unfortunately you lost me after trying to prove 3+2=5 not always being true.

In the first example you are claiming that the objects added might be fractioned and hence the formula is wrong. In reality at the moment of applying the “+” the “=” will hold true. In that point of time you have 2 whole units moved together with 3 other whole units and you end up with 5 whole units. After that point in time if the objects react to it and become distorted in any way that is a separate event and requires a sperate math formula that reflects the event.

In your second example you are talking about relative speed although the formula you apply does not reflect the relativity of time. If the formula is not applied and we assume time is constant your equation would hold true as the distance traveled will be constant and since time on the train would be the same as time on ground you will have the same units.

The third example is very similar to the first. You provide mathematical equation that dscribes an event and then a separate event occures that changes the value of goods, but yet you do not provide math to reflect this event. Here is an example similar to yours: There are 3 people in a room and 2 more enter in the room at 9pm. How many people are in the room at 10pm? According to mathematics it should be 5. That is wrong there are 4 people because 1 already exited at half past 9, so math is wrong…?

Math is only concerned with numbers which are abstract in their nature. To apply math to practical situations you need units of measurment which are going to be assigned numerical values and those values should adhare to all laws of mathematics. The units themselves do not, they are products of different siences which have their own formulae to convert units like fractions of breakble objcts, time relativity and price depreciacion. Mathematics should be fundamentally true as the units it uses are defined by the laws of mathematics itself and nothing else effects them.

I believe that you misunderstood the purpose of those examples. They were not intended to prove that “3+2=5 is not always true”. They were simply intended to show that 3+2=5 is a model that sometimes works (provides an accurate model) and sometimes doesn’t work (because it models things poorly). Furthermore, the examples were intended to illustrate that we cannot prove that math IS true simply by finding examples of things that it models well.

Hello! My name is Kevin, I would like to tell you that if I have 2 icecreams, I will be happy to share one with you if you are my friend. My dad is JQ and he is born is in the India in the USA in the China, I think in the Australia too! I like Maths a lot because I am happy to know numbers

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you are wrong

I very much appreciate your statement “Clearly ’3′, ‘+’ , ’2′, ‘=’ and ’5′ are not objects in the physical universe.” I am still pondering about it.

The origin of numbers is zero….without numbers, you have no math…no calculations. The premise of math originates from nothingness (zero) so it seems like it might not be true? Look at the number line and its origin. How can you have ‘nothing’ and ‘something’ at the same time. Without zero you can not have 1. Perhaps it is just a point of reference in order to understand the concept of 1? Doesn’t math then originate from ‘nothingness…..being the number zero? I know, I’m high!

If one keeps definitions consistent, 3+2=5 will always be true. In mathematics, the statement 3+2=5 is for the situation where the units are kept the same. Just look, here are three ls: lll. Now here are two: ll. Put them together and you get this: lllll. How many ls is that? This is definitely true according to the definition of “2,” “5,” “+,” and “=.” There are no unprovable axioms used here. Furthermore, axioms are self-evidently true, that is, it would possible for those axioms to exist if they weren’t true.

Look here: ll – two ls. lll – three ls. Put them together. How many ls is that? lllll – five ls. 3+2=5 is definitely true because of the definition of “3,” “+,” “2,” “=,” and “5.” I appreciate that you mentioned that, but then you go on to confuse yourself. It’s as if you are doing mental gymnastics to avoid an obvious conclusion. The closest thing I’ve seen to it is how politicians behave. Mathematicians also hold the units constant when issuing the statement; in fact that is part of the definition of the statement. Finally, axioms are self-evidently true because they could not possibly exist if they were not true. The very definition of “3” is three ones, that is 1+1+1. What do we do when we put sticks next to each other? We pick up one stick at a time, and then put them together.

I want to say 2 things:

– The concepts of “true” and “false” are two mathematical concepts.

– More broadly the concepts of “logic”, “reason”, even “proof” are mathematically-infected, if I may say so.

If you want to prove that math is false you already lost.

This is just the wrong question.

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The reason why i beleive maths are real and independent from the physical world is that we can reach conclusions (theorems) which cannot refer to the physical world. Forma example the euler formula, or the fundamental theorem of arithmetics. Physics and maths have in common that both are based on axiomatised logical systems. And again, formal logic remain abstract and thus separate from the physical realm.

George says: mathematical “axioms are self evidently true.” Here is the entire breakdown of the word “true”. It’s not the math we should be questioning, but logic and reasoning. What does this concept of “self evidident” mean. Self evident simply means because we can’t see it any other way, this must be the only answer. This is way the human brain is wired. The Funny thing is that human brains are also wired for time, yet Einstein tells us that our brains are wrong and we only think time is self evident when it is not. How is a self-evident axiom any different than our ancestors believing that the future must follow the present. Or, them believing that and object (particle) can only be in one place at one time. I can help thinking of the Old Testament Bible verse where God says: “…as the heavens are higher than the earth so are my thoughts higher than your thoughts…” Perhaps are brains are not capable of understanding “truth” but we can use math to approximate it.

yeah, really why expect math to be true? first you need to define what does it mean math to be true

I really appreciate your explanation. I am a 17 year old boy. One day, when i was studying Electromagnetism , I wondered why all the formulas that were developed hold true and we can predict what could possibly happen if we change the condition and i took it as a serious issue for myself. From that day i can’t stop thinking about it. I researched it over internet. Sometime I thought that i have reached to my conclusion but i couldn’t feel that to be true. I think that like we have developed a way of talking to ourselves by creation of alphabets and then joining them to make words this also didnt exist in our world before. Mathematics is also an another stuff that was created and developed in the same way as alphabets. What if human didnt developed ‘3’ there could be possibility that while counting thing he would make three straight line which also signify the same.

We can know certain axioms are true via their proofs.

For example, take Euclid’s axioms. The Pythagorean Theorem is a theorem that can be deducted back into Euclid’s axioms. Also, The Pythagorean Theorem has more proofs than any other theorem. (Just Google “Pythagorean Theorem proofs” and you’ll find more than a fair share. Even Einstein himself created a proof for it! Neat stuff).

Anyways, how we know what is true is based on the basic line of axioms -> proofs -> theorems. An axiom is non-deductive. It cannot be reduced further. A point. A line, etc. These are fundamental, self-evident axioms. The Pythagorean Theorem can be built from Euclid’s axioms, and therefore deduced back into Euclid’s axioms. Plato himself showed how the Pythagorean Theorem is in of itself self-evident, using the Socratic Method on Meno’s Slave. (Just Google “Meno’s Slave” to see it in action.) Thus, the Pythagorean Theorem is itself, self-evident, built from self-evident axioms, and the methods used to prove the Pythagorean theorem, are used as the fundamental “language” to prove theorems everywhere.

So, if you’ve stayed with me after all of this, we can thus know what is true, by the very nature of Euclid’s axioms, and building all we know up from that. All theorems, that are true, can (should) be able to be deduced all the way back down into Euclid’s axioms, or proofs of Pythagoras.