Explore a few mind-boggling examples which demonstrate that not all arithmetic principles are absolute truths.
Entering a New Mathematical Realm
For centuries, mathematicians had tried to simplify Euclid’s system—one of the most elegant systems in mathematics, and the basis for the geometry taught in schools today. To accomplish this feat, they would have to show how they could derive the fifth postulate from the other four.
Not only was a solution never found, but they made some bizarre discoveries along the way.
This is a transcript from the video series Redefining Reality: The Intellectual Implications of Modern Science. Watch it now, on The Great Courses Plus.
In the first half of the 19th century, mathematicians like Nikolay Lobachevsky of Russia realized that they had found something incredibly deep and troubling. They had in their hands a new geometry, a different geometry—a non-Euclidean geometry.
This strange world was a new mathematical realm—a parallel mathematical universe. If we have two geometries, which one is true? When we had only Euclid’s, we assumed that it gave us the absolute truth about the nature of shape and space.
If there’s a possible alternative, we can’t hold its truth to be absolute. We need a new sort of evidence to justify our belief in what seemed indubitable.
But, what kind of evidence could this be? We can’t simply say that the alternatives are too weird—being weird doesn’t make it false.
To make matters worse, more systems were created by denying other postulates and combinations of postulates. Possible geometric systems were popping up right and left.
Which one was true? Which one was the real geometry? How do we know? What had been the most secure place on the entire intellectual landscape for more than a thousand years was now suddenly without a foundation.
Learn more about the shocking discoveries of non-Euclidean geometries
Were Arithmetics Still on Solid Ground?
Mathematicians were not happy, but at least we had the other side of the mathematical house. Arithmetic was still safe and secure; one plus one is two. There can’t be any reason to doubt that.
We had thought the numbers were well behaved, that they obeyed certain undeniable first truths. Think back to Euclid’s first axiom: Equals added to equals yields equals.
But let’s now think of the fifth axiom—the whole is greater than the parts. If I have an amount of money and in my will, I leave some of it to a relative and the rest to a charity, neither heir gets as much as I previously had in sum; by getting part, both get less than I had in aggregate.
This seems trivial and obvious. Of course, it’s always true.
In the second half of the 19th century, the German mathematician Georg Cantor showed this is not the case. Suppose we have a mutual friend named George and next week is George’s birthday.
We want to get him something we know he’ll love as a gift, but what to get him? You remind me that he’s an avid collector of numbers. We talk to his wife and she tells us that his collection now includes all of the positive integers.
He has 1, 2, 3, 816, 9,674,217—he’s got them all. If he has all the numbers you could count from 1 forward, we’ll also get him the one before 1—we’ll give him 0.
The big day comes and after blowing out his candle, he opens his gift, and sees his new 0, a number he didn’t have before. Overjoyed he looks at us and says, “Thanks for nothing.”
George’s number collection now has one more than it had before his birthday. The pre-birthday collection is only a part of the whole, so the whole is larger than the part, right?
Wrong; he still has the same number of numbers.
Learn more about the underlying reality that governs the universe
Rethinking the Nature of Sets
Suppose we go to a movie theater and we want to see if the show is sold out, undersold, or oversold. Since all people have but one backside and we use that backside to sit in but one seat, we could count the number of seats in the theater, count the number of backsides, and see which number is bigger or if they’re equal.
But Cantor realized we could do it in a way without counting at all. Ask everyone to sit down—that is, match every available seat to an available person, one to one. Then see if there is a remaining seat, a remaining individual, or neither.
This is a way to compare the size of sets without counting. Two sets are of equal size if there exists a way to map the members of one set onto the members of the other so that each element in the first set corresponds to one, and only one, member of the second with none left over.
Let’s do this with George’s numbers: If we take each of the numbers in his post-birthday collection, can we map it to one, and onto only one, from his pre-birthday collection? Take each number in the collection after his birthday and map it onto that number plus 1 in his old collection.
That is, 0 goes onto 1, 1 goes onto 2, 2 goes onto 3, and so on. In this way, in the end, there will be no number in one that does not have a correlated number in the other.
This maps one set perfectly onto the other set. The two collections are the same size.
George’s extra number has made his collection no larger even though it now includes a new element above and beyond what it had. For an infinite number—the number of counting numbers—that infinite number plus one yields the same number.
All right, that’s weird but we might think, “It’s all because it’s an infinite number of numbers. You can’t make infinity bigger. It is already infinite. How can you make it bigger?”
But all we showed is that infinite amounts are as big as one can get. You can’t have a smaller or larger infinity. This is where Cantor’s work gets fun.
Sizes of Infinity
Consider the numbers between zero and one. Some of these are what we call rational numbers, that is, they can be written as ratios: one half is ½, three-quarters is ¾. These can be written equivalently as decimals: one half is 0.5, three-quarters is 0.75.
Interestingly, all of these numbers will have one of two properties. Either they will terminate, that is like ½ will end, 0.5, done. Or, they will repeat—one-third is 0.3333333 and as far you go, there will always be more threes.
One-seventh, when written as a decimal, is 0.142857142857142857, repeats infinitely. There will always be another 142857. Such is the case with all ratios: they terminate or they repeat.
Then there are the numbers that do not repeat or terminate when we write them out as decimals. These are what we call the irrational numbers—not because they’re crazy, but because they can’t be written as a ratio of two counting numbers.
The most famous irrational number, of course, is pi—3.14159 and off it goes forever, always another digit, never repeating endlessly like the 3’s of 1/3 or the 142857’s of 1/7. If you take the rational numbers and combine them with the irrational numbers, you get what we call the real numbers.
Suppose George has an older brother, Frank, who has been collecting numbers even longer. He’s amassed all the rational numbers between 0 and 1: ½, ¼, 9/16, all of them.
Frank then orders from an online retailer the set of irrational numbers between 0 and 1. When it arrives, he now has both the infinite set of rational numbers between 0 and 1 and the infinite set of irrational numbers between 0 and 1. That is, he has all of the real numbers between 0 and 1.
We might think that, like George on his birthday, Frank’s new set with more numbers is the same size as it was before. Infinity is infinity; you can’t have more than infinity.
But you would be wrong. Cantor proved with absolute certainty that Frank’s new set is a bigger infinity. There are sizes of infinity.
If we map Frank’s old set onto his new set, Cantor showed that there’s a simple way to demonstrate that the new set has at least one number that can’t be in the old set, which means the new set is bigger.
Some infinite subsets are the same size as the sets that include them, and other infinite sets are bigger than others. There would be an infinite set of infinite numbers and these obey different rules than the finite numbers.
Learn more about the paradoxical subject of quantum mechanics
Are Any Number Rules True?
Which number rules are true? The easy answer would be that there’s one set of rules for finite numbers and another set for infinite ones.
That conclusion was the line mathematicians pursued until 1931 when the Austrian mathematician Kurt Gödel proved that we could not set out a complete set of rules for arithmetic.
Gödel showed that we could use any set of possible rules to create sentences similar to the sentence, “This sentence is false.” If it is true, then it is false, but if it is false, then it is true.
Any attempt to create rules would either allow sentences like, “This sentence is unprovable,” to be proven and so we would have sentences that can be proved but are false. We would have just proven the sentence that says it cannot be proven.
Alternatively, we could strengthen our rules to exclude these sentences, but then because we can no longer prove the sentence, the sentence “This sentence is unprovable,” would be true.
We would have true sentences that we can’t prove in our system, making our system incomplete. Any set of rules would be either unsound—that is, include false sentences—or incomplete—not allow all true sentences to be proved.
The days of mathematics as the epitome of human rational understanding seemed to close at the end of the 19th and beginning of the 20th century. It was the canary in the intellectual coal mine.
Common Questions About Mathematics and Absolute Truth
There are absolute truths in mathematics such that the axioms they are based on remain true. Euclidean mathematics falls apart in non-Euclidean space and different dimensions result in changes. One could say that within certain jurisdictions of mathematics there are absolute truths.
Mathematics was not invented. The Kemetic priests of Egypt taught a wholistic concept of number and sound which became a cult led and taught by Pythagoras to the Greeks. Around 300 B.C.E., the axiomatic system we still use to discover mathematical insights was developed by Euclid.
Mathematics appears to exist as a part of this universe. There is a mathematical universe hypothesis by Max Tegmark that posits that the universe itself is a mathematical structure. It is possible in other universes that we could not understand them and they would not be mathematical; however, our perception within this universe is mathematical and so even considering these other possibilities is difficult.