Is mathematics one big tautology?

Question

Is mathematics one big tautology? Let me put the question in clearer terms:

Mathematics is a deductive system; it works by starting with arbitrary axioms, and deriving therefrom "new" properties through the process of deduction. As such, it would seem that we are simply creating a string of equivalences; each property can be traced back logically to the axioms. It must be that way, that's how deductive systems work!

If that be the case, then in what sense are we introducing novel or new ideas? It would seem that everything is simply equivalent to the fundamental set of axioms that we choose to start with. Is there a precise step in mathematical derivation that we can isolate as going beyond pure logic? If so, then how does this fit in with the fact that mathematics is deductive? Must we change our view of mathematics as purely deductive? And if not, then is there a way to reconcile the feeling of creativity in mathematics with the fact that it boils down to pure logic?

I'm trying to figure out the true nature of what's going on here.

Answer 1

Disclaimer: different people view this differently. I side with Lakatos: Logic is a tool. Proofs are a way to verify one's intuition (and in many cases to improve one's intuition) and it is a tool to check the consistency of theories in a process of refining the axioms. The fact that every proof boils down to a tautology is true but irrelevant to mathematics.

Here is an isomorphic question to the question you posed: A painting is just blobs of paint of different colour on canvas. So, are we to deduce from this fact that the art of painting is reduced to just placing paint on canvas? Technically, the answer is yes. But the painter does much more than that. In fact, it is clear that while the painter must possess quite a large amount of skill in placing paint on canvas, this skill is the least relevant (while absolutely necessary) for the creative process of painting.

So it is with mathematics. Being able to prove is essential, but is the least relevant skill for doing mathematics. In mathematics we don't deduce things from axioms. Rather we try to capture a certain idea by introducing axioms, check which theorems follow from the axioms and compare these results against the idea we are trying to capture. If the results agree we are happy. If the results disagree, we change the axioms. The ideas we try to capture transcend the deductive system. The deductive system is there to help us find consequences from the axioms, but it does not tell us how to gauge the validity of results against the idea we try to capture, nor how to adjust the axioms.

This is my personal point of view of what mathematics is (or at least what a sizable portion of it is). It is very close to what physics is. Physics is not just some theories about matter and its interactions with stuff. Rather it is trying to model reality. So does mathematics, it's just not entirely clear which reality it is trying to model.

Answer 2

It doesn't matter whether every provable theorem is, logically, a tautology. That's just a slightly provocative restatement of the definition of provable.

What matters is via which means such proofs are found. This is where the novel ideas are introduced - in, for example, an ingenious construction, or some clever definition. Or often by going off seemingly on a tangent, proving a few lemmata which at first glance seems to have nothing to do with the theorem in question, only to turn around later, combing the lemmata, and voila, there the theorem.

Note that you can, in some formal systems, actually remove such detours from proofs. That property is called cutfree-ness, and basically states that every proof can be brought into a form where it doesn't take any detours and doesn't proof anything that isn't strictly necessary (that's a very rough description, I know). For such systems, you could argue that it's indeed hard to find creative value in a proof, if the same fact can also be proved in a straight-forward (and boring) way. Luckily for us, it turns out that the property of being cut-free itself has to be something that is very hard to proof. One can show that being cut-free automatically makes a deduction system consistent. Thus, from Gödels theorem, it follows that actually proving cutfree-ness of something like PA has to use methods which themselves go beyond PA's capabilities. Which of course makes cutfree-ness not applicable to the proof of cutfree-ness itself, and this thus always leaves some room for creativity. On just has to pick a strong enough formal system.

Answer 3

Your statement that "Mathematics is a deductive system: it works by starting with arbitrary axioms, and deriving therefrom "new" properties through the process of deduction" is one that many mathematicians (though I presume not all) would disagree with; I am one of them. There is a famous "chairs, tables, and beer-mugs" remark attributed to Hilbert that encourages this point-of-view, but my own view is that the very formal attitude to mathematics that this remark suggests is reflective more of a particular period in mathematics (one when foundational issues were at the forefront, for various reasons) than of the essence of mathematics.

I share the veiw of Ittay Weiss, namely that the ideas come first, and the axioms are just a way to model them. Reasoning, too, often proceeds by working with the ideas first; as the argument develops, eventually it will be molded into something more formal, but (in my experience) this is not how arguments begin their lives.

Answer 4

Tautology is a term used in logic to denote a statement that is true regardless of the values assigned to its variables, or its interpretation.

Mathematical statements aren't statements of logic (at last not the ones that consist of stuff other than logic, when we consider logic to be a branch of mathematics). Logic is used to connect statements in mathematics.

An algebraic equation E1, for instance, is connected to logic insofar as that equation is either true or false. We can derive another true equation E2 based on E1, but that is derivation and not logical deduction.

And so then we can look at this and make a logical statement such as "if E1, then E2".

But this statement is not a tautology. It is false whenever E1 is true and E2 is false.

Of course, it is not the case that E1 is true and E2 is false because we have established E1, and then mathematically derived E2 from it. But all that has to do with the interpretation that we impose upon "if E1, then E2" and not with its logical form.

But these are interpretations which are not the consequence of the logical form of "if E1 then E2". That form itself isn't, well, tautological.

Secondly, tautology is not a pejorative term in logic. The word is used pejoratively in the sense of rhetoric tautology: some kind of obviously true statement that doesn't advance a discourse.

Logical tautologies are darn useful. Take, for instance, De Morgan's laws. The field of logic probably has tautologies that are large, complicated and surprising.

Answer 5

You've found a property of mathematics that might be philosophically interesting, but has very little to do with math itself. You're right, but that doesn't add any interesting knowledge.

The interesting part of mathematics is which deductions you make, and how science applies these deducted theorems in understanding reality. Even if it cannot be proven that they are true in an absolute sense.

You can be aware that it is impossible to know anything for sure, and yet live a happy life of learning.

To know that you do not know is the best. To pretend to know when you do not know is a disease.

Lao Tsu

Answer 6

One could extend this argument ad absurdam: English, as a language, consists of words whose meanings are defined in a dictionary; one could read a dictionary and know the meanings of all the words there are. Having done so, what need would there be to read anything else? Poetry, literature, discourse; it all just consists of the same words in varying order.

The same is true of many things - modern engineers construct new devices out of gears, cams, rods and parts that would be familiar to a Victorian engineer; computers are powered by logic gates, resistors, capacitors and components that have been around for decades, and so on.

A mathematical proof is something of a gestalt; as a whole it can explain things that go beyond the immediate implications of its constituent parts. Admittedly, it was just as true before anyone discovered it as after, but there is still great skill in producing it. Atoms existed in medieval times as they do today, but that doesn't take anything away from the achievement of the physicists who proved their existence and determined their structure. Foreign continents like America or Australia had been sitting beyond the horizon for thousands of years before sailors got there, but we still remember the names of the ships that made the first crossings. Likewise, every mathematical proof that humanity will ever produce is already true, but proving so is still worthwhile. The ideas may not be new in a fundamental sense, but they are new to us.

Answer 7

I think you'll find Chapter 1 (in particular) of Bertrand Russell's Principles of Mathematics a delightful read. I'm positive that Russell's work will help you "figure out the true nature" of what's going on. Forgive me for not giving an explanation in my own words, but I firmly believe that I could not put it any better than Russell himself.

Answer 8

Mathematics is one big tautology -- there's no escaping that. If you imagine MATH as a person, and you were to state any theorem to MATH, MATH would say "well obviously."

But humans don't work like math. If you axiomatize HUMAN as a belief/logical system, we lack several properties that MATH enjoys. Just because we believe {A_i} doesn't mean we necessarily believe anything that can be derived from {A_i}. Just because we believe A doesn't mean we don't believe !A.

Any theorem that we prove is completely trivial to MATH, but often non-trivial and novel to HUMAN.

Answer 9

In a way, yes it is. In fact, although I rarely bother, I find pleasure in reducing proofs to more and more fundamental terms.

But I think, as Ittay Wiess says, mathematics is more than that. It is a picture, a painting.

If we were beings of infinite intelligence, then perhaps mathematics would be a much less prominent field because all of the paths from the axioms would be trivially obvious. If we were simply vastly more intelligent, then, it could be used for communication, but like casual language, would be a much more "mundane" experience - perhaps the objectivity would allow us to engulf ourselves in the intangible aspects of the experience even less. It would be a mere descriptor.

For example imagine if we as human being, had flawless memory. There would be no reason to paint a landscape. It is possible one might paint to convey an experience, but what meaningfulness would this have to someone with an infinite memory? Someone with flawless memory could simply relive the experience of seeing the picture; why paint it? It would knock the hec out of certain aspects of art, perhaps (not that it would annihilate it for sure; I'm not sure what infinite memory would do to our immediate sense of aesthetics! Perhaps our sense of aesthetics evolved as a 'shortcut' way of encouraging us to modify our environment. But that's another question...)

And that is where the human element really comes in. We must perceive what we do (I hope.) As a discipline, mathematics, input into a willing and receptive mind, forms a way of thinking about things, of subdividing problems, in a way that is reflective of reality. It can complement our minds in much the same way our complex use of words in thought might give us a richer intellectual experience than our nearest evolutionary cousin's, though few have been the subject of a difference so prominent.

Think of it as a medium of thought, (for more on that term try the book "Rise of the image, fall of the word," by Mitchell Stephens.) It gives us a way of perceiving things; the logic is an art unto itself, and one that some of us have been so blessed as to be able to see the beauty of.

Also, in say, a different universe, there might be different fundamental axioms, of a kind we cannot even conceive of - this would lead to completely different mathematics!

Answer 10

Edit: expanded

The purely formal process of combining axioms with laws of deduction to move from one "truth" to another "truth" is tautological.

But the meaning which we assign to mathematical statements exists outside of this formal system.

To illustrate the difference, consider the following thought experiment. Suppose there existed an oracle that told us which statements are true (say, in ZFC). Then our job as mathematicians would be made easier in some sense, but still it would by no means be rendered obsolete.

First of all, mathematics is not just the quest for all truths. Rather, mathematicians work within a dynamic field of meaning, a constantly evolving language consisting of concepts that they find useful. An oracle, on the other hand, would produce statements in a fixed language, and without regard for their "meaning".

Secondly, mathematicians want to know why things are true. And they are not content with just any proof -- they want, if possible, a proof that illuminates, that has a potential for shedding insight, or for being generalizable or for embodying a technique that can be used in other problems. If we liken mathematics to a novel, then we are not content with just a list of facts or of minute details of cause-and-effect -- we are interested in a larger insight into the relationships between the characters.

Finally, as has been commented upon, a large part of mathematics is in finding the "right" questions to ask -- the ones which we find interesting and which could lead to further insight.

I hope this illustrates some of the differences between the "meaningless" nature of a truth-producing oracle and the meaningful, or meaning-seeking, nature of mathematics

Answer 11

This reminds me of a story told by Gelfand at his seminar: a drill sergeant at the ROTC training for math students used to tell them: "this is not algebra, here you have to actually think!" According to Gelfand, the phrase was true: algebra is trivial because the LHS is always the same as the RHS, while analysis is non-trivial because it deals with inequalities and estimates.

Tautology or not, mathematics is useful for expressing and gaining knowledge about the world we live in.

Moreover, saying that it is a tautology is like saying that since the all fish consists of cells, ichthyology can be reduced to cytology, which, in turn, can be reduced to chemistry.

PS. Philosophy of science is as useful to scientists as ornithology is to birds. (attributed to Feynman).

Answer 12

This question has been accurately explained in Wittgensteins' Tractatus. In §6.1 he said: "The propositions of logic are tautologies." Later: "It is clear that one could achieve the same purpose by using contradictions instead of tautologies."

Mathematics seem to be a way of thinking and talking; of building propositions and sentences. Moreover we're fascinated by the idea that the universe actually adheres to our thinking. That we've found "the basic language of the nature".

Here are two practical ideas why mathematics are (maybe) not just a big tautology:

• still under development;
• an extremely valuable system for individuals and cultures.

Mathematics is a part of our culture, and developed further with each new generation. For example, the Romans yet had no zero and hence no higher mathematics. Although logic has no intrinsic value, other than more logic, it has a meaning (for us) which transcends itself.

Answer 13

The answer to this question requires that to be correct does not contain any mathematical tool: no theorem, no logical proposition, no axiom, and no structure isomorphic.

Gödel would be happy to show you this grand illusion.

Answer 14

There are a few things to mention here; without checking what others have already said, I'll quickly sum them up.

First, there are more than only one set of axioms in Mathematics. Indeed, the beauty, and diversity, of mathematics arises from the diversity, and beauty, of the different axiomatic systems that have been introduced thus far. Certainly, it is not a novel act to introduce such a system, but it is so if the system is itself novel or interesting. Since, axioms are supposed to be, by definition "fundamental," introducing two sets of logically equivalent axioms (which differ, in the end, only in presentation) will not result in two (wholly, perhaps I should clarify this more) distinct theories. On a related note, it may very well be difficult to establish the logical equivalence of two sets of axioms. For more on the discussion of axiomatic systems, I recommend you study the development of and current set theory and formal logic.

Secondly, you must understand the difference between logical equivalence and identity. For example, the statement $P \implies Q$ is logically equivalent to $ \neg P \lor Q$, however, they are not the same statement. Indeed, both statements say the same thing, but they differ, as mentioned prior, in presentation; you may use one over the other, however, you may not be able to derive the same conclusions depending on your choice. For more on the difference between the two, see this and this.

Thirdly, and perhaps the most obvious to mention here, there are certain propositions in Mathematics that can neither be proved nor disproved. Indeed, this is Godel's theorem.

To answer your question about the "creation" of new and novel truths, theorems can simply be interpreted as new ways of looking at combinations of axioms. For example, $${12 \over \pi^2}\sum_{k = 1}^\infty {1\over{k^2}} = 2 \tag{1}$$ is just another way (I would say new, but that is not really the case) of saying $$2 = 2 \tag{2}.$$ And, certainly, there is nothing novel about saying $(1)$ in this light. But! The point of the exercise is to draw relations that are not immediately obvious, and relations such as they may be useful in drawing other relations and equivalences. It is after all, as you said, a matter of drawing a string of equivalences, but again, keep in mind we are not limited to one set of axioms.

Answer 15

In 1958 at the Edinburgh ICM I heard Raoul Bott remark: "Alexander Grothendieck is prepared to work very hard to make things tautological!"

To do this, often meant developing new concepts, which explained why something was true. But Grothendieck also wrote of "the difficulty of bringing new concepts out of the dark!"

I like this quotation from the play "A Midsummer Night's Dream", of which the first line describes one of the roles of a mathematician:

"As imagination bodies forth the forms of things unknown,

The poet's pen turns them to shapes, and gives to airy nothing

A local habitation and a name."

New names, concepts, and notations, i.e. new language, are just some of the characteristics of mathematics.

Answer 16

Usually it is. That's what makes it useful. I could say: "I have 5 gallons of gasoline so I can go to Mars." and mathematics can tell you whether you are right or wrong and help you avoid embarking on the journey if you are wrong.

But not all statements can be decided to be true or false. That's what makes it interesting. The search in math is to identify which useful statements are provably true, which are provably false, and most interestingly those to which we must say "I don't know."

The "I don't know" bit makes it not a tautology; not one big "duh!". It also makes it a human artistic effort because the box of "I don't knows" includes both those that are genuinely unprovable and those that we do not have the skill to prove yet.

Answer 17

First of all, I want to thank everyone for their contributions and comments, there's some really great stuff here, and I've (and I feel anyone who has viewed this question would agree) definitely gained a lot from all that has been said. I've given my question a lot of thought, and I feel compelled to give an answer that encompasses some of the more elegant ideas expressed by other users. Here goes:

Mathematics, in its most essential form, is simply a set of ideas about quantity. These ideas are ways to view the world through the lens of measurement and of dimension. They have a unique "flavor", and in an exactly analogous manner, every other science (or field of study) is a group of ideas that help us view the world in a "flavored" way. Mathematicians discover these ideas, invent them, use them to find other ideas, clarify them and bring them to precise formulation, work with them, and this list may go on. We have a body of knowledge, and that is the key.

Thus far I've said nothing of deduction and for good reason: Deduction is the tool we use in order to ground our ideas in a firm, unshakable system. We strive to find axioms that will properly serve as the "seeds" for all of the mathematical ideas we'd like to have in our system. We wish to build up a logical fortress of implications and equivalences that will house our mathematical ideas. The deductive system is the means to derive and discuss the ideas of mathematics. Let me put this in different words: In order to further our quantitative knowledge, it becomes necessary to clearly define our abstract structures; it also becomes necessary to clearly perceive the properties of these structures, to manipulate them, and ultimately use them to arrive at other structures. This is the nature of the mathematical-deductive system, and the benefit is twofold: Firstly, we have a precise and logical way to house our intuitive quantitative ideas which form the main body of math. Secondly, once the dust has settled and we have worked out our precise deductive system to the last jot and tittle, it becomes possible to create new ideas, to reach novelties! That is the magic of the deductive system; the precision is a way to formalize our intuition, and the new-found rigor may even result in new ideas hitherto unsuspected. Once you have a logical system, novelties might be reached by utilizing the system, playing by its logic rules of derivation. So the true mathematics is a combination of both intuition and rigor.

It doesn't matter that, in some technical sense, everything might be traced back to axiomatic roots. That is just how deduction works, you can't fault a logical system for being logical. But at each derivative step, we have a new idea, that is the essential point! Even in an equality which seems to be the biggest triviality, the LHS and the RHS represent two different ideas, and their being equal a third. To reiterate, the deduction is a rigorous, "tautological" (not in the strict sense) system; but in each each step that we take, that brings us further and further away from the axioms, a new idea is birthed or represented. It is the concepts contained in each theorem or identity that are at the heart of math, and not the logical steps between them. Those logical steps are just due to the nature of deduction, and are of necessity if we are to employ the tool of deduction.

I hope I have added something new to this (in my opinion) intriguing discussion, or at least synthesized what's already here. Thanks for reading.

Answer 18

This is a great question that I have pondered in my own way many times.

When I was in Real Analysis, I proposed to my teacher that truth could be defined as the largest set of all statements which can coexist without contradicting one another.

As we all know, when one lies, one cannot go on forever without reaching a contradiction. But the truth can go on forever, or at least we can postulate that it can, without ever reaching a contradiction. Hence, any set of lies would have to be smaller than the set containing all true things. And all those sets of lies would in a sense be embedded within the set of truth because the true version of each lie is the very fact that the lie is a lie.

My teacher responded that this was far too philosophical and had no place in a math class. But I still wonder if that would be a workable definition for truth.

Answer 19

I think we can compare this with physics. In physics we develop a theory. To check the validity of this theory, we make experiments. In Mathematics, we make ideas, conjectures, using maybe intuition. To check these ideas, we use rigor (proofs). An employer who doesn't understand anything in Physics may participate in an experiment designed to check a certain physics theory. For example he may check daily some data in computer using software without even knowing what these data represent. It may appear to him that he does the same work everyday, which is natural because he does not know the meaning of these data, unlike the physicists how understand these data and are so exited doing or observing the experiments to check if they agree with their theories.

Answer 20

I would simply say that the space of axiomatic systems and theorems deduceable from those systems is definitely infinite. All those axiomatic systems and theorems /exists/ already in a sense. And they are tautologic.

The creative process is finding a way through that system. And not an arbitrary way of random theorems, but a way with lot of connections to other such paths, and with connections to the art of thinking or to common mathematical problems or to beauty or to practical usability.

And as coffee_table says, often a mathematician has an idea of what he hopes to find.

Answer 21

It is technically true that every theorem is equivalent to the axioms; that is the point, or else the theorem would be wrong. But that doesn't mean the axioms and the theorems are the same.

I quite like the term Douglas Hofstadter uses: implicit truths and explicit truths. While all the theorems are truths which are implicit in the axioms, it is the job of mathematics to make them explicit. You might enjoy reading the book Godel, Escher, Bach by Douglas Hoftsadter.

Answer 22

I think the answer you might be looking for is that truth within a mathematical model is an axiom or combination of axioms, augmented with a number of definitions. There can be no other truths. Whether math represents truth in general is discussed in

http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=truth;action=display;num=1268924603;start=

Answer 23

That will probably be a matter of opinion. The importance also will be a matter of what school you belong. If your're in logicism, than it will matter a lot; if in intuitionism, not very much. Other people already wrote on this subject, you should see what Bertrand Russell says of Wittgenstein and Ramsey in "My philosophical development". And also "The foundations of mathematics" from Ramsey.

Have a nice day.

Answer 24

Mathematicians seem to be persons that:

• study mathematics;
• pick up new ideas and formulate consistent theories that capture the essence of these ideas;
• formulate and examine statements and try to prove them or find counterexamples;
• publish their discoveries and results to be reviewed and used by other mathematicians;
• teach mathematics to others.

I conclude that mathematics is a process of accumulation of knowledge about mathematical theories on an increasing number of topics.

About the relation between mathematics and tautologies, Kurt Gödel said (Jean van Heijenoort, 1967, From Frege to Gödel s. 601):

"The class of provable formulas is defined to be the smallest class of formulas that contains the axioms and is closed under the relation 'immediate consequence', i.e. formula $c$ of $a$ and $b$ is defined as an immediate consequence in terms of modus ponens or substitution."

Which in my opinion makes it clear why proved theorems doesn't have counterexamples.

Logic is the trivial part of a mathematical proof and that's the tricky part which is mathematics.