November 2025

My intention for this article is to provide a look into how the decimal expansion of some fractions yields the Fibonacci sequence. Let us first take a look at the decimal expansion of 1/89:

Now, for a quick refresher on the Fibonacci sequence. You start with the numbers 0 and 1, and every number after that is the sum of the two before it. This gives us the sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…

(A small note on notation: Fₙ = Fib(n) = nth Fibonacci number)

After looking at the Fibonacci sequence, look back at the decimal expansion of 1/89 and try to spot any similarities. You would see

Now, at first glance, this pattern falls apart. The next digit should have been 8, which is Fib(6). However, if we take a closer look, you can see that we get 9 from 8 and the tens place of 13, which is Fib(7).

This gives us the intuition that each Fibonacci number is given one digit of space, so it starts to spill over to the previous digit when the size of a Fibonacci number exceeds one digit. This makes it extremely difficult to trace out which Fibonacci values are responsible for which decimal places.

(We will solve the digit overflowing problem at the end.)

Working Backwards

Although we have seen a pattern, we do not know whether this will hold for the infinite expansion of 1/89, so we need to prove this relationship. A complete proof would start with an infinite summation of Fibonacci numbers divided by increasing powers of 10 and prove that the expression is equal to 1/89. So, we can start out with the expression:

By dividing by 10 we get:

By subtracting the first expression from the second, we get:

And the proof is complete.

Generalizing

Now, a true mathematician would ask “What if we wanted to generalize this to make a more common expression? What if we wanted n digits of space for every Fibonacci number?” This leads us to the following expression.

Hence, we have derived an expression that will generate the Fibonacci sequence as a decimal expansion. Let’s take it for a spin and see if it works:

First, we can plug in n=1, and get the result we saw earlier.

Now, let’s try n = 5

And we can see the pattern continue clearly.

Limitations

To be a fair and impartial judge, I would like to go over the downsides of this method. To figure out the nth Fibonacci number, you would have to find the size of the number, so you can use the right n value. Otherwise, you are going to have to deal with digits carrying over, which is a headache.

Benefits

This method of calculating the nth Fibonacci number can be computed in the same speed as the fastest known method(Matrix Exponentiation). Also, this is a super interesting way of finding the nth Fibonacci number, because unlike Binet’s Formula, this method can be done by hand.

I will be writing another article in the future analyzing the computation side of this formula, but that’s all from now. Goodbye.

(Remark: One technical detail that I glossed over is that for us to set the summation equal to a finite quantity S, we need to prove that the sum converges. That is beyond the scope of this article, but can be proven using Binet’s formula.)

Understanding Pierre de Fermat’s observation about prime numbers

If p is a prime and a is any integer not divisible by p, then p divides aᵖ⁻¹ – 1.

This property of numbers discovered by Pierre de Fermat in 1640 essentially says the following: Take any prime p and any number a not divisible by that prime. Say, p = 7 and a = 20. By Fermat’s little theorem, we then find that:

Now, we don’t care much about the actual number that results from this calculation. Rather, we care about the fact that without having to do the calculation at all, the theorem tells us that a whole number, an integer, has to result from it.

Introduction

Among Pierre de Fermat’s many correspondences with mathematicians around the world in the 1600s, the most consequential for number theory would be that which he shared with French mint official Bernard Frénicle de Bessy (1605–1675). As the story goes, de Bessy was renowned in France for his gift for calculating large numbers:

“On hearing that Fermat had proposed the problem of finding cubes that when increased by their proper divisors become squares, as is the case with 7³ + (1 + 7 + 7²) = 20², [De Bessy] immediately gave four different solutions, and supplied six more the next day.”Excerpt, Elementary Number Theory (Burton, 2011)

De Bessy himself would later be best remembered for his publication Des quarrez ou tables magiques (“Finding the square equivalent of magic tables”), a treatise on magic squares published after his death in 1693, in which he provides all 880 different magic squares of order 4. A magic square is an n × n grid filled with distinct positive integers such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is the same.

Although in no way Fermat’s equal as a mathematician, as a number theorist no contemporary could challenge Fermat like De Bessy (Burton, 2011). The correspondence between the two in the mid-1600s would result in some of the most striking discoveries in number theory, including the cube property of the number 1729 (now known as the Hardy-Ramanujan number, or taxicab number, for Srinivasa Ramanujan’s 1919 comment to G.H. Hardy that the latter’s taxicab number was “a very interesting number, […] the smallest number expressible as the sum of two cubes in two different ways”).

The most striking result of the correspondence between the two was however Fermat’s following statement in a letter dated October 18th, 1640:

Correspondence from Fermat to De Bessy (1640)
Tout nombre premier mesure infailliblement une des puissances – 1 de quelque progression que ce soit, et l’exposant de la dite puissance est sous-multiple du nombre premier donné – 1 ; et, après qu’on a trouvé la première puissance qui satisfait à la question, toutes celles dont les exposants sont multiples de l’exposant de la première satisfont tout de même à la question.

Essentially stating that “every prime number p divides necessarily one of the powers minus one of any geometric progression”. The result has since become known as Fermat’s Little Theorem:

Fermat’s Little Theorem (1640)
If p is a prime and a is any integer not divisible by p, then p divides aᵖ⁻¹ – 1.

Fermat added “de quoi je vous envoierois la démonstration, si je n’appréhendois d’être trop long”, claiming, as he famously also did with his proposition of Fermat’s Last Theorem, that he would have sent De Bessy a proof of his claim “if I did not fear its being too long”.

Proofs

Nearly one hundred years later Euler would be the first to provide a proof to Fermat’s little theorem, in a 1736 paper entitled Theorematum Quorundam ad Numeros Primos Spectantium Demonstratio (“A proof of certain theorems regarding prime numbers”) in the Proceedings of the St. Petersburg Academy. However, it was later discovered that Leibniz had found virtually the same proof in an unpublished manuscript sometime before 1683, but that there is no way for Euler to have known about it (Burton, 2011).

Today, numerous proofs of the theorem are known. The proofs generally rely on two simplifications: First, the assumption that a is in the range 0 ≤ a ≤ p − 1. Second, that it is sufficient to prove that Fermat’s little theorem holds for values of a in the range 1 ≤ a ≤ p − 1.

Proof using the binomial theorem

Euler’s first proof (rediscovered after Leibniz) is a very simple application of the multinomial theorem, which describes how to expand a power of a sum in terms of powers of the terms in that sum:

The multinomial theorem

The summation is taken over all sequences of nonnegative integer indices k₁ through kₐ so that the sum of all kᵢ is n. If we express a as a sum of 1s raised to the power of p (1 + 1 + 1+ … 1ₐ)ᵖ, we obtain:

If p is prime and kⱼ is not equal to p for any j, we have:

If p is prime and kⱼ is equal to p for some j, we have:

Since there are exactly a elements such that kⱼ = p, the theorem follows.

Proof as a corollary of Euler’s Theorem

Another proof of the theorem appears as a consequence of that fact that Euler’s theorem is a generalization of Fermat’s little theorem. It states that for any modulus n and integer a coprime to n (the only positive integer that divides both is 1), one has:

where φ(n) is Euler’s totient function, which counts the integers from 1 to n that are coprime to n. If n is a prime number, Fermat’s little theorem appears, as φ(n) = n − 1. A proof of the theorem hence follows from the proof of Euler’s theorem, which is typically done using group theory.

Proof using modular arithmetic

The following proof, using modular arithmetic, was originally discovered by James Ivory in 1806, before being rediscovered by Dirichlet in 1828.

Proof of Fermat’s Little Theorem (Burton, 2011)
We begin by considering the first p – 1 positive multiples of a; that is, the integers a, 2a, 3a, …, (p – 1)a. None of these numbers are congruent modulo p to any other, nor is any congruent to zero. Indeed if it happened thatr × a ≡ s × a (mod p), 1 ≤ r < s ≤ p – 1then, cancelling a on both sides would give r ≡ s (mod p), which is impossible because r and s are both between 1 and p – 1. Therefor, the previous set of integers must be congruent modulo p to 1, 2, … , p – 1. Multiplying these congruences together, one finds thata × 2a × 3a × … × (p – 1) × a ≡ 1 × 2 × 3 × … × (p – 1)(mod p)meaningaᵖ⁻¹ × (p – 1)! ≡ (p – 1)!(mod p).Cancelling (p – 1)! from both sides of this expression, we obtainaᵖ⁻¹ ≡ 1 (mod p), which stated differently equals aᵖ⁻¹ – 1, Fermat’s Little Theorem.

Proof using group theory

To prove Fermat’s little theorem using group theory, recognize that the set G = {1, 2, …, p − 1} with the operation of multiplication forms a group. Of the four group axioms, the only that requires effort to verity is the fourth, namely that the elements in G are invertible.

If we assume that every element in G is invertible, assume that a is in the range 1 ≤ a ≤ p − 1, that is, assume a is an element of G. Let k be the order of a, i.e. the smallest positive integer such that aᵏ ≡ 1 (mod p) is true. Then the numbers 1, a, a², …, aᵏ⁻¹ reduced modulo p, form a subgroup of G whose order is k. Therefor, by Lagrange’s theorem, k divides the order of G, which is p − 1. So p − 1 = km for some positive integer m, and:

To prove that every element b of G coprime to p is invertible, this identity can help us as follows. Because b and p are coprime, Bézout’s identity assures that there are integers c and d such that bc + pd = 1. Modulo p, this implies that c is an inverse for b, because bc ≡ 1 (mod p). Because b is invertible, so is every other element in G, and so G must be a group.

Several other proofs are available on Wikipedia.

Application: Primality test

Fermat’s little theorem would become the basis for the Fermat primality test, a probabilistic method of determining whether a number is a probable prime. If we for instance want to find out whether n = 19 is prime, randomly pick 1 < a < 19, say a = 2. Calculate n − 1 = 18, and its factors: 9, 6. We check by calculating 2¹⁸ ≡ 1 (mod 19), 2⁹ ≡ 18 (mod 19) and 2⁶ ≡ 7 (mod 19) and find that 19 must be prime.

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As I finished reading The Fault in Our Stars by John Green, perhaps what stood out the most was the phrase, “Some infinities are bigger than other infinities.” As Hazel Grace looked back at the time that Augustus Waters and her had left together, she had struck a reference to a seemingly enigmatic idea — that there are neither one nor two infinities, rather, there are an infinite number of infinities.

Perhaps of all the disciplines we are aquatinted with, mathematics stands to be seen as the most definitive. Governed by rules, proofs, theorems, and logic, the subject has afforded scientists a new language with which to grasp our conceptual understandings thus far. For example, the power of mathematics can be seen in research. Statistical tests to confirm hypotheses serve to sway the course of scientific relevance in a society fueled by innovation and discovery. However, the same discipline upon which discoveries in chemistry and physics are founded upon is the same discipline which has many enigmatic properties that are not explained by mathematics’ definitive nature. Today, I will be discussing one such enigma: the paradox of infinity.

An Exploration of Infinity

Infinity is the representation of the upper-limit of all real numbers, however, the differences in its behavior from real numbers’ could not be greater. One way we can see this is in the following examination: how many real numbers are between one and two? Infinite. How many real numbers are between one and three? Infinite. The interval between one and three is twice as long as that between one and two, hence, the latter infinity must be twice as large as the former, yet they are both infinity. In the former case, ∞ = ∞, which fits our logic. In the latter, however, 2∞ = ∞. In theory, infinity multiplied by any scalar should yield simply infinity itself, however, this is where we see its deviant behavior. The equation 2∞ = ∞ can be seen as simplifying to 2 = 1 (this is not exactly what happens as you cannot divide infinity by infinity, however, for the sake of Hazel Grace’s argument, let us reduce this to a comparison of the intervals’ sizes). Clearly, that is illogical and does not obey the course of mathematical logic. It is here that we see mathematics sway from its seemingly definitive nature of which we are introduced throughout our lives.

So, why does this matter? See, infinity is a concept, not a definitive mathematical rule. It is a concept assigned to explain the unfathomable bounds of real numbers. However, this concept also bounds our ability to think. Srinivasa Ramanujan (1887–1920) is regarded as one of the best mathematicians of our time. He is most notably regarded as the man who proved that the summation of all positive integers equals -1/12. Had he applied the concept of infinity to his work, that theorem would have stopped at the first step, in which the summation of all positive integers would be assigned the answer of infinity. However, he looked past this concept and was able to prove an idea that has served practical purpose in fields such as string theory.

Conclusion

The idea of infinity is abstract. It adds a layer of uncertainty to mathematics that almost makes it uncomfortable to explore. However, Ramanujan’s work begs me to wonder, is this abstraction that defies the course of real number logic simply an obstacle to understanding our world? String theory was proposed as a means to understanding our universe, however, that theory was built with the help of an idea that ignored the existence of infinity. It is almost philosophical when we ponder whether some infinities can be larger than others, yet that philosophy breaks down when we apply simple mathematics. Perhaps infinity is the bridge between the known and the unknown. Perhaps that is what Hazel Grace grappled with in her last days with Augustus Waters.

Many philosophers and logicians have analysed the forms of arguments and shown us that the same forms occur across many different cases. They’ve also shown us that when it comes to what are called argument-forms, the semantic content of the premises doesn’t (really) matter as long as the argument-form (which involves the proper use of the symbols for the premises and logical operations) is adhered to.

This piece also deals with the form (or syntax) of the premises (or sentences) themselves. It attempts to show the philosophical consequences of premises (or sentences) being taken to be without semantic content. This is to deal with something more than the simple fact that the semantic content of premises (or sentences) can be ignored by (formal) logicians: it’s an argument that in some cases (such as the sentence “This sentence is false”) the premise (or sentence) doesn’t have semantic content in the first place (i.e., even after that content is analysed or determined). More specifically, an attempt is made to show that if a sentence in an argument has no semantic content, then that may be at least one reason why it can lead to a semantic paradox. (Note the irony of the adjective “semantic”.)

The first section deals with a purely syntactic logic as it applies to argument-forms. In the second section, Roger Penrose’s position on mathematical formalismis discussed in order to see if it can shed light on what may be deemed to be logical formalism. The third section deals with the syntax of premises (or sentences) mainly as they are found in logical arguments. It also comments on the philosophical consequences of dealing with premises (or sentences) in purely syntactical terms. The Final section pays exclusive attention to the “liar sentence” and defends the position that because it can be taken to have no semantic content at all (therefore it is treated purely syntactically), then that may be at least one reason why it leads to a paradox.

Empty Syntax

Noam Chomsky’s well-known example of a syntactically- and grammatically-correct surreal sentence.

The broad position of this piece is at least partly expressed by Stephen Read in the following:

“What must be acknowledged is that belief that every valid argument is valid in virtue of form is a myth, and exclusive concentration on the study of pure forms of argument does a disservice both to logic and to those who can be helped by it.”

That’s Stephen Read’s broad philosophy-of-logic position. And in the following piece a similar way of thinking is also applied to the premises (or sentences) actually found within arguments.

Read then goes on the apply his (as it were) non-formalist position to the actual technical detail of valid arguments. He continues:

“Validity is a question of the impossibility of true premises and false conclusion for whatever reason, and there are arguments which are materially valid and where that reason is not purely logical.”

It can be argued that Read goes too far in the first passage above. That passage is a clearly a philosophical take on logic. Yet one can fully accept a purely syntactical (or formal) logic because such a thing clearly has its place. However, it’s also true — at least in part — that such a logic is indeed (to use Read’s words again) “a disservice to those who can be helped by it”. That is, for those interested in the philosophy of logic and in using logic in their philosophical arguments (i.e., logical arguments which aren’t actually about logic), the purely syntactical approach may seem odd or even pointless. On the other hand, syntactic (or formal) logic has had a great impact on metamathematics and mathematics itself. In addition, the purely formal way of looking at logic has been of great help to the development of computer programmes, computers themselves and other technologies.

So one doesn’t need to reject syntactic (or formal) logic. One only needs to accept that a purely syntactic logic is one thing; and a logic with important or relevant semantic content is another. Nonetheless, there’s more to it than that. If we recognise the important difference between syntactic logic and semantic logic, then that may help us explain much about the so-called semantic paradoxesand sentences like “This sentence is false”. (It may even help us develop a position on Gödel sentences.)

Stephen Read makes the formalist position on logic very clear when he states the following:

“Logic is now seen — now redefined — as the study of formal consequence, those validities resulting not from the matter and content of the constituent expressions, but from the formal structure.”

We can now ask:

What is the point of a logic without material, semantic or relevant content?

Wouldn’t all the premise, proposition, predicate, etc. symbols — not the purely logical symbols —used simply be self-referential in nature? (Thus all the p’s, q’s, x’s, F’s, G’s etc. would be autonyms.) And what would be left of logic if that were the case? Clearly we could no longer really say that it’s about argumentation at all… or could we? That is, we can still learn about argumentation from argument-forms which are purely formal in nature. The dots don’t always — or necessarily — need to be filled in.

An Example

The following has been deemed (by some logicians and philosophers) to be both a valid and a sound argument (i.e., I’ve simply changed the premise and conclusion):

Football is a sport.— — — — — — —

∴ Snow is white.

To the layperson the above will seem both wrong and silly. However, classical logic allows premises and conclusions to be completely unrelated. That is, it is the form (hence argument-forms) of the argument that matters to logicians.

Now form is just as much a matter of premises (or sentences) as it is about arguments. As stated, the argument-form above is both valid and sound. But what of the premise (or antecedent sentence) itself? This premise is true. So is the conclusion. But what about this argument? –

This sentence is false.— — — — — — — — — 

∴ Football is a sport.

The premise above is perfectly acceptable — according to classical logic. The conclusion is also true. But is the premise also true… or false for that matter? If it is true, then the argument is valid. If it is false, then the argument isn’t sound. What if the premise is both true and false? Then, according to the principle of explosion, “anything and everything follows from it”. Thus the conclusion mustalso follow from it.

But what if the premise is neither true nor false, or both true and false? And what if its status as a proposition (i.e., as used as a premise) is questionable in the first place?

Roger Penrose on Formalist Mathematics

Roger Penrose and Plato.

A parallel to this issue of a purely formal logic can be found in mathematics — or at least in the philosophy of mathematics (or in metamathematics). For example, here’s the mathematical physicist and mathematician Roger Penrose on mathematical formalism:

“The point of view that one can dispense with the meanings of mathematical statements, regarding them as nothing but strings of symbols in some mathematical system, is the mathematical standpoint of formalism.”

Penrose has a serious problem with the “point of view” that is (or was) mathematical formalism. He goes on to say that “[s]ome people like this idea, whereby mathematics becomes a kind of ‘meaningless game’”. Penroseconcludes:

“It is not an idea that appeals to me, however. It is indeed ‘meaning’ — not blind algorithmic computation — that gives mathematics its substance. Fortunately, Godel dealt formalism a devastating blow!”

One doesn’t need to be a Platonist to understand how formalism can also be a problem for logic. As it is, one can believe that both mathematics and logic can indeed have a purely formal aspect. Indeed one can see maths (though not logic) as always being purely formal (i.e., the contrary of Penrose’s position). However, in logic (unlike in mathematics) we have logical arguments expressed in a natural language. And these arguments contain sentences which include names and predicates. That is, they contain premises (or sentences) which have extensional, referential and existential import. None of this is true of maths — unless, that is, one believes that numbers, sets and even functions (as abstract objects) are the referents or extensions of the numerals and other symbols!

Penrose on Mathematical Syntax

In the above the word “syntax” has been used a few times. So what exactly do I mean by the word “syntax”? Let Penrose again (though he’s talking about mathematics) explain:

“[‘S]yntactically correct’ essentially means ‘grammatically’ correct — i.e. satisfying all the notational rules of the formalism, such as brackets being paired off correctly, etc. — so that P has a well-defined true or false meaning.”

Indeed this “formalism” is given even more importance by Penrose because he places it within the context of David Hilbert’s quest for a systematic and secure grounding of all mathematics. Penrose continues:

“If Hilbert’s hope could be realized, this would enable us to dispense with worrying about what the propositions mean altogether! P would just be a syntactically correct string of symbols.”

Now Penrose rejects this formalism for mathematics. And here I am applying it to logic. Prima facie, if formalism can be challenged in the case of maths, then it seems to be far easier to challenge it when it comes to logic. After all, logic (among other things) makes explicit use of the words “true” and “false”. In addition, logic’s Ps and Qs are meant to stand in for propositions, which themselves can be (or are) expressed in natural-language sentences which contain names, predicates and suchlike. None of this can be found in maths. Indeed in maths the words “true” and “false” are rarely used in the symbolism itself. Any use of the words “true” and “false” that can be found are usually found in the domains of metamathematics and the philosophy of mathematics, not in mathematics itself.

So, as with logic, maths can be seen purely as a “game” (to use Penrose’s word just quoted above). Alternatively, maths can be given an interpretation or a philosophical account. Thus it can be said that formalists are like the shut-up-and-calculate brigade in quantum mechanics who’re only concerned with the mathematical formalism and how that formalism helps with experiments, predictions and observations. However, other physicists are also keen on the “interpretation” of quantum mechanics. That means that we can conclude by saying that Penrose is keen on the interpretation of maths. (His interpretation is essentially Platonic — see here.)

As for logic, rather than maths.

Robert S. Tragesser (who discusses Gottlob Frege) explains logical formalism in a similar way to which Penrose explained mathematical formalism. He wrote:

“Frege believed that the principle virtue of such formal-syntactical reconstructions of inferences — as validly moving on the basis of the meanings of the signs for the logical operations alone — was that it eliminated dependence on intuition and let one see exactly on what our inferences depended.”

The important point here being that it’s only “the signs for the logical operations alone” that have “meanings”. In other words, it doesn’t matter what conditions, events, facts, etc. A and B (in this case) stand for (or, in other cases, which natural-language sentences the symbols P and Q are meant to stand for), or what the conditional sign ⊃ “means” (i.e., beyond its syntactic role), what matters is the “formal-syntactical reconstructions of inferences” which (according to Frege at least) allow us to bypass any use of “intuition”.

The Logical Syntax of Sentences and Premises

One problem with standard (or classical) logic is that the content of a premise is irrelevant. What matters is its form. So just as form is important in logical arguments, so form is also important when it comes to what are taken to be the premises (or sentences) found in logical arguments. This last aspect of logical formality hasn’t been discussed as much as the former.

Of course all premises are taken to have a semantics by formal logicians. If this weren’t the case, then they wouldn’t be seen to be premises at all. Yet the precise semantics isn’t the concern of the formal logician (if there is such a pure being). He doesn’t himself establish the references and extensions of the premises he uses. That is, premises are taken to have already-established extensions and references — otherwise they couldn’t be taken to be either true or false in the first place.

So logic always has a semantics or an interpretation.

In addition, there’s the (sorta) semantics of the symbols and their operations themselves. That is, a semantics of the symbols qua symbols and of the operations qua operations. In this case, the symbols and operations — taken in and of themselves — are taken to have “meanings”.

The heart of the problem can be seen when it comes to the logician himself. Specifically, it’s the distinction most (formal) logicians make between a sentence which is written down (or spoken), and the proposition that the written (or spoken) sentence is said to express. Such logicians are only interested in the proposition. Thus the proposition can be seen as being “underneath” or “behind” the sentence. Or, less metaphorically, the proposition is seen as being the idealisation of the sentence.

This emphasis on the proposition is but a means to an end. Put simply, propositions (therefore symbols) are much easier to logically manipulate than natural-language sentences (as with sentences/statements which are given Gödel numbers).

All this will depend on what we take a proposition to be. And that may end up being partly — or even wholly — a stipulational matter. For example (as we shall see), it can be argued that if the sentence “This sentence is false” is without reference and extension, then perhaps it can’t be a genuine proposition at all. Having said that, in logic it is taken to be a bona fide proposition.

The (as it were) formalist position on sentences (i.e., not on arguments) is put very well by Bryson Brown in the following:

“These rules are based on the syntactic structure of the sentences, that is the symbols and how they are arranged in each sentence, rather than on an account of their truth conditions.”

So here, rather than concentrate on the syntactic structure of an argument-form, Bryson Brown talks of the “syntactic structure of a sentence”. In other words, the sentences (or premises) in this particular “consequence relation” can be seen as being purely syntactical too.

So why this stress on syntax? Bryson continues:

“One advantage of this approach to consequence relations is that it focuses our attention on the process of reasoning, rather than on ‘meanings’ that are taken to lie behind that process.”

This means that one can reason without a semantics; just as one can do arithmetic without worrying about the meanings — or the ontology — of the numbers one is using.

So let’s tackle one logically-acceptable (though still philosophically problematic) sentence.

The Liar Sentence

The following is the liar sentence:

This sentence is false.

One problem is that the liar sentence certainly looks like other sentences. Grammatically, it appears to be in very good shape. It’s in good shape in a similar way to which Noam Chomsky’s well-known “colorless green ideas sleep furiously” (see image above) looks like a grammatically-acceptable sentence. (There are, of course, clear distinctions which can be made between the liar sentence and Chomsky’s surreal sentence.)

But what about the following? –

This sentence is true.

By inference, if one accepts the sentence “This sentence is true”, then one must also accept its negation — namely, “This sentence is false”. Yet it’s the case that the sentence “This sentence is true” is less problematic than the sentence “This sentence is false”. That is, the former doesn’t engender a paradox. Nonetheless, it’s still as empty as the sentence “This sentence is false”.

But firstly, let’s get something out of the way.

Take the liar sentence again:

This sentence is false.

Is that really the logical form of the following? –

“All Cretans are liars.”

Aren’t there obvious differences between “This sentence is false” and “All Cretans are liars”? That may not matter because the former is still taken to be the logical form of the latter.

We can summarise the sentence “All Cretans are liars” in the following ways:

1) The sentence is spoken by a human person.
2) It includes the extension that is the set Cretan liars.
3) It is spoken by a member of the set Cretan liars. 
4) It includes a reference to the psychological (behavioural) act of lying.

The sentence “This sentence is false”, on the other hand, has no extension or reference. So in what sense can it really be said to be the logical form of the sentence “All Cretans are liars”? If anything, the sentence “All Cretans are liars” is more acceptable than “This sentence is false” because the latter has no content and is paradoxical; whereas the former both has content and is paradoxical.

Of course one can argue that the extension (or reference) of “This sentence is false” is the sentence itself (i.e., even if the sentence itself has no references or extensions). Thus the sentence “This sentence is false” can only have an empty inscription as its reference or extension. But an empty inscription can’t be a proposition (or statement). It’s a mere collection of words with an acceptable grammar or syntax. That is, a grammar or syntax without any semantic content.

Here’s another take. One possibility is that the statement

“All Cretans are liars.”

should actually be something like the following:

“Except for myself, all Cretans are liars.”

The above is a more natural — and less problematic — expression of the words “All Cretans are liars”. However, the universal quantifier “all” ( or ∀ in logic) is negated by the proceeding clause “Except for myself”. This also has the consequence of making the entire sentence no longer paradoxical.

Of course all this hinges on the quantifier “all” and the problems self-reference throw up. In logic, it’s often agreed that quantifiers nearly always have a restricted range (or domain) which is determined by specific contexts. Does that mean that the word “all” in “All Cretans are liars” also has a restricted range? Is the speaker of the words “All Cretans are liars” that very restriction (or exception) himself? Yet if we take the word “all” literally, then he can’t be. However, if we take the word “all” contextually or as a quantifier with a restricted range, then the Cretan liar may well be that very exception. After all, in natural-language terms (therefore also in terms of context), many people would be happy to accept that when a person says that “All people are evil”, then he may well be exempting himself from that universal generalisation. Indeed if someone were to say (out loud) that “All people always remain silent”, then (by definition) he must be an exception to his own universal generalisation.

Following on from all that, it’s also the case that the liar sentence can be seen as being neither true nor false. But here again it depends on which version we’re talking about. It seems easier to believe that the sentence “All Cretans are liars” is neither true nor false than the sentence “This sentence is false”. Why is that? Because it hardly makes sense to say of an empty sentence that it is neither true nor false. On the other hand, it makes more sense to say that the sentence “All Cretans are liars” is neither true nor false in that this may make sense of its paradoxical nature.

So there’s no actual truth-gap when it comes to the sentence “This sentence is false” because — arguably — the issue of truth or falsehood can’t arise in the first place for an empty sentence. Again, a truth-gap result may make much more sense for the sentence “All Cretans are liars” than it does for “This sentence is false”.

Strictly speaking, then, if any approach to logic is purely syntactical in nature, then truth and falsehood (at least as most people see such things) simply aren’t the issue. Having said that, if the symbols and operations in such a logic are correctly adhered to, then why can’t we call its results (or conclusions) “true” and “false”? After all, since what’s called “the nature of truth” has created controversy throughout the history of philosophy, then why can’t we be at least a little logically ad hoc about this matter?

If you are looking for the answer to the question “which one is best suited philosophy of mathematics?” I’m afraid this article can’t give it to you. But I do encourage you to work on it. In fact, to develop a good answer to the question “what the entire mathematical reasoning is about?” you will probably need to outrun the three main contemporary philosophies of mathematics: formalism, logicism, and intuitionism. Being the three leading scientists of each: Hilbert (formalist), Frege (logicist), and Poincaré (intuitionist). Or, it’s possible you will choose one of them when you finish reading this article. Let’s see.

These three philosophies emerged shortly after Peano’s five axioms of arithmetic:

“(1) zero is a number. (2) The successor of any number is a number. (3) No two numbers have the same successor. (4) zero is not the successor of any number. (5) Any property which belongs to zero, and also to the successor of every number which has the property, belongs to all numbers.” Bertrand Russell in Introduction to Mathematical Philosophy (1919).

Peano found, indeed, a good way to reduce the entire arithmetic to a few axioms, and his work was deeply respected by a great part of the scientific community. However, Frege continued to ask questions such as: what about the entire mathematics? Are numbers synthetic or analytical objects? In his most famous work Frege stated,

“For a proposition to be true is just not the same thing as for it to be thought.” Frege in The Foudations of Arithmetic (1884).

For him, in the sentence “The North Sea is 10,000 square miles in extent”, ’10,000 square’ miles would be a classification about the North Sea (and by “classification” we could say it means it is a “set”). But in order to establish what kind of set the arithmetical numbers belong to Frege employed a good deal of philosophical and logical definitions. That is, Frege reduced all mathematical entities to a combination of (1) sets of Cantor’s set theory, (2) a few formations rules stipulated by Frege himself, and (3) some philosophical ideas. For him, for example, the North Sea only need to exist in order to be true (it’s a synthetic kind of object), while any concept of number, or any mathematical truth, needs to follow rules in order to be true (therefore, mathematical objects would be analytical).

Frege, as you may notice, was not an empiricist. But was he wrong? Chess, for example, doesn’t represent quite well a war between two kingdoms, but we may define what’s a truly good move in a chess match based on logical rules (that is, based on the rules of the game).

Let’s continue.

Frege was doing a very formidable work for which he is now considered one of the main names of propositional calculus, and the father of analytical philosophy.

However, Russell, in 1902, sent a very famous letter to Frege in which he noticed a paradox in Cantor’s Set Theory (and consequently in Frege’s ideas too). Frege not only admitted there was a paradox, but he also included a note about Russell’s letter on his second tome of The Basic Laws of Arithmetic (which was about to be published/impressed at the occasion).

The paradox was, in fact, previously discovered by Zermelo, but it was Russell who noticed how important it was. This is how we achieve the paradox:

“consider the class of all classes that are not members of themselves. Is this class a member of itself? If it is, then it is not. and if it is not, then it is.” Simon Blackburn in The Oxford Dictionary of Philosophy (2008).

After it was brought to evidence by Russell, logicians and mathematicians were asking, “then, what sets can contain, and what they can’t? Or, in what circumstances they can’t contain another set?” Since then, no one knows what to do.

Russell, it’s true, tried to find a solution, but, as Wittgenstein noticed, he was censoring arbitrarily the paradox. Wittgenstein, in response to Russell, wrote:

“only the description of expressions (i.e., any sentence in logic) may be presupposed.” Wittgenstein in Tractatus Logico-Philosophicus (1921).

So Russell and Frege seemed to have failed to build a complete foundation for mathematics — or, at least, their work was incomplete somehow —. Then, everything was open for discussion and was formed the three main positions on the matter of the foundation of mathematics: logicism, formalism, and intuitionism.

Logicism:

According to Gratann-Guiness,

“the French word ‘Logistique’ was introduced by Couturat and others at the 1904 International Congress of Philosophy, and was used by Russell and others from then on.” Gratann-Guiness in The Search for Mathematical Roots (2000).

The logicist thesis can be summarized as follows,

“pure mathematics is a branch of logic.” Max Black in The Nature of Mathematics (1933).

And some of the most important names of its adepts are Russell, Frege, Couturat, and Zermelo; although, according to Black, we could see in Leibniz

“the germ of the entire logistic conception.” Max Black in The Nature of Mathematics (1933).

The main reason for this interpretation is that the concept of function was introduced by Leibniz, and it turned possible to arithmetize spatial calculations (that is, it was no longer strictly necessary the drawing of geometrical figures to calculate its properties). Let’s see what Leibniz (1646-1716) himself said about it in a letter he sent to Huygens (1629-1695):

“(…) it is often difficult to analyze the properties of a figure by calculation, and still more difficult to find very convenient geometrical demonstrations and constructions, even when the algebraic calculation is completed. But this new characteristic (functions), which follows the visual figures, cannot fail to give the solution, the construction, and the geometric demonstration all at the same time, and in a natural way and in one analysis, that is, through determined procedure.” Leibniz in Gottfried Wilhelm Leibniz Philosophical Papers and Letters (1989).

Leibniz’s functions gave arithmetic the possibility to represent big geometrical calculations. Because of this, mathematicians were discovering new ways to use more symbolical calculations. They were discovering, for example, new possibilities for logic (see Boole’s logic). And then the logicists thought those functions could help describe the foundations of the entire mathematical reasoning.

Formalism:

According to Black, formalists thought pure mathematics was “the science of the formal structure of symbols.” And they rejected the idea that

“mathematical concepts can be reduced to logical concepts.” Max Black in The Nature of Mathematics (1933).

Formalists were, in fact, creating an opposition between logic and mathematics in the heated discussions of the foundations of mathematics.

The leading formalist was David Hilbert. He was the first to use the term ‘formalism’; although, initially, he had no intention to refer to a philosophical position in mathematics with it. According to Sinaceur,

“In his essays on the foundations of mathematics, Hilbert did use the German word ‘Formalismus’, but not to characterize a philosophical attitude towards questions on the nature of mathematical objects or practice. ‘Formalismus’ meant ‘formal system’ or ‘formal language’, both technical concepts of mathematical logic.” The term was used, according to him (my note), “only in the technical sense and” and he explained (my note) “that the use of formulas, i.e. formalization, is a necessary tool of logical investigation.” Sinaceur in Logicism, intuitionism, and formalism: What has become of them? (2019).

‘Formalism’ was first taken as a philosophical posture by Luitzen Brouwer. For Brouwer, Hilbert was grounding mathematics

“in a logical, i.e. a non-mathematical, conviction of legitimacy.” Sinaceur in Logicism, intuitionism, and formalism: What has become of them? (2019).

Brouwer refused Hilbert’s philosophy, which was summarized by Hilbert himself as follows

“If the arbitrarily given axioms do not contradict one another, then they are true, and the things defined by the axioms exist.” David Hilbert in Gottlob Frege: Philosophical and Mathematical Correspondence (1980).

Brouwer’s criticisms were indeed pertinent, however, Hilbert’s methods achieved the first consistent foundation for the Euclidean geometry (for which the entire scientific community was really pleased).

Here are two other important names of the formalist philosophy: Carnap and Quine.

• Carnap is considered one of the greatest logic positivists (or logic empiricists), and he also

“was one of the originators of the new field of ‘philosophy of science’ and later a leading contributor to semantics and inductive logic.” Leitgeb and Carus in The Stanford Encyclopedia of Philosophy: Rudolf Carnap (2020).

His opinion on the foundations of mathematics was:

“The formalist view is right in holding that the construction of the system can be effected purely formally, that is to say without reference to the meaning of the symbols.” Carnap in The Logical Syntax of Language (1937).

• Quine, on the other hand, is considered one of the most influential philosophers of the second half of the 20th. In his texts about ontology, philosophy of language, and epistemology he achieved (in a quite indirectly way) a philosophy of mathematics. Let’s see. He rejected

“the attempt to ground knowledge of the external world in allegedly transcendent and self-validating mental experience.” Duignan in Encyclopedia Britannica: Willard Van Orman Quine (2019).

For him,

“The proper task of a ‘naturalized epistemology’, as he saw it, was simply to give a psychological account of how scientific knowledge is actually obtained.” Duignan in Encyclopedia Britannica: Willard Van Orman Quine (2019).

And he understood that

“mathematical theories are part and parcel of scientific theories, they too are confirmed by experience.” Horsten in The Stanford Encyclopedia of Philosophy: Philosophy of Mathematics (2019).

So Quine thought ‘experience’ would be responsible to decide if a proposition is true or false in the formal science which is mathematics. This thesis was part of his promise of a holistic unification between analytical and synthetic propositions. If this unification were verified, then formalism would only need an empirical basis to ground its fundamental propositions.

Intuitionism:

Brouwer was the first to assign a philosophical meaning to the terms ‘intuitionism’ and ‘formalism’. And he saw no distinction between logicism and formalism. From his point of view, formalists and logicists were attesting only a linguistic existence of mathematical entities.

According to Kreisel,

“the real opposition between Brouwer’s and Hilbert’s approach was (…) between the conception of what constitutes a foundation.” Kreisel in Hilbert’s Programme (1958).

Brouwer believed philosophy of mathematics needed not only consistency (like Hilbert). For Brouwer, it was needed intuitions to justify the “existence” of mathematical entities. However, this intuitionist demand seemed either little pragmatic or little mathematically productive (that is, it does not reach important abstractions that could be linked to the calculation of mathematical entities).

Besides Brouwer, there were other important names in intuitionism like Kronecker (considered a pre-intuitionist), and Heyting. But none of them was more famous than Poincaré. Poincaré was trying to defend Kant from the attacks of the “new logic” (Frege, for example, wrote directly against Kant’s philosophy of mathematic). In fact, Kant’s idea of ‘pure forms of intuition founding the entire rational thought’ was losing popularity with the emergence of some “counter-intuitive” scientific theories like Darwin’s theory of evolution, non-Euclidean geometry, and, later, Einstein’s relativity. So Poincaré was trying to reconcile the pure forms of Kant’s transcendental intuition with what he understood as the inventiveness of scientists.

In short:

• Logicism: the foundation of mathematics can be achieved by logical elements like formation rules, or ‘grammatical’ rules, and some philosophical notions.
• Formalism: formal elements can ground mathematics, but not necessarily logical elements(and I would say the less philosophical the better for them).
• Intuitionism: points out non-formal, but “intuitive” subjects, as fundamental for the foundation of mathematics. And I would say they do not reject deep philosophical questions about the nature of mathematics.

I was born on the 2nd of August, exactly 33 years before my father was born. I always taught the fact of sharing the birthday with my dad was something really unique. I don’t even have two friends who were born on the same day.

I never really thought about the math of two people having the same birthday. If one day a friend of mine hadn’t talked to me about the birthday paradox, I probably never would. He said, “You’re at a party. There are exactly 23 people in the room, what are the odds of two people sharing the same birthday?” I said, “I don’t know but I think they are pretty low.”

“Actually, it is more than 50%.”

“Shut up! No way that’s true!”

As it turns out, it is.

Introduction

To solve this problem we have to answer a simple question:

How many people do we need to have the probability of two of them sharing the same birthday be more than 50%?

Before we begin we have to make some assumptions. First, we consider a year of 365 days (no leap years, sorry). This means that to have a 100% probability we need 366 people. The second assumption is that all the 365 birthdays are equally likely. In reality, this is not true but the results are affected only slightly. Actually, this is the worst case (learn more know here)

Solution

Let’s start simple. What’s the chance that two people share the same birthday? The first person can be born on any day of the year, this means that the probability is 365/365 = 1. The second person has to be born on the same day as the first and there is a 1/365 chance of that happening.

These two events need to happen at the same time so the probability is:

Not very high as we expected.

Now we can consider a group of three people (let’s call them A, B, C). To know the probability of at least two people sharing their birthday we have to calculate:

1. Probability of A and B having the same birthday.
2. Probability of B and C having the same birthday.
3. Probability of A and C having the same birthday.
4. Probability of A, B, and C having the same birthday.

This is not too difficult. However, if we want to do the same for a larger number of people the calculations needed would be too many.

Thankfully, we can use a little trick.

We want to calculate the probability that two people are born on the same day, which we call p(B), but it’s more simple to do the opposite. So we’re going to compute the probability of two people not sharing their birthday, and we call this p’(B).

When we have p’(B), to calculate the probability p(b) all we have to do to get the result is p(B) = 1-p’(B)

Let’s start simple. The probability that two people don’t have the same birthday is p’(B)

The 365/365 term means that the first person can be born on any day of the year. However, if we want that the second person doesn’t share the birthday with the first one, we have to exclude that day from the number of possible birthdays for the second person.

We can do the same for three people and the result is this

You probably already guessed where we’re going with this. If we apply this principle for 23 people the result we get is

This means that the probability of two people not sharing the birthday is 49.3% if there are 23 people.

Now to calculate the probability we simply have to do

There we have it, if we take a group of 23 people, it is more likely that two of them share their birthday than not.

To better visualize the result it is useful to plot in a graph the two probability we have calculated, p’(B) and p(B).

(A) Probability of two people not sharing their birthday (B) Probability of two people sharing the birthday

To be more specific, here are the probabilities of two people sharing their birthday:

• For 23 people the probability is 50.7%
• For 30 people the probability is 70.6%
• For 40 people the probability is 89.1%
• For 50 people the probability is 97.0%
• For 75 people the probability is 99.97%
• As the number of people increases the probability gets more closer to 100%. It is exactly 100% for 366 people.

Conclusion

Now you may be wondering why is this problem a paradox. And you would be right because it is not. However, the fact that there’s more than a 50% chance that two people are born on the same in a small group of 23 people, is really counter-intuitive.

The main reason is that if we are in a group of 23 and we compare our birthday with the others, we think we’re making only 22 comparisons. This means that there are only 22 chances of sharing the birthday with someone.

However, we don’t make only 22 comparisons. That number is much larger and it is the reason we perceive this problem as a paradox.

In fact, the second person was compared with the first one, so he/she has 21 comparisons to make. The third one has to do 20 and so on. To get the total number of comparisons we have to do:

So in total, we make 253 comparisons. Those are a lot more than the 22 we thought we were making at the beginning.

In conclusion, the counter-intuitive nature of this problem is the reason everyone refers to it as the birthday paradox even though it isn’t.

From the Big bang to the Heat death of the universe

It’s not because you’re stupid or weren’t concentrating in school

Two envelopes with different amounts of money in them. Choose the better one with a higher chance than fifty-fifty!

Natural numbers were created by God, everything else is the work of men — Kronecker (1823–1891).

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The nature of subsymmetries

The British writer, mathematician and logician Charles Lutwidge Dodgson (which was Lewis Carroll’s real name) worked in the fields of geometry, matrix algebra, mathematical logic and linear algebra. Dodgson was also an influential logician. (He introduced the Method of Trees; which was the earliest use of a truth tree.

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Who Says Nature is Mathematical?

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Particle Physics

The Discovery of Spontaneous Radioactivity

Sweety, let me see what you got inside.

Group Theory

Group Theory

The nature of symmetry and the symmetry of nature

Philosophy

Is ‘Philosophy of Mathematics’ somehow useful?

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Geometry

Horrocks’ Measurements of How Far Away The Sun Is

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Discrete Mathematics

How Many Distances Must Be Made from N Points?

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“When we met, was I walking to the faculty club or away from it? I’m wondering, because in the latter case I’ve already had my lunch”

American mathematician Norbert Wiener (1894–1964) was by all accounts, a very peculiar man. After graduating from high school at 11 years old, he entered Tufts College and within three years was awarded an A.B. in mathematics. Before the age of 18, Harvard had awarded him a Ph.D. for his dissertation in mathematical logic. Described by author Sylvia Nasar as:

“An American John von Neumann, a polymath of great originality who made stunning contributions in pure mathematics and then embarked on a second and equally astounding career in applied mathematics”

Wiener would be the man to give modern meaning to the word ‘feedback’ through his invention of cybernetics (the study of regulatory systems) which has since birthed important subfields such as artificial intelligence, robotics, neuroscience, computer vision and others.

His career achievements notwithstanding, Wiener is perhaps even more so remembered for his extraordinary personality. According to one of his biographers, the great man spent 30 years “wandering the halls of MIT, like a duck”, one of its most well-regarded and renowned professors of mathematics, especially known for his absent-mindedness:

“His office was a few doors down the hall from mine. He often visited my office to talk to me. When my office was moved after a few years, he came in to introduce himself. He didn’t realize I was the same person he had frequently visited; I was in a new office so he thought I was someone else.”— Phyllis L. Block

“He went to a conference and parked his car in the big lot. When the conference was over, he went to the lot but forgot where he parked his car. He even forgot what his car looked like. So he waited until all the other cars were driven away, then took the car that was left.” — Howard Eves

This is the story of the “myopic parody of an absentminded professor”, Dr. Norbert Wiener.

Early Life (1894–1909)

Norbert Wiener in 1901, at the age of 7 (Photo: Courtesy MIT Museum)

Norbert Wiener was born in Missouri in 1894 to Leo Wiener and Bertha Kahn, both of Jewish origin. His father Leo was by the time Norbert was born already a renowned scholar of history and languages, having attended the University of Warsaw in 1880 and later the Friedrich-Wilhelm-Universtät in Berlin. A polyglot, his father knew languages well. As Norbert recalled in his autobiography Ex Prodigy (1953) multilinguism was the custom at the time when his father grew up:

“German was the language of the family, and Russian that of the State. […] He learned French as the language of educated society; and in Eastern Europe, especially Poland, there were still those who adhered to the Renaissance tradition and used Italian as another language of polite conversation.”

His father however took the custom to an extreme. By the time he was ten years old, Leo reportedly spoke equally as many languages. Over the course of his life, he would reportedly master a total of 34, including Gaelic, multiple American Indian languages as well as the language spoken by the Bantu people of Sub-Saharan Africa.

Bertha and Leo Wiener (Photos: Courtesy MIT Museum left, right)

Leo met Norbert’s mother Bertha while working as a teacher in Kansas City and married her in 1883. Bertha was “a small, pretty woman, […] remembered as a practical, sociable and “folksy” housewife” by the people in the small town where she and her husband later settled. The two married in 1893, a year before Norbert was born. Their son was named after the leading man in Robert Browning’s dramatic poem In a Balcony.

Child prodigy

Even from his earliest years, Norbert showed signs of remarkable mental agility. At eighteen months old, as his father Leo recounted, Norbert learned the alphabet by watching his caretaker draw letters in the sand at a beach:

When he was 18 months old, his nurse-girl one day amused herself by making letters in the sand of the seashore. She noticed that he was watching her attentively, and in fun she began to teach him the alphabet. Two days afterward she told me, in great surprise, that he knew it perfectly.

Norbert at age 9 (Photo: Courtesy of MIT Museum left, right)

His mother Bertha would read to Norbert from a very young age. By age three however, Norbert was the one reading to her. His father Leo — by then professor at Harvard University — would also tutor him on the floor of his study. Young Norbert loved science books and on his third birthday was given a copy of Wood’s Natural History, which he devoured. He was still below school age when his father’s training began. According to Wiener himself, at that point his schooling consisted mainly of informal lectures in his father fields of expertise (languages and literature) including the Greek and Latin classics, followed by Leo’s favorite German poets and philosophers as well as Darwin and Huxley. Norbert was not yet six years old.

Although his son was clearly gifted, Leo was also a strict tutor who upheld very high standards. When Norbert made mistakes, his father reportedly turned “extremely critical and harsh” as Wiener in his autobiography Ex Prodigy (1953) later recalled:

Algebra was never hard for me, although my father’s way of teaching it was scarcely conducive to peace of mind. Every mistake had to be corrected as it was made. He would begin the discussion in an easy, conversational tone. This lasted exactly until I made the first mathematical mistake. Then the gentle and loving father was replaced by the avenger of blood.

Despite still being physically immature, by the time Norbert was seven his father had placed him in the progressive Peabody School in Cambridge, Massachusetts. His young age notwithstanding, he was initially placed in the third grade before swiftly being advanced to the fourth, but still did not fit in. His reading skills were excellent, but, paradoxically, his aptitude for mathematics appeared lacking. Upon learning that this was the result of Norbert’s boredom from “routine drills of memorization”, Leo subsequently pulled his son back out of school and continued on for another three years with his “radical experiment in home schooling” (Conway and Siegelman, 2005).

The world first came to know of Norbert Wiener on October 7th 1906, when a portrait of the boy genius adorned the front page of Joseph Publitzer’s New York Worldunder the inconspicuous headline “The Most Remarkable Boy in the World”. The featured article included interviews with both Norbert and his father, in a tone seemingly approving of his father’s unconventional approach to early child development:

The boy, Norbert, had learned his letters at eighteen months. Under his father’s tutelage, he began reading at three, reciting in Greek and Latin at five, and in German soon after. At seven he took up chemistry, by nine algebra, geometry, trigonometry, physics, botany, and zoology, and that fall, at eleven, he had entered Tufts College in the neighboring town of Medford after only three and a half years of formal schooling.

– Excerpt, “The Most Remarkable Boy in the World” in New York World, October 7th (1906)

Norbert’s Relationship with Leo

“My closest mentor and dearest antagonist”

As described by physicist Freeman Dyson (1923–2020) in his essay “The Tragic Tale of a Genius” in The New York Review of Books (2005):

“While he was growing up and trying to escape from his notoriety as a prodigy at Tufts and Harvard, Leo was making matters worse by trumpeting Norbert’s accomplishments in newspapers and popular magazines” — Freeman Dyson

Young Norbert Wiener (Photo: Courtesy of MIT Museum)

Indeed, Norbert’s father trumped his educational ideas publicly, in addition to the New York World article, in issues of the Boston Evening Record, the American Journal of Pediatrics and American Magazine. Indeed, Leo Wiener “made no bones about his intentional molding of Norbert and his sisters to make them geniuses” (Heims, 1980):

Professor Leo Wiener, of Harvard University […] believes that the secret of precocious mental development lies in early training. […] He is the father of four children, ranging in age from four to sixteen; and he has had the courage of his convictions in making them the subjects of an educational experiment. The results have […] been astounding, more especially in the case of his oldest son, Norbert.

– Excerpt, American Magazine, July (1911)

Regarding his methods, Leo’s public accounts differ significantly from those which he gave his son. For instance, in reading Norbert’s fathers’ own account of how his children came to be so talented, one gets the sense that praise and acknowledgement of his children’s abilities were not a large part of his philosophy (Heims, 1980):

“It is nonsense to say, as some people do, that Norbert and Constance and Bertha are unusually gifted children. They are nothing of the sort. If they know more than other children of their age, it is because they have been trained differently.”

“Norbert is lazy and doesn’t study as much as the average boy his age” — Leo Wiener, 1906

Left: Leo Wiener (Photo: Courtesy of MIT Museum). Right: Norbert’s dedication to his father at the beginning of his best-selling book “The Human Use of Human Beings” (1950)

More than anything, from Norbert’s own writing one gets the sense that he was indeed rather negatively affected by his fathers’ claims: (Wiener, 1953):

I felt that Father himself was not immune to the temptation to grant interviews to the slicks about me and my training […]. In these interviews, he emphasized that I was essentially an average boy who had the advantage of superlative training.

This image of having been “created” by one’s father, combined with a lack of acknowledgement both of his talents, efforts and sacrifices, left a lasting impression on Wiener. “It rendered me more diffident as to my own ability that I would have otherwise been even under my father’s scolding.”

“In short, I had the worst of both worlds”

However, as clear from the interview with Leo in the New York World article, his father indeed did in fact know how gifted his son was, even though he was loathed to admit it to Norbert:

“I hate to talk about the boy, not because I am not proud of him but because it might get to his ears and spoil him. He has a keen analytical mind and a tremendous memory. He doesn’t learn by rote, as a parrot might, but by reasoning.”

Education (1903–1913)

Following his father’s homeschooling, Norbert entered Ayer High School in 1903 at the age of nine, and was soon advanced even further:

It soon became clear that the greater part of my work belonged to the third year of high school, so when the year was over I was transferred to the senior class.

Following his graduation in 1906, his father “decided to send […] him to Tufts College rather than risk the strain of the Harvard entrance examinations”. Norbert, by then twelve years old, as always abided diligently.

Tufts University (1906–1909)

The young Wiener entered Tufts College in Massachusetts in the fall of 1906. There, he studied Greek and German, physics and mathematics, in addition to biology:

In spite of this interest in biology, it was in mathematics that I was graduated. I had studied mathematics every year in college […] found calculus and differential equations quite easy, and I used to discuss them with my father who was thoroughly oriented in the ordinary college mathematics.

He graduated cum laude with an A.B. degree in 1909, at 14 years of age.

Graduation photos from Tufts College in 1909 and Harvard University in 1913 (Photos: Courtesy MIT Museum left, right)

Harvard University (1909–1913)

“I was nearly fifteen years old, and I had decided to make my try for the doctor’s degree in biology”

After graduating from college, Wiener went on to enter graduate school at Harvard University (where his father worked ) to study zoology. This despite the objections of Leo, who “was rather unwilling to concur in it. He had thought it might be possible for me to go to medical school” (Wiener, 1953). However, the emphasis on laboratory work combined with Wiener’s poor eyesight made zoology a particularly difficult specialization for him. His rebellion was not long lasting, and after a time, Wiener decided to follow his father’s advice and instead take up philosophy.

As usual the decision was made by my father. He decided that such success as I had made as an undergraduate at Tufts in philosophy indicated the true bent of my career. I was to become a philosopher.

Wiener was offered a scholarship to the Sage School of Philosophy at Cornell University, and transferred there in 1910. However, after a “black year” (Wiener, 1953) of feeling insecure and out-of-place, he transferred back to Harvard Graduate School in 1911. Originally intending to work with philosopher Josiah Royce (1855–1916) for his Ph.D. in mathematical logic, due to the latter’s onset illness, Wiener had to recruit his former professor at Tufts College — Karl Schmidt — to take his place. Schmidt, who Wiener himself later stated was “then a young man, vigorously interested in mathematical logic” was the person who inspired him to investigate a comparison between the algebra of relatives of Ernst Schroeder (1841–1902) and that of Whitehead and Russell’s Principia Mathematica (Wiener, 1953):

There was a lot of formal work to be done on this topic which I found easy; though later, when I came to study under Bertrand Russell in England, I learned that I had missed almost every issue of true philosophical significance. However, my material made an acceptable thesis, and it ultimately led me to the doctor’s degree.

His dissertation in philosophy, highly mathematical, was in formal logic. The essential results of his dissertation were published the following year in the 1914 paper “A simplification in the logic of relations” in the Proceedings of the Cambridge Philosophical Society. The coming fall, Wiener traveled to Europe to do postdoctoral work in the hopes that he might eventually land a permanent position on the faculty of one America’s most prominent universities.

Postdoctoral Work (1913–1915)

Following his doctoral dissertation, defense and graduation from Harvard, Wiener — then 18 years old — was awarded one of the school’s prestigious one-year graduate fellowships to study abroad. His chosen destination was Cambridge, England.

Cambridge University (1913–1914)

“Leo Wiener hand-delivered his son to Bertrand Russell”

Norbert Wiener first arrived at Trinity College, Cambridge in September of 1913. Traveling with him was his entire family, spearheaded by his father Leo who had seized on the opportunity to take a year of sabbatical from Harvard and join his son in Europe. As Conway & Siegelman (2005) describe, “Young Wiener strode through the great gate of Trinity College, Cambridge, the Mecca of modern philosophy and the new mathematical logic, with his father in lock-step behind him”.

Wiener went to Cambridge to continue his study of philosophy with one of the authors of the Principia Mathematica which had been the focus of his dissertation at Harvard. Lord Bertrand Russell (1872–1970) — at that point in his early forties — was by 1913 considered the foremost philosopher of the Anglo-American world following the praise of his’ and Alfred North Whitehead’s monumental three-volume work, published in 1910, 1912 and 1913. The Principia or “PM” as it is often known, was at that point the most complete and coherent piece of mathematical philosophy to date. Renowned still for its rigor, the work among other efforts, infamously grounded the theory of addition to logic by proving, in no less than thirty pages, the validity of the proposition that 1+1 = 2.

Despite having been brought up at the heft of a polyglot “Harvard Don”, Wiener’s first impression of Russell’s fierce personality left something to be desired, as he would soon communicate to his father in letter-form:

Russell’s attitude seems to be one of utter indifference mingled with contempt. I think I shall be quite content with what I shall see of him at lectures

Russell’s impression of Wiener, or at least what he let him on to believe, appeared mutual. “Apparently, young Wiener did not “sense data” or do philosophy the way the titan of trinity prescribed it” (Conway & Siegelman, 2005):

Excerpt, letter from Norbert to Leo Wiener (1913)
”My course-work under Mr. Russell is all right, but I am completely discouraged about the work I am doing under him privately. I guess I am a failure as a philosopher […] I made a botch of my argument. Russell seems very dissatisfied […] with my philosophical ability, and with me personally. He spoke of my views as “horrible fog”, said that my exposition of them was even worse than the views themselves, and […] accused me of too much self-confidence and cock-sureness […] His language, though he excused himself, it is true, was most violent.”

As with his father Leo, sadly, Russell’s opinion of Norbert, then 18 years old, was not as harsh as he himself had believed. In his private papers, Russell indeed noted approvingly of the boy and after reading Norbert’s dissertation commented that it was “a very good technical piece of work”, giving the young student a copy of the third volume of the Principia as a gift (Conway & Siegelman, 2005). 

Read more about Wiener and Russell’s relationship in a previous newsletter:

PrivatdozentOn Norbert Wiener’s Relationship with Bertrand RussellNorbert Wiener (1894–1964) and Bertrand Russell (1872-1970) The later “Father of Cybernetics” Norbert Wiener (1894–1964) first arrived at Trinity College, Cambridge in September of 1913. Traveling with him was his entire family, spearheaded by his father Leo who had seized on the opportunity to take a year of sabbatical from his professorship at Harvard …Read more3 years ago · 6 likes · Jørgen Veisdal

The single most important take-away of Wiener from his work with Russell, however, was neither physical nor related to philosophy. Rather, it was the Lord’s suggestion that the young Wiener look up four papers from 1905 by physicist Albert Einstein, which he would later make use of. Wiener himself at time time singled out G.H. Hardy (1877–1947) as having the most profound influence on him (Wiener, 1953):

Hardy’s course […] was a revelation to me […] [in his] attention to rigor […] In all my years of listening to lectures in mathematics, I have never heard the equal of Hardy for clarity, for interest, or for intellectual power. If I am to claim any man as my master in my mathematical thinking, it must be G.H. Hardy.

In particular, Wiener credited Hardy for introducing him to the Lebesgue integral which “lead directly to the main achievement of my early career”.

Göttingen University (1914)

One experience richer, Wiener in 1914 continued to Göttingen University. He arrived in the spring after briefly stopping by to visit his family in Munich. Although only staying for a single term, his time there would be crucial to his further development as a mathematician. He assumed the study of differential equations under David Hilbert (1862–1943), perhaps the foremost mathematician of his era whom Wiener would later laud as “the one really universal genius of mathematics”.

Wiener remained in Göttingen until the outbreak of World War I in June of 1914, when he decided to return to Cambridge and continue his studies of philosophy with Russell.

Career (1915-)

Prior to being hired at MIT — an institution he would remain with for the rest of his life — Wiener worked a number of somewhat odd jobs, in various industries and cities in America. He officially returned to the United States in 1915, briefly living in New York City while continuing studies in philosophy at Columbia University with philosopher John Dewey (1859–1952). After that, he went on to teach philosophy courses at Harvard and next accepted a job as an engineer apprentice at General Electric. After that, he joined Encyclopedia Americana in Albany, New York after his father had secured him a job as a staff writer there, “convinced that with my clumsiness I could never really make good at engineering“ (Wiener, 1953). He also worked briefly for the Boston Herald.

With America’s entry into World War I, Wiener was eager to contribute to the war effort, and attended a training camp for officers in 1916, but ultimately failed to earn a commission. In 1917 he tried again to join the military, but was rejected due to his poor eyesight. The next year, Wiener was invited by mathematician Oswald Veblen (1880–1960) to contribute to the war effort by working on ballistics in Maryland:

I received an urgent telegram from Professor Oswald Veblen at the new Proving Ground in Aberdeen, Maryland. This was my chance to do real war work. I took the next train to New York, where I changed for Aberdeen

Mathematicians in uniform at Aberdeen Proving Grounds in 1918, Wiener on the far right (Photo: Courtesy of MIT Museum)

His experiences at the Proving Ground transformed Wiener, according to Dyson (2005). Before arriving there, he was a 24-year-old mathematical prodigy who had been discouraged away from mathematics due to the failings of his first teaching job at Harvard. Afterwards, he was re-invigorated by the applications of his teachings on real world problems:

We lived in a queer sort of environment, where office rank, army rank, and academic rank all played a role, and a lieutenant might address a private under him as ‘Doctor’, or take orders from a sergeant. When we were not working on the noisy hand-computing machines which we knew as ‘crashers’, we were playing bridge together after hours using the same computing machines to record our scores. Whatever we did, we always talked mathematics.

Mathematics

Photo: Courtesy of MIT Museum

In his extensive bibliography of published writings, Wiener’s first two publications in mathematics appeared in the 17th issue of the Proceedings of the Cambridge Philosophical Society in 1914, the latter of which is now lost:

• Wiener, N. (1914). “A Simplification of the Logic of Relations”. Proceedings of the Cambridge Philosophical Society 17, pp. 387–390.
• Wiener, N. (1914). “A Contribution to the Theory of Relative Position”. Proceedings of the Cambridge Philosophical Society 17, pp. 441–449.

The first work, which regarded mathematical logic, was according to Wiener “presented on 23 February 1914 by G. H. Hardy” despite “exciting no particular approval on the part of Russell”. In the note, Wiener introduces the “dissymmetry between the two elements of an ordered pair by using the null set”. The work, which was the main result of his Ph.D. thesis at Harvard, proved how the mathematical notion of relations can be defined by set theory, thereby showing that the theory of relations does not require any distinct axioms or primitive notions.

Wiener’s most well known mathematical contributions were however mostly made between the ages of 25 and 50, in the years 1921–1946. As a mathematician, Chatterji (1994) singles out Wiener’s skillful utilization of the Lebesgue type integration theory (which Hardy had introduced him to in Cambridge) as a unique hallmark of his art. The Lebesgue integral extends the traditional integral to a larger class of functions and domains.

Following the end of World War I, Wiener tried to secure a position at Harvard, but was rejected, likely to the university’s anti-semitism at the time, often attributed to the influence of Department Head G. D. Birkhoff (1884–1944). Instead, Wiener assumed the position of lecturer at MIT in 1919. From that point on, his research output increased significantly.

In the first five years of his career at MIT, he published 29 (!!) single-authored journal papers, notes and communications in various subfields of mathematics, including:

• Wiener, N. (1920). “A Set of Postulates for Fields”. Transactions of the American Mathematical Society 21, pp. 237–246.
• Wiener, N. (1921). “A New Theory of Measurement: A Study in the Logic of Mathematics”. Proceedings of the London Mathematical Society, pp. 181–205.
• Wiener, N. (1922). “The Group of the Linear Continuum”. Proceedings of the London Mathematical Society, pp. 181–205.
• Wiener, N. (1921). “The Isomorphisms of Complex Algebra”. Bulletin of the American Mathematical Society 27, pp. 443–445.
• Wiener, N. (1923). “Discontinuous Boundary Conditions and the Dirichlet Problem”. Transactions of the American Mathematical Society, pp. 307–314.

The Wiener Process (1920-23)

Wiener first became interested in Brownian motion when was in Cambridge studying under Russell, who directed him to the “miracle year” work of Albert Einstein. In his 1905 paper Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhended Flüssigkeiten suspendierten Teilchen (“On the Motion of Small Particles Suspended in a Stationary Liquid, as Required by the Molecular Kinetic Theory of Heat”), Einstein modeled the irregular motion of a pollen particle as being moved by particular individual water molecules. This “irregular motion” had first been observed by botanist Robert Brown in 1827, but had not yet been investigated formally in mathematics.

Wiener approached the phenomenon from the perspective that “it would be mathematically interesting to develop a probability measure for sets of trajectories” (Heims, 1980):

A prototype kind of problem Wiener considered is that of the drunkard’s walk: a drunk man is at first leaning against a lamp post; he then takes a step in some direction-it may be a short step or a long step; then he either stands still maintaining his balance or takes another step in some direction; and so on. The path he takes will in general be a complicated path with many changes in direction. 

[…]

Assuming the man has no a priori preference for any particular direction or particular step size and may move fast or slowly according to his whim, is there some way to assign a probability measure to any particular set of trajectories?

– Excerpt, John von Neumann and Norbert Wiener by Steve Heims (1980)

Example of a one-dimensional Wiener process/Brownian motion

Wiener extended Einstein’s formulation of Brownian motion to describe such trajectories, and so established a link between the Lebesgue measure (a systematic way of assigning numbers to subsets) and statistical mechanics. That is, Wiener provided the mathematical formulation for describing the one-dimensional curves left behind by Brownian processes. His work, now often referred to as the Wiener process in his honor, was published in a series of papers developed in the period 1920–23:

• Wiener, N. (1920). “The Mean of a Functional of Arbitrary Elements”. Annals of Mathematics 22 (2), pp. 66–72.
• Wiener, N. (1921). “The Average of an Analytic Functional”. Proceedings of the National Academy of Sciences 7 (9), pp. 253–260.
• Wiener, N. (1921). “The Average of an Analytic Functional and the Brownian Movement”. Proceedings of the National Academy of Sciences 7 (10), pp. 294–298.
• Wiener, N. (1923). “Differential Space”. Journal of Mathematics and Physics 2, pp. 131–174.
• Wiener, N. (1924). “The Average Value of a Functional”. Proceedings of the London Mathematical Society 22, pp. 454–467.

As Wiener himself testified, although neither of these papers solved physical problems, they did however provide a robust mathematical framework which was later used by von Neumann, Bernhard Koopman (1900–1981) and Birkhoff to address problems in statistical mechanics originally posed by Willard Gibbs (1839–1903).

Wiener with Max Born in Göttingen in 1925 (Photo: George H. Davis, Jr. Courtesy MIT Museum and MIT Historical Collections)

Return to Göttingen (1924–1926)

On the strength of his work in the early 1920s, Wiener returned to Göttingen during the summers of 1924–26, the latter year as a Guggenheim scholar. The height of the so-called golden age of quantum physics, his stays overlapped the visiting stays of both von Neumann and Oppenheimer, the former of which he came to know personally and correspond with.

In the summer of 1925, Wiener lectured on his work to the assembled group of mathematicians at Göttingen, both on faculty and visiting, later writing home that Hilbert had described his work as sehr schön (“very beautiful”). Towards the end of the stay, the head of the mathematics department, Richard Courant (1888–1972), reportedly told him that if he came back the following year he would be given the position of visiting professor.

Wiener–Khinchin Theorem (1930)

Starting after his stay in Göttingen, Wiener also began working in applied mathematics, and in 1930 on so-called autocorrelation functions which provide the correlation between a signal and a delayed copy of that signal, as a function of its delay. The so-called Wiener-Khinchin theorem shows how the autocorrelation function Rₓₓ(τ) is related to the power spectral density Sₓₓ(f) via the Fourier transform:

The result was published the same year Wiener was promoted to associate professor at MIT.

• Wiener, N. (1930). “Generalized Harmonic Analysis”. Acta Mathematica. 55, pp. 117–258.

Left: Portrait of Norbert Wiener. (Photo: Courtesy of MIT Museum). Right: The Table of Contents for Wiener’s celebrated text “Generalized Harmonic Analysis” (1930) from Acta Mathematica 55, pp. 117–258.

Wiener’s Tauberian Theorem (1932)

Although by the early 1930s firmly engaged with signal processing and the early developments in electrical engineering, Wiener continued publishing papers in pure mathematics, including works in the analysis on Lebesgue spaces. Wiener’s tauberian theorem, published in 1932, provides the necessary and sufficient condition under which any function in L₁ or L₂ can be approximated by linear combinations of translations of a given function. His results appeared in the chapter

which Wiener begins by describing how:

“Numerous important branches of mathematics and physics concern themselves with the asymptotic behavior of functions for very large of very small values of their arguments”

His tauberian theorem regarded how to approximate such functions at very large values. The same year that the result was published, Wiener was promoted to full professor of mathematics at MIT.

Norbert in his office at MIT, year unknown (Photo: Tekniske Museet Telehistoriska)

Paley-Wiener theorems (1934)

Wiener supervised relatively few Ph.D. students. In Heims (1980) one of them, Norman Levinson (1912–1975), recounts his experience of collaborating with the great man:

He was a most stimulating teacher. He would actually carry on his research at the blackboard. As soon as I displayed a slight comprehension of what he was doing, he handed me the manuscript of Paley-Wiener for revision. I found a gap in a proof and proved a lemma to set it right. Wiener thereupon sat down at his typewriter, typed my lemma, affixed my name and sent it off to a journal.

– Excerpt, John von Neumann and Norbert Wiener by Heims (1980)

The Paley-Wiener theorems Levinson helped work on are a class of theorems which relate the decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The work appeared in the book

Cybernetics

“Information is information, not matter or energy.”

Left: Wiener testing the special-purpose “correlator” computer. Right: Wiener with Claude Shannon, the so-called “Father of Information Theory” (Photos: Courtesy of MIT Museum, left, right)

The field that is now synonymous with Wiener’s name, came to be largely as a function of the Norbert’s interest in stochastic and mathematical noise processes, which are prevalent in both electrical engineering and communication theory. In a lecture entitled “Men, Machines, and the World About Them” Wiener describes how his pioneering work first came to be as the result of him trying to contribute to the war effort in the 1940s:

There were two converging streams of ideas that brought me into cybernetics. One of them was the fact that in the last war, when it was manifestly coming, but before Pearl Harbor at any rate, when we were not yet in the conflict, I tried to see if I could find some niche in the war effort at that time.

As Wiener himself states in the lecture, his first effort in the nascent theory of digital computing was not considered sufficiently immediate to be effective in that war, and so Wiener went looking for something else. His second initiative related to weaponry, specifically anti-aircraft defense:

I looked around for another thing and the great question that was being discussed at that time was anti-aircraft defense. It was the time of the Battle of England and the existence of the United States as a competent country, the survival of anybody to combat with Germany seemed to depend on anti-aircraft defense.

Now, the anti-aircraft gun is a very interesting type of instrument. In the First World War, the anti-aircraft gun had been developed as a firing instrument, but one still used range tables directly by hand for firing the gun. That meant essentially that one had to do all the computation while the plane was flying overhead. And actually by the time you got in position to do something about it, the plane had already dome something about it and wasn’t there.

Photo: Courtesy of MIT Museum

And so, Wiener goes on, “this lead to some very interesting mathematical theories. I had some ideas that turned out to be useful there”. He worked on the problem with Julian Bigelow (1913–2003). “Early in 1941, the two men took over an empty classroom on the second floor of the mathematics department, Building 2, Room 244, and set to work on the blackboard. Wiener outlined the fire control problem for his new collaborator” (Conway & Siegelman, 2005):

“Wiener and Bigelow considered the observer, gun, airplane, and pilot as an integrated, probabilistic system. The odds favored the pilot: in 1940 only one out of about 2,500 antiaircraft shells scored a hit. In a preliminary report, they explained how they intended “to place the analysis of the problem of prediction upon a purely statistical basis, by determining to what extent the motion of target is predictable on the basis of known facts and history, and to what extent the motion of the target is not predictable”.

– Excerpt, Turing’s Cathedral by George Dyson (2012)

An audio recording of Wiener’s lecture “Men, Machines, and the World About Them” from 1950 is available below. He begins speaking at 13:30:

Wiener Filter (1942)

Wiener’s work on the anti-aircraft fire-control problem resulted in his invention of a filter used to compute a statistical estimate of an unknown signal by using an input and filtering it to produce the estimate of an output. The filter built on several of Wiener’s previous results on integrals and Fourier transforms. Although developed at Radiation Laboratory at MIT, the result was published in a classified document. The first unclassified document describing the filter appeared in Wiener’s 1949 book “Extrapolation, Interpolation, and Smoothing of Stationary Time Series”.

As the war had ended in 1947 he was invited to a congress on harmonic analysis held in Nancy, France. The congress was organized by the secretive French mathematical society ‘Bourbaki’ in collaboration with mathematician Szolem Mandelbrojt (1899–1983), uncle of Benoit Mandelbrot (1925–2010), later discoverer of the Mandelbrot-set. While there, Wiener was offered to write a manuscript on the ‘unifying character of the part of mathematics which is found in the study of Brownian motion and telecommunication engineering’. The following year, he coined the neologism ‘cybernetics’ to denote the study of such ‘teleological mechanisms’. His manuscript would form the basis of the popular science work “Cybernetics: Or Control and Communication in the Animal and the Machine”, published by MIT Press/Wiley and Sons in 1948. The book was heralded (Wikipedia, 2020):

“A beautifully written book, lucid, direct, and, despite its complexity, as readable by the layman as the trained scientist, if the former is willing to forego attempts to understand mathematical formulas” — The Saturday Review (1949)

“Its scope and implications are breathtaking, and leaves the reviewer with the conviction that it is a major contribution to contemporary thought” — Philosophy of Science 22 (1955)

“One of the most influential books of the twentieth century, Cybernetics has been acclaimed as one of the ‘seminal works’ comparable in ultimate importance to Galileo or Malthus or Rousseau or Mill” — The New York Times (1964)

Personality

“He hardly seemed to know where he was”

Wiener being interviewed by Charles Romine for the CBS educational TV-show “The Search” (1954) (Photo: Courtesy of MIT Museum)

His genius and extraordinary scientific contributions notwithstanding, Wiener is perhaps even better known for being famously eccentric. ”His appearance alone was remarkable […] He puffed on fat cigars. He waddled like a duck, a myopic parody of an absentminded professor.” Extremely near-sighted, the daughter of a colleague recalls Wiener being “A fascinating man, but such a child. Everybody took care of him. He used to sleep upstairs and nobody could go to sleep because of his snoring. It was horrible. Then, in the morning, he’d come padding into the wrong room, wandering around, because he couldn’t see” (Conway & Siegelman, 2005).

“While lecturing, he might be seen picking his nose “energetically,” with no apparent concern for his lack of social grace.”

Indeed, numerous accounts exist of various people interacting with and/or observing the great man, deep in thought, oblivious of his surroundings:

“It is vivid in my memory that Professor Wiener would always come to class without any lecture notes. He would first take out his big handkerchief and blow his nose very vigorously and noisily. He would pay very little attention to his class and would seldom announce the subject of his lecture. He would face the blackboard, standing very close to it because he was extremely near-sighted. Although I usually sat in the front row, I had difficulty seeing what he wrote. Most of the other students could not see anything at all.”

– Excerpt, Recollections of a Chinese Physicist by C.K. Jen (1990)

“On at least one occasion, he strode into the wrong classroom and delivered a rousing lecture to an audience of baffled students.”

In the essay collection “Mathematical Conversations – Selections from the Mathematical Intelligencer”, writer and mathematician Steven G. Krantz (1951-) recounts the following similar anecdote as additional evidence of Wiener’s modus operandi:

“When he walked the halls of MIT he invariably read a book, running his finger along the wall to keep track of where he was going. One day, engaged in this activity, Wiener passed a classroom where a class was in session. It was a hot day and the door had been left open. But of course Wiener was unaware of these details – he followed his finger through the door, into the classroom, around the walls (right past the lecturer) and out the door again.”

– Excerpt, Mathematical Conversations by Krantz (2001)

Wiener Testimonial Party in 1961 (Photo: Courtesy of MIT Museum)

Biographers Conway & Siegelman (2005 p. 31) trace the permission for Wiener to indulge in his eccentricities back to his time as a postdoctoral fellow at Trinity College, Cambridge, where he first “saw all around him that stately bastion of high intellect and fading aristocracy a range of eccentricity that raised peculiarity to an art form”. Unlike Harvard, which Wiener later said “has always hated the eccentric and the individual”, in Cambridge “eccentricity is so highly valued that those who do not really possess it are forced to assume it for the sake of appearances”. The same sentiment was echoed by biographer Sylvia Nasar when describing the hot-house atmosphere of MIT’s math department in the 1950s:

“Showing off wasn’t regarded as a crime if you knew your stuff. Lack of social graces was considered part and parcel of being real mathematicians. “Their attitudes were famously nonbourgeois, exhibitionistic, dissolute,” Felix Browder recalled. If anything, all of them placed a certain premium on eccentricity and outrageousness.”

– Excerpt, A Beautiful Mind by Sylvia Nasar (1998)

“An MIT alumnus was driving in New Hampshire and stopped to help a tubby-looking man with a flat tire. He recognized Norbert Wiener and asked if he could help. Wiener asked if [the alumnus] knew him. Yes, he said, he had taken calculus with him. `Did you pass?’ asked Wiener. `Yes.’ `Then you can help me,’ said Wiener. — Robert K. Weatherall, Vice President for Alumni at MIT”

Of course, his eccentricity served only to fuel the legend of Professor Norbert Wiener at MIT:

“MIT legends depicted him scribbling away furiously and ambidextrously on math department blackboards, solving simultaneous sets of complex equations, one with each hand” (Conway & Siegelman, 2005)

Depression

“Norbert Wiener was a great thinker and a tortured person.” — Benoit Mandelbrot

Gregarious as he was, Wiener’s mental life cannot have been a simple one. Starting when he was a child (in addition to being extraordinarily gifted) Norbert was often described as both careful and sensitive. He once described being “filled with gloom” from walking passed a hospital for “incurables” and even at an advanced age recalled how he was “instilled an abhorrence of suffering” from visiting a blacksmith whose toe had been crushed by a horse (Conway and Siegelman, 2005).

If a child or a grandchild of mine should be as disturbed as I was, I should take him to a psychoanalyst, not with confidence that the treatment would be successful in some definitive way, but at least with the hope that there might be a certain understanding and a certain measure of relief — Norbert Wiener

From reading the descriptions by people who knew Wiener and from his own autobiographies, it is clear that he struggled with feelings of inferiority. Likely not unrelated to his upbringing at the hands of his father Leo, these feelings extended beyond purely mathematical endeavors and also into other parts of his life (Krantz, 2001):

“When he played bridge at lunch with a group of friends, he would invariably say, every time he bid or played, “Did I do the right thing? Was that a good play? His partner Norman Levison, would patiently reassure him each time that he couldn’t have done any better.”

– Excerpt, Mathematical Anecdotes by Stephen G. Krantz (1990)

Indeed, Wiener was notoriously insecure. According to Nasar (1998), he would ask anxiously if his name appeared in whatever books people were reading:

“At his low points, he fell prey to paralyzing depressions that drove him to threaten suicide frequently in the confines of his home and family, and at times among his MIT colleagues.” — Nasar (1998)

“When he became famous, he hounded his faculty colleagues to learn “what others at MIT thought of him.” When the talk turned to people at other institutions, “his first question was ‘What do they think of my work?’” — Conway & Siegelman (2005)

According to renowned Nobel Laureate economist Paul Samuelson (1915–2009), also of MIT, the lack of acceptance from Harvard did not help:

“The exodus from Harvard dealt a lasting psychic trauma to Norbert Wiener. It did not help that his father was a Harvard professor […] or that Norbert’s mother regarded his move as a cruel comedown in life” — Paul Samuelson, 1964

Wiener himself, perhaps not unsurprisingly, believed that “society’s understanding of mental illness could benefit greatly from the lessons afforded by the new brain-like computing machines” (Conway & Siegelman, 2005):

He suggested that many human “functional” (as opposed to organic) mental disorders were “fundamentally diseases of memory, of the circulating information kept by the brain in the active state”

[…] 

He explained how modern problems such as “malignant worry,” anxiety attacks, and other classic neurotic disorders might start with a relatively trivial concern and “build itself up into a process totally destructive to the ordinary mental life,” just as a computer caught up in a logical paradox “may go into a circular process which there seems to be no way to stop”.

– Excerpt, Dark Hero of the Information Age by Conway & Siegelman (2005)

Personal Life

“We are not the stuff that abides, but patterns that perpetuate themselves.”

Wiener playing chess with his daughter Peggy (Photos: Courtesy of MIT Museum)

Wiener’s parents arranged his marriage in 1926, to a German immigrant named Margaret Engemann. Despite this circumstance, the two remained together for the rest of their lives, and had two children, Barbara and Margaret “Peggy”. They lived in Cambridge, Massachusetts. His insecurities, absent-mindedness and propensity for depression notwithstanding, by all accounts Wiener was a good father and a great friend:

“Wiener took his paternal responsibilities seriously, and in particular sought to avoid the pattern of education imposed on him by his father. In appraising him as a father, his alter-emotional life, which must have made him difficult, have to be balanced against his lively, imaginative companionship and his warm interest in the welfare of his daughters.”

– Excerpt, John von Neumann and Norbert Wiener by Steve J. Heims (1980)

In fact, stories of Wiener’s gentle and caring demeanor are many.

“Perhaps because of his own psychological struggles, Wiener had an acute empathy for other people’s trials. […] When a younger colleague was writing a book but couldn’t afford a typewriter, Wiener showed up at his door with a Royal portable under his arm.” — Nasar (1998)

The widow of his graduate student Norman Levinson described how in the fall of 1933, Wiener “arranged for Levinson to spend the next academic year in England studying higher mathematics, as he had, with G. H. Hardy at Cambridge”, even caring for his parents when he was gone:

“Norbert Wiener went to the little working class slum where Norman’s parents lived to reassure them. The whole time Norman was in England, Norbert came to their house, usually on Saturdays, and he would talk to them, not about his theorems, but about nice, practical things, about England, landladies, tea time, high table”.— Fagi Levinson

Wiener with his wife Margaret, daughters “Peggy” and Barbara and son-in-law Gordon Raisbeck (Photo: Courtesy of MIT Museum)

Death (1964)

Norbert Wiener died on March 18th, 1964 of a heart attack in Stockholm, Sweden where he gave a lecture at the Royal Academy of Sciences. He was 69 years old. As the news reached MIT,

“Work came to a halt as people gathered to share the news and their memories, and the institute’s flags were lowered to half staff in honour of the fallen institute professor who had roamed its halls for forty-five years.”

– Excerpt, Dark Hero of the Information Age by Conway & Siegelman (2005)

Those interested in reading more about the life of Norbert Wiener are encouraged to look up two books: Dark Hero of the Information Age* by Conway & Siegelman (2005) and Harmonies of Disorder* by Leone Montagnini (2017). 

References

• Chatterji, S.D. 1994. The Mathematical Work of Norbert Wiener (1894–1964). Kybernetes 26 (6/7). p. 34.
• Conway, F. & Siegelman, J. 2005. Dark Hero of the Information Age. Basic Books.
• Dyson, F. 2005. The Tragic Tale of a Genius. The New York Review of Books.
• Dyson, G. 2012. Turing’s Cathedral. Pantheon Books. New York, NY.
• Heims, S.J. 1980. John von Neumann and Norbert Wiener. The MIT Press.
• Kranz, S.G. 2001. Mathmatical Anecdotes in Mathematical Conversations: Selections from the Mathematical Intelligencer by Wilson & Gray. Springer-Verlag, New York, NY.
• Montagnini, L. 2017. Harmonies of Disorder. Springer.
• Nasar, S. 1998. A Beautiful Mind. Simon & Schuster.
• Wiener, N. 1953. Ex-prodigy. The MIT Press.

* This essay contains Amazon Affiliate Links

In 1896, Becquerel had registered the very first spontaneous radioactivity while working on x-rays. A mere accident allowed him to detect remarkable, not-seen-before phenomenon.

To some degree, all discoveries are like rolling a dice. What, once thought out, is kind of funny; because all initial conditions must be met, without any knowledge about them. Like walking around a place would surprisingly lead to discovering Atlantis. From radioactivity through penicillium and far beyond, this scenario repeats all the time.

Becquerel’s experimenting set was:

• photographic plates wrapped in thin black paper to prevent appearing of casual images;
• fluorescent mineral, for they can absorb radiation: potassium uranyl sulfate.

He used this mineral to expose it to the sunlight for absorbing radiation, which became much brighter with time; I am sure you have played with fluorescent materials in your lifetime. Once it was bright enough, he put it on the plate wrapped in the protective black paper and waited. He let it be that way for a moment, after which he took the orb aside and unwrapped the plate from its protective layer. Unsurprisingly, the orb’s shadow image appeared on there, meaning emission of the earlier absorbed radiation had to get through the protective layer; at least, he thought that was the case.

However, Becquerel was about to change his mind. On 26th February 1896, he had prepared probes, as usual, but decided to abandon testing as it was an overcast day, so he put them in a drawer. He had not known yet, but this decision was his life’s work.

As he was unwrapping the plate from protective paper, he was expecting a very dim image at best, and I can only imagine how shocking seeing the bright one had to be for him.

There is only one explanation for this fact; namely, the orb had to radiate itself. Becquerel had added two to two and put it forward as a hypothesis.

To work it out, Becquerel changed the experiment slightly. He enclosed the orb inside a lead block that had a narrow but deep opening made to let it in. If the hypothesis was right, an image should appear on the screen only above it. And indeed, it was the case.

But, it is not over yet. At that time, Maxwell’s equations were few decades old, and so, he knew that magnetic fields influence paths of electrical charges.

He had repeated the experiment in the presence of a magnetic field and discovered that a single point on the screen now had appeared at three places. Meaning that a single beam had to triple: today are known as α, β, γ rays.

This way, he discovered spontaneous radioactivity for which got a Nobel Prize in 1903, along with Marie and Pierre Curie, who advanced investigation upon it.