Today the usual definition of a perfect number is in terms of its divisors, but early definitions were in terms of the ‘aliquot parts’ of a number.

An aliquot part of a number is a proper quotient of the number. So for example the aliquot parts of 10 are 1, 2 and 5. These occur since 1 = ¹⁰/₁₀, 2 = ¹⁰/₅, and 5 = ¹⁰/₂. Note that 10 is not an aliquot part of 10 since it is not a proper quotient, i.e. a quotient different from the number itself. A perfect number is defined to be one which is equal to the sum of its aliquot parts.

The four perfect numbers 6, 28, 496 and 8128 seem to have been known from ancient times and there is no record of these discoveries.

6 = 1 + 2 + 3,
28 = 1 + 2 + 4 + 7 + 14,
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064

The first recorded mathematical result concerning perfect numbers which is known occurs in Euclid’s Elements written around 300BC. It may come as a surprise to many people to learn that there are number theory results in Euclid’s Elements since it is thought of as a geometry book. However, although numbers are represented by line segments and so have a geometrical appearance, there are significant number theory results in the Elements. The result which is if interest to us here is Proposition 36 of Book IX of the Elements which states [2]:-

If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect.

Here ‘double proportion’ means that each number of the sequence is twice the preceding number. To illustrate this Proposition consider 1 + 2 + 4 = 7 which is prime. Then

(the sum) × (the last) = 7 × 4 = 28,

which is a perfect number. As a second example, 1 + 2 + 4 + 8 + 16 = 31 which is prime. Then 31 × 16 = 496 which is a perfect number.

Now Euclid gives a rigorous proof of the Proposition and we have the first significant result on perfect numbers. We can restate the Proposition in a slightly more modern form by using the fact, known to the Pythagoreans, that

1 + 2 + 4 + … + 2^k-1 = 2^k – 1.

The Proposition now reads:-

If, for some k > 1, 2^k – 1 is prime then 2^k-1(2^k – 1) is a perfect number.

The next significant study of perfect numbers was made by Nicomachus of Gerasa. Around 100 AD Nicomachus wrote his famous text Introductio Arithmetica which gives a classification of numbers based on the concept of perfect numbers. Nicomachus divides numbers into three classes: the superabundant numbers which have the property that the sum of their aliquot parts is greater than the number; deficient numbers which have the property that the sum of their aliquot parts is less than the number; and perfect numbers which have the property that the sum of their aliquot parts is equal to the number (see [8], or [1] for a different translation):-

Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed to one another; as for those that occupy the middle position between the two, they are said to be perfect. And those which are said to be opposite to each other, the superabundant and the deficient, are divided in their condition, which is inequality, into the too much and the too little.

However Nicomachus has more than number theory in mind for he goes on to show that he is thinking in moral terms in a way that might seem extraordinary to mathematicians today (see [8], or [1] for a different translation):-

In the case of the too much, is produced excess, superfluity, exaggerations and abuse; in the case of too little, is produced wanting, defaults, privations and insufficiencies. And in the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort – of which the most exemplary form is that type of number which is called perfect.

Now satisfied with the moral considerations of numbers, Nicomachus goes on to provide biological analogies in which he describes superabundant numbers as being like an animal with (see [8], or [1]):-

… ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms, or having too many fingers on one of its hands….

Deficient numbers are compared to animals with:-

a single eye, … one armed or one of his hands has less than five fingers, or if he does not have a tongue…

Nicomachus goes on to describe certain results concerning perfect numbers. All of these are given without any attempt at a proof. Let us state them in modern notation.

(1) The nth perfect number has n digits.
(2) All perfect numbers are even.
(3) All perfect numbers end in 6 and 8 alternately.
(4) Euclid’s algorithm to generate perfect numbers will give all perfect numbers i.e. every perfect number is of the form 2^k-1(2^k – 1), for some k > 1, where 2^k – 1 is prime.
(5) There are infinitely many perfect numbers.

We will see how these assertions have stood the test of time as we carry on with our discussions, but let us say at this point that assertions (1) and (3) are false while, as stated, (2), (4) and (5) are still open questions. However, since the time of Nicomachus we do know a lot more about his five assertions than the simplistic statement we have just made. Let us look in more detail at Nicomachus’s description of the algorithm to generate perfect numbers which is assertion (4) above (see [8], or [1]):-

There exists an elegant and sure method of generating these numbers, which does not leave out any perfect numbers and which does not include any that are not; and which is done in the following way. First set out in order the powers of two in a line, starting from unity, and proceeding as far as you wish: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096; and then they must be totalled each time there is a new term, and at each totalling examine the result, if you find that it is prime and non-composite, you must multiply it by the quantity of the last term that you added to the line, and the product will always be perfect. If, otherwise, it is composite and not prime, do not multiply it, but add on the next term, and again examine the result, and if it is composite leave it aside, without multiplying it, and add on the next term. If, on the other hand, it is prime, and non-composite, you must multiply it by the last term taken for its composition, and the number that results will be perfect, and so on as far as infinity.

As we have seen this algorithm is precisely that given by Euclid in the Elements. However, it is probable that this methods of generating perfect numbers was part of the general mathematical tradition handed down from before Euclid’s time and continuing till Nicomachus wrote his treatise. Whether the five assertions of Nicomachus were based on any more than this algorithm and the fact the there were four perfect numbers known to him 6, 28, 496 and 8128, it is impossible to say, but it does seem unlikely that anything more lies behind the unproved assertions. Some of the assertions are made in this quote about perfect numbers which follows the description of the algorithm [1]:-

… only one is found among the units, 6, only one other among the tens, 28, and a third in the rank of the hundreds, 496 alone, and a fourth within the limits of the thousands, that is, below ten thousand, 8128. And it is their accompanying characteristic to end alternately in 6 or 8, and always to be even.

When these have been discovered, 6 among the units and 28 in the tens, you must do the same to fashion the next. … the result is 496, in the hundreds; and then comes 8128 in the thousands, and so on, as far as it is convenient for one to follow.

Despite the fact that Nicomachus offered no justification of his assertions, they were taken as fact for many years. Of course there was the religious significance that we have not mentioned yet, namely that 6 is the number of days taken by God to create the world, and it was believed that the number was chosen by him because it was perfect. Again God chose the next perfect number 28 for the number of days it takes the Moon to travel round the Earth. Saint Augustine (354-430) writes in his famous text The City of God :-

Six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect…

The Arab mathematicians were also fascinated by perfect numbers and Thabit ibn Qurra wrote the Treatise on amicable numbers in which he examined when numbers of the form 2ⁿp, where p is prime, can be perfect. Ibn al-Haytham proved a partial converse to Euclid’s proposition in the unpublished work Treatise on analysis and synthesis when he showed that perfect numbers satisfying certain conditions had to be of the form 2^k-1(2^k – 1) where 2^k – 1 is prime.

Among the many Arab mathematicians to take up the Greek investigation of perfect numbers with great enthusiasm was Ismail ibn Ibrahim ibn Fallus (1194-1239) who wrote a treatise based on the Introduction to arithmetic by Nicomachus. He accepted Nicomachus’s classification of numbers but the work is purely mathematical, not containing the moral comments of Nicomachus. Ibn Fallus gave, in his treatise, a table of ten numbers which were claimed to be perfect, the first seven are correct and are in fact the first seven perfect numbers, the remaining three numbers are incorrect. For more details of this impressive work see [6] and [7].

At the beginning of the renaissance of mathematics in Europe around 1500 the assertions of Nicomachus were taken as truths, nothing further being known concerning perfect numbers not even the work of the Arabs. Some even believed the further unjustified and incorrect result that 2^k-1(2^k – 1) is a perfect number for every odd k. Pacioli certainly seems to have believed in this fallacy. Charles de Bovelles, a theologian and philosopher, published a book on perfect numbers in 1509. In it he claimed that Euclid’s formula 2^k-1(2^k – 1) gives a perfect number for all odd integers k, see [10]. Yet, rather remarkably, although unknown until comparatively recently, progress had been made.

The fifth perfect number has been discovered again (after the unknown results of the Arabs) and written down in a manuscript dated 1461. It is also in a manuscript which was written by Regiomontanus during his stay at the University of Vienna, which he left in 1461, see [14]. It has also been found in a manuscript written around 1458, while both the fifth and sixth perfect numbers have been found in another manuscript written by the same author probably shortly after 1460. All that is known of this author is that he lived in Florence and was a student of Domenico d’Agostino Vaiaio.

In 1536, Hudalrichus Regius made the first breakthrough which was to become common knowledge to later mathematicians, when he published Utriusque Arithmetices in which he gave the factorisation 2¹¹ – 1 = 2047 = 23 . 89. With this he had found the first prime p such that 2^p-1(2^p – 1) is not a perfect number. He also showed that 2¹³ – 1 = 8191 is prime so he had discovered (and made his discovery known) the fifth perfect number 2¹²(2¹³ – 1) = 33550336. This showed that Nicomachus’s first assertion is false since the fifth perfect number has 8 digits. Nicomachus’s claim that perfect numbers ended in 6 and 8 alternately still stood however. It is perhaps surprising that Regius, who must have thought he had made one of the major breakthroughs in mathematics, is virtually unheard of today.

J Scheybl gave the sixth perfect number in 1555 in his commentary to a translation of Euclid’s Elements. This was not noticed until 1977 and therefore did not influence progress on perfect numbers.

The next step forward came in 1603 when Cataldi found the factors of all numbers up to 800 and also a table of all primes up to 750 (there are 132 such primes). Cataldi was able use his list of primes to show that 2¹⁷- 1 = 131071 is prime (since 750² = 562500 > 131071 he could check with a tedious calculation that 131071 had no prime divisors). From this Cataldi now knew the sixth perfect number, namely 2¹⁶(2¹⁷ – 1) = 8589869056. This result by Cataldi showed that Nicomachus’s assertion that perfect numbers ended in 6 and 8 alternately was false since the fifth and sixth perfect numbers both ended in 6. Cataldi also used his list of primes to check that 2¹⁹ – 1 = 524287 was prime (again since 750² = 562500 > 524287) and so he had also found the seventh perfect number, namely 2¹⁸(2¹⁹ – 1) = 137438691328.

As the reader will have already realised, the history of perfect numbers is littered with errors and Cataldi, despite having made the major advance of finding two new perfect numbers, also made some false claims. He writes in Utriusque Arithmetices that the exponents p = 2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37 give perfect numbers 2^p-1(2^p – 1). He is, of course, right for p = 2, 3, 5, 7, 13, 17, 19 for which he had a proof from his table of primes, but only one of his further four claims 23, 29, 31, 37 is correct.

Many mathematicians were interested in perfect numbers and tried to contribute to the theory. For example Descartes, in a letter to Mersenne in 1638, wrote [8]:-

… I think I am able to prove that there are no even numbers which are perfect apart from those of Euclid; and that there are no odd perfect numbers, unless they are composed of a single prime number, multiplied by a square whose root is composed of several other prime number. But I can see nothing which would prevent one from finding numbers of this sort. For example, if 22021 were prime, in multiplying it by 9018009 which is a square whose root is composed of the prime numbers 3, 7, 11, 13, one would have 198585576189, which would be a perfect number. But, whatever method one might use, it would require a great deal of time to look for these numbers…

The next major contribution was made by Fermat. He told Roberval in 1636 that he was working on the topic and, although the problems were very difficult, he intended to publish a treatise on the topic. The treatise would never be written, partly because Fermat never got round to writing his results up properly, but also because he did not achieve the substantial results on perfect numbers he had hoped. In June 1640 Fermat wrote to Mersenne telling him about his discoveries concerning perfect numbers. He wrote:-

… here are three propositions I have discovered, upon which I hope to erect a great structure. The numbers less by one than the double progression, like

1  2  3   4   5   6    7    8    9    10    11    12    13
1  3  7  15  31  63  127  255  511  1023  2047  4095  8191

let them be called the radicals of perfect numbers, since whenever they are prime, they produce them. Put above these numbers in natural progression 1, 2, 3, 4, 5, etc., which are called their exponents. This done, I say

1. When the exponent of a radical number is composite, its radical is also composite. Just as 6, the exponent of 63, is composite, I say that 63 will be composite.
2. When the exponent is a prime number, I say that its radical less one is divisible by twice the exponent. Just as 7, the exponent of 127, is prime, I say that 126 is a multiple of 14.
3. When the exponent is a prime number, I say that its radical cannot be divisible by any other prime except those that are greater by one than a multiple of double the exponent…
   

Here are three beautiful propositions which I have found and proved without difficulty, I shall call them the foundations of the invention of perfect numbers. I don’t doubt that Frenicle de Bessy got there earlier, but I have only begun and without doubt these propositions will pass as very lovely in the minds of those who have not become sufficiently hypocritical of these matters, and I would be very happy to have the opinion of M Roberval.

Shortly after writing this letter to Mersenne, Fermat wrote to Frenicle de Bessy on 18 October 1640. In this letter he gave a generalisation of results in the earlier letter stating the result now known as Fermat’s Little Theorem which shows that for any prime p and an integer a not divisible by p, a^p-1- 1 is divisible by p. Certainly Fermat found his Little Theorem as a consequence of his investigations into perfect numbers.

Using special cases of his Little Theorem, Fermat was able to disprove two of Cataldi’s claims in his June 1640 letter to Mersenne. He showed that 2²³ – 1 was composite (in fact 2²³ – 1 = 47 × 178481) and that 2³⁷ – 1 was composite (in fact 2³⁷ – 1 = 223 × 616318177). Frenicle de Bessy had, earlier in that year, asked Fermat (in correspondence through Mersenne) if there was a perfect number between 10²⁰ and 10²². In fact assuming that perfect numbers are of the form 2^p-1(2^p – 1) where p is prime, the question readily translates into asking whether 2³⁷ – 1 is prime. Fermat not only states that 2³⁷ – 1 is composite in his June 1640 letter, but he tells Mersenne how he factorised it.

Fermat used three theorems:-

(i) If n is composite, then 2ⁿ – 1 is composite.

(ii) If n is prime, then 2ⁿ – 2 is a multiple of 2n.

(iii) If n is prime, p a prime divisor of 2ⁿ- 1, then p – 1 is a multiple of n.

Note that (i) is trivial while (ii) and (iii) are special cases of Fermat’s Little Theorem. Fermat proceeds as follows: If p is a prime divisor of 2³⁷ – 1, then 37 divides p – 1. As p is odd, it is a prime of the form 2 × 37m+1, for some m. The first case to try is p = 149 and this fails (a test division is carried out). The next case to try is 223 (the case m = 3) which succeeds and 2³⁷ – 1 = 223 × 616318177.

Mersenne was very interested in the results that Fermat sent him on perfect numbers and soon produced a claim of his own which was to fascinate mathematicians for a great many years. In 1644 he published Cogitata physica mathematica in which he claimed that 2^p – 1 is prime (and so 2^p-1(2^p – 1) is a perfect number) for

p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257

and for no other value of p up to 257. Now certainly Mersenne could not have checked these results and he admitted this himself saying:-

… to tell if a given number of 15 or 20 digits is prime, or not, all time would not suffice for the test.

The remarkable fact is that Mersenne did very well if this was no more than a guess. There are 47 primes p greater than 19 yet less than 258 for which 2^p – 1 might have been either prime or composite. Mersenne got 42 right and made 5 mistakes. A suggestion as to the rule he used in giving his list is made in [9].

Primes of the form 2^p- 1 are called Mersenne primes.

The next person to make a major contribution to the question of perfect numbers was Euler. In 1732 he proved that the eighth perfect number was 2³⁰(2³¹ – 1) = 2305843008139952128. It was the first new perfect number discovered for 125 years. Then in 1738 Euler settled the last of Cataldi’s claims when he proved that 2²⁹ – 1 was not prime (so Cataldi’s guesses had not been very good). Now it should be noticed (as it was at the time) that Mersenne had been right on both counts, since p = 31 appears in his list but p = 29 does not.

In two manuscripts which were unpublished during his life, Euler proved the converse of Euclid’s result by showing that every even perfect number had to be of the form 2^p-1(2^p – 1). This verifies the fourth assertion of Nicomachus at least in the case of even numbers. It also leads to an easy proof that all even perfect numbers end in either a 6 or 8 (but not alternately). Euler also tried to make some headway on the problem of whether odd perfect numbers existed. He was able to prove the assertion made by Descartes in his letter to Mersenne in 1638 from which we quoted above. He went a little further and proved that any odd perfect number had to have the form

(4n+1)^4k+1 b²

where 4n+1 is prime. However, as with most others whose contribution we have examined, Euler made predictions about perfect numbers which turned out to be wrong. He claimed that 2^p-1(2^p – 1) was perfect for p = 41 and p = 47 but Euler does have the distinction of finding his own error, which he corrected in 1753.

The search for perfect numbers had now become an attempt to check whether Mersenne was right with his claims in Cogitata physica mathematica. In fact Euler’s results had made many people believe that Mersenne had some undisclosed method which would tell him the correct answer. In fact Euler’s perfect number 2³⁰(2³¹ – 1) remained the largest known for over 150 years. Mathematicians such as Peter Barlow wrote in his book Theory of Numbers published in 1811, that the perfect number 2³⁰(2³¹ – 1):-

… is the greatest that ever will be discovered; for as they are merely curious, without being useful, it is not likely that any person will ever attempt to find one beyond it.

This, of course, turned out to be yet one more false assertion about perfect numbers!

The first error in Mersenne’s list was discovered in 1876 by Lucas. He was able to show that 2⁶⁷ – 1 is not a prime although his methods did not allow him to find any factors of it. Lucas was also able to verify that one of the numbers in Mersenne’s list was correct when he showed that 2¹²⁷ – 1 is a Mersenne prime and so 2¹²⁶(2¹²⁷- 1) is indeed a perfect number. Lucas made another important advance which, as modified by Lehmer in 1930, is the basis of computer searches used today to find Mersenne primes, and so to find perfect numbers. Following the announcement by Lucas that p = 127 gave the Mersenne prime 2^p – 1, Catalan conjectured that, if m = 2^p – 1 is prime then 2^m – 1 is also prime. This Catalan sequence is 2^p – 1 where

p = 3, 7, 127, 170141183460469231731687303715884105727, …

Of course if this conjecture were true it would solve the still open question of whether there are an infinite number of Mersenne primes (and also solve the still open question of whether there are infinitely many perfect numbers). However checking whether the fourth term of this sequence, namely 2^p – 1 for p = 170141183460469231731687303715884105727, is prime is well beyond what is possible.

In 1883 Pervusin showed that 2⁶⁰(2⁶¹- 1) is a perfect number. This was shown independently three years later by Seelhoff. Many mathematicians leapt to defend Mersenne saying that the number 67 in his list was a misprint for 61.

In 1903 Cole managed to factorise 2⁶⁷ – 1, the number shown to be composite by Lucas, but for which no factors were known up to that time. In October 1903 Cole presented a paper On the factorisation of large numbers to a meeting of the American Mathematical Society. In one of the strangest ‘talks’ ever given, Cole wrote on the blackboard

2⁶⁷ – 1 = 147573952589676412927.

Then he wrote 761838257287 and underneath it 193707721. Without speaking a work he multiplied the two numbers together to get 147573952589676412927 and sat down to applause from the audience. [It is worth remarking that the computer into which I (EFR) am typing this article gave this factorisation of 2⁶⁷ - 1 in about a second - times have changed!]

Further mistakes made by Mersenne were found. In 1911 Powers showed that 2⁸⁸(2⁸⁹ – 1) was a perfect number, then a few years later he showed that 2¹⁰⁷- 1 is a prime and so 2¹⁰⁶(2¹⁰⁷- 1) is a perfect number. In 1922 Kraitchik showed that Mersenne was wrong in his claims for his largest prime of 257 when he showed that 2²⁵⁷- 1 is not prime.

We have followed the progress of finding even perfect numbers but there was also attempts to show that an odd perfect number could not exist. The main thrust of progress here has been to show the minimum number of distinct prime factors that an odd perfect number must have. Sylvester worked on this problem and wrote (see [20]):-

… the existence of [ an odd perfect number] – its escape, so to say, from the complex web of conditions which hem it in on all sides – would be little short of a miracle.

In fact Sylvester proved in 1888 that any odd perfect number must have at least 4 distinct prime factors. Later in the same year he improved his result to five factors and, over the years, this has been steadily improved until today we know that an odd perfect number would have to have at least eight distinct prime factors, and at least 29 prime factors which are not necessarily distinct. It is also known that such a number would have more than 300 digits and a prime divisor greater than 10⁶. The problem of whether an odd perfect number exists, however, remains unsolved.

Today 46 perfect numbers are known, 2⁸⁸(2⁸⁹- 1) being the last to be discovered by hand calculations in 1911 (although not the largest found by hand calculations), all others being found using a computer. In fact computers have led to a revival of interest in the discovery of Mersenne primes, and therefore of perfect numbers. At the moment the largest known Mersenne prime is 2^43112609 – 1 (which is also the largest known prime) and the corresponding largest known perfect number is 2^43112608(2^43112609 – 1). It was discovered in August 2008 and this, the 45th such prime to be discovered, contains more than 10 million digits. If you wonder why we have not included the number in decimal form, then let me say that it contains about 150 times as many characters as this whole article on perfect numbers. Also worth noting is the fact that although this is the 45th to be discovered, it is not be the 45th largest perfect number as not all smaller cases had been ruled out. A month after this discovery the 46th (but smaller) Mersenne prime was discovered.

Article by: J J O’Connor and E F Robertson

May 2009

MacTutor History of Mathematics
[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Perfect_numbers.html]

My work is possible because of a stunning fact we often overlook: the world can be understood mathematically. The party goer’s misconception reminds us that this fact is not obvious. In this article I want to discuss a big question: “why can maths be used to describe the world?”, or to extend it more provocatively, “why is applied maths even possible?” To do so we need to review the long history of the general philosophy of mathematics — what I will loosely call metamaths.

What is applied maths?

A stunning fact: the world can be understood mathematically.
Before we go any further, we should be clear on what we mean by applied mathematics. I will borrow a definition given by an important applied mathematician of the 20th and 21st centuries, Tim Pedley, the GI Taylor Professor of Fluid Mechanics at the University of Cambridge. In his Presidential Address to the Institute of Mathematics and its Applications in 2004, he said “Applying mathematics means using a mathematical technique to derive an answer to a question posed from outside mathematics.” This definition is deliberately broad — including everything from counting change to climate change — and the very possibility of such a broad definition is part of the mystery we are discussing.

The question of why mathematics is so applicable is arguably more important than any other question you might ask about the nature of mathematics. Firstly, because applied mathematics is mathematics, it raises all the same issues as those traditionally arising in metamaths. Secondly, being applied, it raises some of the issues addressed in the philosophy of science. I suspect that the case could be made for our big question being in fact the big question in the philosophy of science and mathematics. However, let us now turn to the history of metamaths: what has been said about mathematics, its nature and its applicability?

Metamathematics
The long history of mathematics generally lacks a distinction between pure and applied maths. Yet in the modern era of mathematics over, say, the last two centuries, there has been an almost exclusive focus on a philosophy of pure mathematics. In particular, emphasis has been given to the so-called foundations of mathematics — what is it that gives mathematical statements truth? Metamathematicians interested in foundations are commonly grouped into four camps.

Formalists, such as David Hilbert, view mathematics as being founded on a combination of set theory and logic (see Searching for the missing truth), and to some extent view the process of doing mathematics as an essentially meaningless shuffling of symbols according to certain prescribed rules.

Logicists see mathematics as being an extension of logic. The arch-logicists Bertrand Russell and Alfred North Whitehead famously took hundreds of pages to prove (logically) that one plus one equals two.

Intuitionists are exemplified by LEJ Brouwer, a man about whom it has been said that “he wouldn’t believe that it was raining or not until he looked out of the window” (according to Donald Knuth ). This quote satirises one of the central intuitionist ideas, the rejection of the law of the excluded middle. This commonly accepted law says that a statement (such as “it is raining”) is either true or false, even if we don’t yet know which one it is. By contrast, intuitionists believe that unless you have either conclusively proved the statement or constructed a counter example, it has no objective truth value. (For an introduction to intuitionism read Constructive mathematics.)

Plato and Aristotle as depicted in Raphael’s fresco The school of Athens.

Moreover, intuitionists put a strict limit on the notions of infinity they accept. They believe that mathematics is entirely a product of the human mind, which they postulate to be only capable of grasping infinity as an extension of an algorithmic one-two-three kind of process. As a result, they only admit enumerable operations into their proofs, that is, operations that can be described using the natural numbers.

Finally, Platonists, members of the oldest of the four camps, believe in an external reality or existence of numbers and the other objects of mathematics. For a platonist such as Kurt Gödel, mathematics exists without the human mind, possibly without the physical universe, but there is a mysterious link between the mental world of humans and the platonic realm of mathematics.

It is disputed which of these four alternatives — if any — serves as the foundation of mathematics. It might seem like such rarefied discussions have nothing to do with the question of applicability, but it has been argued that this uncertainty over foundations has influenced the very practice of applying mathematics. In The loss of certainty, Morris Kline wrote in 1980 that “The crises and conflicts over what sound mathematics is have also discouraged the application of mathematical methodology to many areas of our culture such as philosophy, political science, ethics, and aesthetics [...] The Age of Reason is gone.” Thankfully, mathematics is now beginning to be applied to these areas, but we have learned an important historical lesson: there is to the choice of applications of mathematics a sociological dimension sensitive to metamathematical problems.

What does applicability say about the foundations of maths?
The logical next step for the metamathematician who bothers to think about the applicability of mathematics would be to ask what each of the four foundational views has to say about our big question. Discussions along this line have been written by a number of mathematicians and scientists, such as Roger Penrose in the book The road to reality, or Paul Davies in his book The mind of god.

I would like to take a different path here by reversing the “logical” next step: I want to ask “what does the applicability of mathematics have to say about the foundations of mathematics?” In asking this question I take for granted that there is no serious disagreement about whether mathematics is applicable: the entire edifice of modern science and technology, depending heavily as it does on the mathematisation of nature, bears witness to this fact.

So what can a formalist say to explain the applicability of mathematics? If mathematics really is nothing other than the shuffling of mathematical symbols in the world’s longest running and most multiplayer game, then why should it describe the world? What privileges the game of maths to describe the world rather than any other game? Remember, the formalist must answer from within the formalist worldview, so no Plato-like appeals to a deeper meaning of maths or hidden connection to the physical world is allowed. For similar reasons, the logicists are left floundering, for if they say “well, perhaps the universe is an embodiment of logic”, then they are tacitly assuming the existence of a Platonic realm of logic which can be embodied. This turns logicism into a mere branch of platonism, which, as we will shall see below, comes with its own grave problems. Thus for both formalists and non-platonist logicists the very existence of applicable mathematics poses a problem apparently fatal to their position.

Neither logicism nor formalism is widely believed any more, despite the cliché that mathematicians are platonists during the week and formalists at the weekend. Both perspectives fell out of favour for reasons other than the potentially fatal one about the applicability of mathematics, reasons largely connected with the work of Gödel, Thoralf Skolem, and others. (See Gödel and the limits of logic.)

Is the world inherently mathematical or is maths a construct of the human mind?
The third proposed foundation, intuitionism, never really garnered much support in the first place. To this day, it is muttered about in dark tones by most working mathematicians, if it is considered at all. What is seen as a highly restricted toolkit for proofs and a bizarre notion of limbo, in which a statement is neither true nor false until a proof has been constructed one way or the other, make this viewpoint unattractive to many mathematicians.

However, the central idea of the enumerable nature of processes in the universe appears to be deduced from reality. The physical world, at least as we humans perceive it, seems to consist of countable things and any infinity we might encounter is a result of extending a counting process. In this way, perhaps intuitionism is derived from reality, from the apparently at-most-countably infinite physical world. It appears that intuitionism offers a neat answer to the question of the applicability of mathematics: it is applicable because it is derived from the world.

However, this answer may fall apart on closer inspection. For a start, there is much in modern mathematical physics, including for example quantum theory, which requires notions of infinity beyond the enumerable. These aspect may therefore lie forever beyond the explicatory power of intuitionistic mathematics.

There is one modern idea which could benefit from the finitist logic of the intuitionists: so-called digital physics. It holds that the Universe is akin to a giant computer. The fundamental particles, for example, are described by the quantum state they happen to be in at a given moment, just as the bit from computer science is defined by its value of 0 or 1. Just like a computer, the Universe is based on information about states and its evolution could in theory be simulated by a giant computer. Hence the digital physics motto, “It from bit”.

But this world view too fails to be truly intuitionistic and seems to sneak in some platonic ideas. The bit of information theory seemingly posits a platonic existence of information from which the physical world is derived.

But more fundamentally, intuitionism has no answer to the question of why non-intuitionistic mathematics is applicable. It may well be that a non-intuitionistic mathematical theorem is only applicable to the natural world when an intuitionistic proof of the same theorem also exists, but this has not been established. Moreover, although intuitionistic maths may seem as if it is derived from the real world, it is not clear that the objects of the human mind need faithfully represent the objects of the physical Universe. Mental representations have been selected for over evolutionary time, not for their fidelity, but for the advantage they gave our forebears in their struggles to survive and to mate.

Created in the image of mathematics
Formalism and logicism have failed to answer our big question. The jury is out on whether inuitionism might do so, but huge conceptual difficulties remain. What, then, of Platonism?

Galileo Galilei, who believed that the world was written in the language of maths, facing the Roman Inquisition for proclaiming that the Earth moved around the Sun. Painting by Cristiano Banti.
Platonists believe that the physical world is an imperfect shadow of a realm of mathematical objects (and possibly of notions like truth and beauty as well). The physical world emerges, somehow, from this platonic realm, is rooted in it, and therefore objects and relationships between objects in the world shadow those in the platonic realm. The fact that the world is described by mathematics then ceases to be a mystery as it has become an axiom: the world is rooted in a mathematical realm.

But even greater problems then arise: why should the physical realm emerge from and be rooted in the platonic realm? Why should the mental realm emerge from the physical? Why should the mental realm have any direct connection with the platonic? And in what way do any of these questions differ from those surrounding ancient myths of the emergence of the world from the slain bodies of gods or titans, the Buddha-nature of all natural objects, or the Abrahamic notion that we are “created in the image of God”?

Indeed, the belief that we live in a divine Universe and partake in a study of the divine mind by studying mathematics and science has arguably been the longest-running motivation for rational thought, from Pythagoras, through Newton, to many scientists today. “God”, in this sense, seems to be neither an object in the space-time world, nor the sum total of objects in that physical world, nor yet an element in the platonic world. Rather, god is something closer to the entirety of the platonic realm. In this way, many of the difficulties outlined above which a platonist faces are identical with those faced by theologians of the Judeo-Christian world — and possibly of other religious or quasi-religious systems.

The secular icon Galileo believed that the “book of the universe” was written in the “language” of mathematics — a platonic statement begging an answer (if not the question) if ever there was one. Even non-religious mathematical scientists today regularly report feelings of awe and wonder at their explorations of what feels like a platonic realm — they don’t invent their mathematics, they discover it. Paul Davies goes further in The Mind of God, and highlights the two-way nature of this motivation. Not only may a mathematician be driven to understand mathematics in a bid to glimpse the mind of God (a non-personal God like that of Spinoza or Einstein), but our very ability to access this “key to the universe” suggests some purpose or meaning to our existence.

In fact, the hypothesis that the mathematical structure and physical nature of the universe and our mental access to study both is somehow a part of the mind, being, and body of a “god” is a considerably tidier answer to the questions of the foundation of mathematics and its applicability than those described above. Such a hypothesis, though rarely called such, has been found in a wide variety of religious, cultural, and scientific systems of the past several millenia. It is not natural, however, for a philosopher or scientist to wholeheartedly embrace such a view (even if they may wish to) since it tends to encourage the preservation of mystery rather than the drawing back of the obscuring veil.

Penrose’s three-worlds diagram.
Roger Penrose has most lucidly illustrated some of this mystery with a three-worlds diagram. The platonic, physical, and mental worlds are the three in question, and he sketches them as spheres arranged in a triangle. A cone then connects the platonic world with the physical: in its most general form, the diagram shows the narrow end of the cone penetrating the platonic world and the wider part penetrating some of the physical world. This is to show that (at least some of) the physical world is embedded in at least some of the platonic world. A similar cone connects the physical to the mental world: (at least some of) the mental world is embedded in the physical world. Finally, and most mysteriously, the triangle is completed by a cone from the mental to the platonic world: (at least some of) the platonic world is embedded in the mental world. Each cone, each world, remains a mystery.

We seem to have reached the rather depressing impasse in which none of the four proposed foundations of mathematics can cope unambiguously with the question of the applicability of mathematics. But I want you to finish this essay instead with the feeling that this is incredibly good news! The teasing out of the nuances of the big question — why does applied mathematics exist? — is a future project which could yet yield deep insight into the nature of mathematics, the physical universe, and our place within both systems as embodied, meaning-making, pattern-finding systems.

from: +magazine

 Every mathematical idea has a story. To remember the idea, just recall the story. In mathematics, the stories are proofs and the endings are theorems. The more you turn a proof into a story, the easier it is to remember the ending. Can you tell me what you did last summer? Of course you can. Did you memorize that? Surely not; there is a context and one thought leads to another. Of course it can get a little tedious recalling a story a hundred times just to get to the ending, so sooner or later one just knows the ending. This is the kind of memorizing that a student should do with mathematics.

Building a solid foundation in math requires a systematic approach. Too many children do not get the broad introduction and ongoing practice that builds confidence and deep understanding. The primary mistakes that parents make in teaching/coaching math are:

• having too narrow a focus. Parents tend to overemphasize arithmetic and
  overlook the other math areas.
• reviewing math concepts out of sequence.

This article will discuss what is an adequately broad approach to teaching math and present how Time4Learning, an online learning system, provides such a foundation for each grade.

A strong math elementary math curriculumteaches these five math strand (yes, there are many other ways of grouping these areas into as few as four and as many as eight different areas but we like this approach):       
* Number Sense and Operations – Arithmetic and place value.                                       * Algebra – From the youngest age, learning to recognize
patterns and sets (“pick the small red fish”) creates the groundwork for working
with unknowns and algebraic variables.
* Geometry and Spatial Sense – When children build on their knowledge of basic shapes, they increase their ability to reason spatially, read maps, visualize objects in space, and eventually use geometry to solve problems.                       * Measurement - Learning how to measure and compare is an important life skill that encompasses the concepts of length, weight, temperature, capacity, time, and money.                                                                                           * Data Analysis and Probability - Using charts, tables, and graphs will help children learn to share and organize information about the world
around them.

What is Time4Learning?                                                                   Time4Learning.com is an online subscription site popular for homeschooling, afterschool, and summer use. As an example, let’s survey how this curriculum builds a a broad math foundation through the shifting it’s focus among these diverse math strands.

Foundation Building – PreSchool and Kindergarten Math – The preschool program combines language arts and preschool math into one integrated learning sequence. It starts with the basics such as following simple instructions given verbally by cartoon
characters such as “Click on the Crayon”. Once the children are interacting successfully, they will learn through a fun set of learning games the basic concepts such as similar and different, quantity, sequence, comparisons, and shapes. Notice that the focus is on learning about sets and features which is pre-algebra. The features and patterns get more complicated and basic geometry is introduced. Then at the end of preschool and in the kindergarten math program, the concepts of comparative quantity and greater and less than are introduced. The focus is not on the simple question of having the kids learn to count up to ten although it is taught.

The Basics of Arithmetic – First to Third Grade Math –Advancing to first grade children will turn their primary focus to numbers and operations. They will learn to add and subtract numbers to one hundred.First grade math will include learning
more about geometrical figures and objects, measurement of length, weight, capacity, time, and temperature, use of money, graphs and charts used for data analysis and prediction, and algebraic patterns. In second grade math children will compare and order whole numbers to one thousand, they will group objects into hundreds, tens and ones, relating the groupings to a written numeral. In numbers up to 1000, the children should know the place value of any designated digit. Second grade math introduces fractions.
By the end of second grade and in the third grade math program, reinforcing math skills met in previous years, children will move on to a more rigorous structure. Third grade word
problems can combine multiple skills in the same problem. Children will work with numbers through the hundred-thousands or more. They learn about decimals in the context of money and get experience with fractions up to 100. Third grade math opens them up to a greater understanding of measurement techniques, geometry and algebraic thinking. It will be a challenging year as they are presented with many new and complex concepts.

Not Just Arithmetic – Fourth and Fifth Grade Math – The major math strands for the fourth grade math curriculum are number sense and operations, algebra, geometry and spatial sense, measurement, and data analysis and probability. This year they are expected to know basic multiplication and division. They will recognize that two fractions are equivalent or non-equivalent and learn to add and subtract fractions using drawings, story
problems and algorithms. During fourth grade, math students use a wide variety of tools and procedures to measure length, area, volume, and perimeter. They investigate angle measures, learning about the common angles of 45°, 90°, and 180° (straight angle). They’ll learn to use these angles as reference for measurement of other angles. During fifth grade math, students master the concepts and mechanics of multiplication and division including the commutative, associative and distributive properties. They are expected to learn to factor and recognize prime numbers to 100 and recognize squares. Fifth grade math students are taught to find factors of numbers including the rules of divisibility and to determine if they are prime or composite. They express whole numbers as products of prime factors and determine the greatest common factor or the least common multiple of
two numbers up to 100 or more. In fifth grade they multiply by powers of 10, demonstrating patterns. They identify and apply rules of divisibility for 2, 3, 4, 5, 6, 9, and 10, and use models to identify perfect squares to 144.

Sixth to Eight Grade Math – More sophisticated geometry, problems, and algebra. In sixth grade math, students build on what they learned in fifth grade math, which led them to the decimal base-10 number system, finding factors of numbers to 100, and multiplication of decimals to hundredths. They compare decimals to fractions, and add, subtract, multiply and divide decimals and fractions. It will be an exciting year full of new,
complex math concepts. During seventh grade math children learn about decimals, percents, exponents, scientific notation, ratios, and square roots. Seventh grade math
opens them up to a greater understanding of measurement techniques, geometry and algebraic thinking. In eighth grade math, students work with positive and negative numbers, exponents, and the order of operations, as well as scientific and standard notation. They learn more about working with whole numbers, fractions, mixed numbers, decimals, and integers.

Summary

There are many successful approaches to curriculum from hands-on applied
mathematics to strict standards-based education. However, all the best curriculum take a broad approach to the math curriculum and struggle to provide
a coherent sequence with ongoing review and reinforcement of previously-learned
skills and concepts. Time4Learning’s math curriculum provides a good example of how to broadly develop skills through the elementary and middle school years.

from: math goodies

Historically, it was regarded as the science of quantity, whether of magnitudes (as in geometry) or of numbers (as in arithmetic) or of the generalization of these two fields (as in algebra). Some have seen it in terms as simple as a search for patterns.

During the 19th Century, however, mathematics broadened to encompass mathematical or symbolic logic, and thus came to be regarded increasingly as the science of relations or of drawing necessary conclusions (although some may see even this as too restrictive).

The discipline of mathematics now covers – in addition to the more or less standard fields of number theory, algebra, geometry, analysis (calculus), mathematical logic and set theory, and more applied mathematics such as probability theory and statistics – a bewildering array of specialized areas and fields of study, including group theory, order theory, knot theory, sheaf theory, topology, differential geometry, fractal geometry, graph theory, functional analysis, complex analysis, singularity theory, catastrophe theory, chaos theory, measure theory, model theory, category theory, control theory, game theory, complexity theory and many more.

source:www.storyofmathematics.com