Praise be to Allaah.

The wording of the hadeeth is not as mentioned in the question. We will quote it here:

It was narrated from Ibn ‘Umar (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) said:  “We are an unlettered nation, we do not write or calculate. The month is such-and-such or such-and-such – meaning sometimes it is twenty-nine and sometimes it is thirty.” (Narrated by al-Bukhaari, 1814; Muslim, 1080)

This hadeeth was narrated concerning the issue of the beginning of the lunar month, and it indicates that there is no need to resort to astronomical calculations in order to know when the month begins. We should rely on visual sighting of the moon when it is new, then we will know that the month has begun. The hadeeth was quoted to explain that we should rely on sighting of the moon and not on calculations. It did not come to urge the ummah to remain ignorant and not to learn arithmetic and all other useful sciences. So this hadeeth does not contradict what Muslims are doing nowadays of seeking education in different branches of science which will benefit their worldly interests. Islam is the religion of knowledge; it calls for knowledge and obliges every Muslim to learn what Allaah has enjoined upon him and to learn whatever rulings of sharee’ah he needs to know about acts of worship and dealings with others. With regard to worldly sciences such as medicine, engineering, agriculture, etc., Muslims have to learn whatever the ummah needs to know. If the Muslims need to manufacture a needle, there has to be someone among them who can learn how to make this needle.

Shaykh al-Islam Ibn Taymiyah wrote a very useful discussion on this hadeeth; there follows a selection of his comments on it:

“The phrase ‘We are an unlettered nation’ is not telling them to be like that. They were unlettered before Islam came, as Allaah says (interpretation of the meaning):

‘He it is Who sent among the unlettered ones a Messenger (Muhammad) from among themselves’

[al-Jumu’ah 62:2]

‘And say to those who were given the Scripture (Jews and Christians) and to those who are illiterates (Arab pagans): Do you (also) submit yourselves (to Allaah in Islam)?’

[Aal ‘Imraan 3:20]

Although this is how they were before the Prophet (peace and blessings of Allaah be upon him) was sent to them, they were not commanded to become like that. Some of the implications of being unlettered were to remain, but as we shall see, they were not commanded to remain as they were in every sense…

 Among the nation to which Allaah sent him were some who could read and write a great deal, just as there were among his Companions. There were also some who could calculate. The Prophet (peace and blessings of Allaah be upon him) was sent with some obligations which involve calculation. It was narrated that when the person he had appointed to take care of the zakaah, Ibn al-Latabiyyah, came to him, he  calculated the total amount collected. He had a number of scribes, such as Abu Bakr, ‘Umar, ‘Uthmaan, ‘Ali, Zayd and Mu’aawiyah, who would write down the Revelation, and would write contracts and write his letters to the people to whom Allaah had sent him, the kings of the earth and the leaders of groups, and to his workers, governors, etc. Allaah says in His Book (interpretation of the meaning):

“… that you might know the number of the years and the reckoning”

[Yoonus 10:5; al-Israa’ 17:12]

This is mentioned in two places in the Qur’aan, where Allaah tells us that He has created [the sun and moon] so that the calculations may be known. 

The word ummi (unlettered or illiterate) comes from the same root as the word ummah (nation). It refers to one who is not distinguished from the rest of the people by knowing how to read or write. It is akin to the word ‘aammi which is used to describe a regular person (one of the ‘aammah or masses) who is not distinguished from them by having knowledge which they do not have. It was also said that the word ummi is connected to the word umm (mother), because he has learned no more than his mother taught him, and so on. 

The distinction which takes him out of the ranks of the regular, unlettered people into the ranks of those who have specialized knowledge may be a virtue in and of itself, as in the case of those who read Qur’aan and understand its meaning, or it may be a means which helps to reach that level of virtue, as when a person is distinguished from them by his being able to write and read. So the one who uses that knowledge to pursue perfection is to be praised, whilst the one who neglects it or uses it for evil purposes is to be condemned. Whoever does away with it in favour of something that is more beneficial is more perfect, and if you can achieve the aim without using these means, then that is even better. 

Thus it is clear that being distinct from the unlettered is of two types. The first nation to which the Prophet (peace and blessings of Allaah be upon him) was sent was the Arabs, through whom the Message was conveyed to all the other nations, because it was sent in their tongue. They were mostly illiterate, and they had no advantage of having knowledge or a scripture etc., although by nature they were more ready to learn than other nations. They were like a plowed field that is ready to be sown, but there was no one to do the task and they had no scripture they could read that had been revealed from Allaah, as the People of the Book had, or any analytical science like the Sabaeans and others had. Writing existed among them, but to a very small extent. They had some knowledge of the type that is acquired by common sense, but which does not make an individual distinct from any other unlettered person; for example, they knew of the Creator, and they held good characteristics in high esteem, and they had some knowledge of astronomy, genealogy and poetry. So they deserved to be called unlettered in all senses of the word, as Allaah said concerning them (interpretation of the meaning):

“‘He it is Who sent among the unlettered ones a Messenger (Muhammad) from among themselves’

[al-Jumu’ah 62:2]

‘And say to those who were given the Scripture (Jews and Christians) and to those who are illiterates (Arab pagans): Do you (also) submit yourselves (to Allaah in Islam)?’

[Aal ‘Imraan 3:20]

So the unlettered or illiterate were described as being opposite to the People of the Book, for the kitaabi (one of the People of the Book) is something other than the ummi (unlettered).

 When he was sent among them, it became obligatory upon them to follow the Book that he brought and to ponder its meanings, understand it and act upon it. Allaah made this Book an explanation for all things, and their Prophet taught them everything, even (the etiquette of) defecation. They became people of a Book and people of knowledge, indeed, they became the most knowledgeable of people and the best of them in beneficial knowledge. They lost the feature of blameworthy and imperfect illiteracy, which is the lack of knowledge and a revealed Book, and they gained knowledge of the Book and wisdom. They inherited the Book, as Allaah said concerning them (interpretation of the meaning):

“He it is Who sent among the unlettered ones a Messenger (Muhammad) from among themselves, reciting to them His Verses, purifying them (from the filth of disbelief and polytheism), and teaching them the Book (this Qur’aan, Islamic laws and Islamic jurisprudence) and Al-Hikmah (As-Sunnah: legal ways, orders, acts of worship of Prophet Muhammad). And verily, they had been before in manifest error”

[al-Jumu’ah 62:2]  

They were unlettered in every sense of the word, but when he taught then the Book and al-Hikmah, Allaah said concerning them (interpretation of the meaning):

“Then We gave the Book (the Qur’aan) as inheritance to such of Our slaves whom We chose (the followers of Muhammad). Then of them are some who wrong their ownselves, and of them are some who follow a middle course, and of them are some who are, by Allaah’s Leave, foremost in good deeds”

[Faatir 35:32]

“And this is a blessed Book (the Qur’aan) which We have sent down, so follow it and fear Allaah (i.e. do not disobey His Orders), that you may receive mercy (i.e. saved from the torment of Hell).

Lest you (pagan Arabs) should say: ‘The Book was sent down only to two sects before us (the Jews and the Christians), and for our part, we were in fact unaware of what they studied.’

Or lest you (pagan Arabs) should say: ‘If only the Book had been sent down to us, we would surely, have been better guided than they (Jews and Christians).’”

[al-An’aam 6:155-157]

 And Allaah answered the prayer of al-Khaleel (Ibraaheem) for them, when he said (interpretation of the meaning):

“Our Lord! Send amongst them a Messenger of their own (and indeed Allaah answered their invocation by sending Muhammad), who shall recite unto them Your Verses and instruct them in the Book (this Qur’aan) and Al-Hikmah (full knowledge of the Islamic laws and jurisprudence or wisdom or Prophethood)

[al-Baqarah 2:129]

 So there are kinds of illiteracy which are haraam, kinds which are makrooh and kinds which are a shortcoming and failure to do that which is better. A person who does not know how to read al-Faatihah or any part of the Qur’aan, is called by the fuqahaa’ in their discussion on al-salaah (prayer), ummi (unlettered), the opposite of whom is a qaari’ (one who reads or recites). They say: it is not correct for a qaari’ to be led in prayer by an ummi, but it is permissible for an ummi to lead another ummi in prayer; and they discuss other issues of the same nature. What they mean by ummi here is one who cannot read what is necessary, regardless of whether he can write and calculate or not.

 This illiteracy also includes neglecting what is obligatory, for which a man will be punished if he is able to learn it but he does not do so.

 There is also the kind of illiteracy which is blameworthy, such as that which Allaah described in the case of the People of the Book, when He said (interpretation of the meaning):

“And there are among them (Jews) unlettered people, who know not the Book, but they trust upon false desires and they but guess”

[al-Baqarah 2:78] 

This is a description of one who does not understand the word of Allaah or act upon it; all he does is to recite it. Al-Hasan al-Basri said: The Qur’aan was revealed to be acted upon, so recitation implies acting upon it. The ummi in this sense may read the letters of the Qur’aan, etc., but he does not understand it; when he speaks about matters of knowledge, he speaks superficially, based on conjecture. This person is also regarded as ummi and deserves to be blamed, just as Allaah condemned him for his lack of obligatory knowledge, whether this knowledge is fard ‘ayn (obligatory on each individual) or fard kafaayah (obligatory on the community as a whole but not on each individual)

 There is also the kind of ‘illiteracy’ which is better and more perfect, such as the one who only reads a part of the Qur’aan and he only understands what he has learned. He only understands as much of sharee’ah as he has to know. Such a person is also called ummi (illiterate), and others who learn and act upon the Qur’aan are better and more perfect than him.

 These matters which distinguish a person are virtues, and if a person fails to attain them, he is missing out on something that is obligatory either on him as an individual (fard ‘ayn) or is obligatory on the community (fard kafaayah) or is mustahabb (recommended). These attributes of perfection and virtue may be attributed to Allaah and His Prophets in general terms. Allaah is All-Knowing and All-Wise, and He combines knowledge and beneficial speech in all that He wants, tells and wills. The same is true of His Prophets and our Prophet, the leader of the knowledgeable and wise.

 With regard to the distinguishing features which are means to attaining virtues, but which one can do without and use alternative means, these are things like writing and numeracy. If a person does not have these abilities, knowing that virtue cannot be achieved without them, not having them is a sign of imperfection. If a person acquires them and uses them to perfect himself – like the one who learns how to read and then uses that to read the Qur’aan and read useful books or write other things that will be of benefit to people – then this is a virtue in his case. But if he uses it for purposes that will harm him or other people – like the one who reads misguided books or writes things that will harm people, like forging the writing of rulers, judges and witnesses – then this will be bad for him, and will be a sin and imperfection. Hence ‘Umar forbade teaching women how to write. If it is possible to do without it yet still achieve perfect knowledge, then this is better, and this was the case with our Prophet (peace and blessings of Allaah be upon him), of whom Allaah said (interpretation of the meaning):

“Those who follow the Messenger, the Prophet who can neither read nor write (i.e. Muhammad) whom they find written with them in the Tawraat (Torah) and the Injeel (Gospel)…”

[al-A’raaf 7:157]

His being unlettered did not mean that he was lacking in knowledge or the ability to recite by heart, for he is the leader of all imaams in that regard. What it meant was that he was not able to write or to read anything that was written, as Allaah said of him (interpretation of the meaning):

“Neither did you (O Muhammad) read any book before it (this Qur’aan), nor did you write any book (whatsoever) with your right hand…”

[al-‘Ankaboot 29:48]

 [Then he (Ibn Taymiyah – may Allaah have mercy on him) went back to discussing the hadeeth “We are an unlettered nation, we do not write or calculate…” and stated that there is a corroborating report which refers to what is meant by this hadeeth. Then he said:]

 When this is compared with his comment that the month may be thirty or twenty-nine days, it becomes clear that what is meant is that with regard to the new moon (beginning of the month), we have no need for writing or calculations, because sometimes it is (thirty days) and sometimes it is (twenty-nine days). What differentiates between them is the sighting (of the new moon), there is nothing else that differentiates between them, such as writing or calculation.

Thus is becomes clear that the “illiteracy” mentioned here is a characteristic which is praiseworthy in several senses:

it means doing without writing and calculations in favour of something whichis clearer and more apparent, namely, (sighting) the new moon.

-Writing and calculation may involve errors…

Etc.