How deep is math?

What is the basis for reason? And mathematics?

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Eodnhoj7
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Re: How deep is math?

Post by Eodnhoj7 » Fri Dec 08, 2017 8:43 pm

Averroes wrote:
Fri Dec 08, 2017 7:23 pm
Eodnhoj7 wrote:
Wed Dec 06, 2017 4:02 am
Interesting post
Thank you my friend. But this is because it is the Holy Qur'an. Holy Qur'an is interesting. So all praises and thanks be to God, The Almighty.

There are some other interesting patterns as well, but from here, there is the prerequisite of understanding the subtleties of the Arabic language. And with respect to the subject of this thread, it would be called statistical frequency analysis. This involves counting the occurrence a particular word or phrase in the Holy Qur'an.

1. Land and Sea.

For example if we count the singular form of the words "the land"(البر) and "the sea" (البحر) in the Holy Qur'an, then we will get a count of 12 occurrences for "the land" and 32 occurrences for "the sea". Adding these value, we get a total of 44. The percentage of "the land" to this total is 100*12/44, which is about 27.3%, and the percentage of "the sea" to this total is about 72.7%. If you check the latest statistics, you will see that the percentage of water on the earth is about 72%, and the rest is land!

From livescience.com: " Some 72 percent of Earth is covered in water"

Now in this count I have not included the plural form of "sea" like for example "the two seas" (البحرين) and also "a sea" (بحر), but I have included exactly "the sea" (البحر). Anyway, one thing that I can mention here is that for some of the verses which contain "the two seas" (البحرين), it is in fact a reference to a scientific statement which has just recently been discovered by modern science. I just mention here that the Holy Qur'an was revealed more than 1400 years ago.

And He is the One Who has released the two seas, one palatable and sweet and the other salty and bitter, and He has placed a barrier between them, a partition that is forbidden (to be passed). [Holy Quran, interpretation of meaning 25:53]

The substance of the above verse is repeated in some other Quranic verses where the phrase "the two seas" is mentioned. Now, from the experts in this field I have the following information:

"Modern science has discovered that in the places where two different seas meet, there is a barrier between them. This barrier divides the two seas so that each sea has its own temperature, salinity, and density (Principles of Oceanography, Davis R., 1972, p.92)"
Please check the site for further details on this here.

2. Frequency analysis with names of Prophets.

We can also do frequency analysis with the names of Prophets. For example, we can do a frequency analysis with the name of Prophet Jesus and compare it with the name of Prophet Adam (peace be upon them).

God, The Almighty says in the Holy Qur'an (interpretation of meaning):

Indeed, the likeness of Isa(Jesus) with Allah is like that of Adam. He created him from dust; then He said to him, "Be", and he was. [Holy Qur'an, interpretation of meaning 3:59]

We know that Prophet Jesus (pbuh) was born without a father, but he (pbuh) had only a mother. And Prophet Adam(pbuh) had neither a father nor a mother, but he was created from dust by God, The Almighty.

Now, the name of both Jesus and Adam (pbut) occurs 25 times each in the whole of Qur'an! Furthermore, in the verse in which God, the Almighty compares Prophet Jesus with Prophet Adam, it is the seventh occurrence of each of these names from the start of the Qur'an. The nineteenth occurrence of both these names occurs simultaneously in the nineteenth chapter of the Qur'an! And the nineteenth chapter of the Qur'an bears the name of the mother of Jesus (pbuh), i.e. Mary (may Allah be pleased with her). In the Qur'an, Mary (may Allah be pleased with her) is described as the best of all women.

And when the Angels said, "O Maryam! Indeed, Allah has chosen you and purified you and preferred you over the women of the worlds." "O Maryam! Be obedient to your Lord and prostrate and bow down with those who bow down." [Qur'an, interpretation of meaning 3:42-43]

We can do this frequency analysis with other words of Qur'an as well, and the results are all very interesting as well.

Another mathematical example that can be mentioned here is: a networking of verses for a whole chapter of the Holy Qur'an forming a ring structure! Recently, a linguist professor, Dr Raymond Farrin, has written an academic paper on this amazing structure. The following YouTube video presents in visual format the gist of that paper: https://www.youtube.com/watch?v=KLKheiGXHZg
Interesting post again. I remember, and you can correct me on this, that one of the "proofs" muslim theologians argued for the Koran was the nature of how it was written...the language so to speak. I remember reading through it briefly and the nature of "frequency" (as you observed) came to my mind in how it was written...so what you said makes alot of sense (to me at least) in regards to the logical/mathematical foundations the Muslims provided.

Now what I was reading at the time was an english translation, however I later read the the Arabic translation maintains this "poetic frequency" (I call it), much better. Is there any work in regards to arabic language and mathematics/logic itself?

Averroes
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Re: How deep is math?

Post by Averroes » Mon Dec 11, 2017 10:21 pm

Eodnhoj7 wrote:
Fri Dec 08, 2017 8:43 pm
Interesting post again.
It is very kind of you, thank you. I apologize for the late reply, I had to do some research and gather some materials before I could provide a decent reply to your post.
Eodnhoj7 wrote:
Fri Dec 08, 2017 8:43 pm
I remember, and you can correct me on this, that one of the "proofs" muslim theologians argued for the Koran was the nature of how it was written...the language so to speak.
The language of the Qur'an is indeed inimitable and magnificent. Even when the Arabic language was at its peak, no one could imitate the Holy Qur'an. Everyone who is knowledgeable in the subject agrees on that point; not just the Muslims but even the staunchest enemies of Islam at that time had recognized that! So on this point there is no disagreement from anyone. Now, even the Western academics who study the Arabic language and the Holy Qur'an nowadays are also saying that today!
Eodnhoj7 wrote:
Fri Dec 08, 2017 8:43 pm
I remember reading through it briefly and the nature of "frequency" (as you observed) came to my mind in how it was written...so what you said makes alot of sense (to me at least) in regards to the logical/mathematical foundations the Muslims provided.
Well, I am glad that it made sense to you my friend. :) But here I have to point out something important. There is consensus among the Muslim scholars on the linguistic aspect of the Holy Qur'an in being beyond human ability to imitate. The statistical frequency analysis though does not enjoy such a consensus among the Muslim scholars. Which means that for at least some scholars, frequency analysis is not on the same level as the linguistic properties of the Holy Qur'an. But anyway, this is to point out to you that the linguistic and mathematical aspects are distinct perspectives from which the Qur'an can be approached. One can also approach the Qur'an from other perspectives also; such as from a scientific, or legal, or even from an economical perspective as well. I have a link to a free ebook on the scientific statements in the Holy Qur'an as well, if you are interested.

For my part, I believe that God, the Almighty created each of us with different complementary qualities/abilities, in such a way that we would be able to live together for our mutual benefit; after all we are all children of Prophet Adam(peace be upon him). Now with each of our different qualities that God, the Almighty gave us, I believe that God, the Almighty also allowed us to use these qualities/capabilities to approach and study His Book. In this way, firstly each of us can develop a personal relationship with the Holy Qur'an and hence with God, the Almighty; and secondly through the study of the Holy Qur'an, mankind in their diversity is united through the Book of God, the Almighty.

Let us take ourselves as examples. Observe how you were much appreciative and getting insight into things when we were talking the language of mathematics! This is, I believe, because you and mostly anyone who would be drawn to this section of the forum, have a predisposition towards mathematics. Numbers, equations, relations etc speak to you; and this is, I believe, a gift/blessing from God, the Almighty. All praises and thanks be to God, the Almighty. So now, with these abilities that God, the Almighty gave us, we can better appreciate the beauty of a mathematical relationship, more so than someone who might not have such a predisposition. And conversely, where we do not have the abilities, we might not get what others are getting. And all this is good, as we can then all share our experiences and let the others benefit too; and thus we get to know each other better and on the way we also enrich our individual perspectives.

God, the Almighty says in the Holy Qur'an (interpretation of meaning):

O mankind! Indeed, We have created you from a male and a female and made you into nations and tribes that you may know one another. Indeed, the most noble of you in the sight of Allah is the most righteous among you. Indeed, Allah is All-Knower, All-Aware. [Qur'an 49:13]

Averroes
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Re: How deep is math?

Post by Averroes » Mon Dec 11, 2017 10:37 pm

Eodnhoj7 wrote:
Fri Dec 08, 2017 8:43 pm
Now what I was reading at the time was an english translation, however I later read the the Arabic translation maintains this "poetic frequency" (I call it), much better.
Just a very slight correction my friend, instead of "Arabic translation", I think you meant "Arabic original". No need to worry, I also make such kind of mistakes sometimes. And if we were not talking about the Holy Qur'an, I would not even have mentioned it.

The point you are trying to make is not only correct, but it is in fact an understatement! There are many semantic and linguistic subtleties which are not conveyed through the various translations of the Holy Qur'an. And Muslims do not consider the translations to be the Holy Qur'an. The Holy Qur'an is in the Arabic language only. And through the translations, a lot of the beauty of Qur'an is not conveyed. But as I said, the translations also provide a wealth of information concerning the message of the Holy Qur'an to those who do not yet understand the Arabic language. And I have benefited and still immensely benefit from the translations of the Holy Qur'an.

Eodnhoj7 wrote:
Fri Dec 08, 2017 8:43 pm
Is there any work in regards to arabic language and mathematics/logic itself?
Yes, there are. But for either the materials addressing the Arabic language and the mathematical patterns of the Holy Qur'an, at the very least, you will have to know the Arabic alphabet. Otherwise, it will be difficult for you to understand the subject matter of these books or articles. So, if you agree, what I would propose to you is some materials for you to start learning and memorizing the Arabic Alphabet first of all. I also give you some advice on how to make this study as effortless as possible, if God wills.

1. Arabic language.

Please, watch the following YouTube videos to get started with the Arabic Alphabet. You will have to watch them several times, preferably each day for about 15 minutes and for about two weeks at least until you feel comfortable with the Arabic Alphabet. When you are familiar with the Arabic Alphabet, then thank God, the Almighty and ask Him to guide you and make things easy for you. He, the Almighty is the Most Merciful and the Most Generous; He gave us life and He provides for us all.

1. Arabic alphabet lesson 1: https://www.youtube.com/watch?v=vid0zqC2UvU
2. Arabic alphabet lesson 2: https://www.youtube.com/watch?v=7GfnPA5W_NY

It may appear difficult at first, but after the two weeks of intermittent exposure at most, if God wills, it becomes easy to recognize and pronounce the letters. With a little more time, it becomes second nature, if God wills.

If you have an android mobile phone or tablet, then you can go to Google Play Store and install the application "Arabic Alphabet" by ArabicPro. Install also "Quran learning" by "2KT Soft". And each day, for about 15 minutes you can go through the alphabet. If God wills, after sometime, it is absorbed by the mind. And the Arabic language journey can begin, if God wills. :)

If you go through these short courses and you succeed in memorizing the Arabic alphabet and you find yourself wanting to further your knowledge of the Arabic language, then I have more advanced free resources which can ease your way through the study of the Arabic language. You can send me a private message, there is no worry on my side.

For now, you will not be able to appreciate fully the linguistic beauty of the Holy Qur'an as you do not know much Arabic. But you can get a glimpse of the beauty of the Holy Qur'an, if you have a look at the following compilation of short linguistic lessons by a US based talented Arabic teacher: https://www.youtube.com/watch?v=j-ULa2JzPG0

2. Mathematics in the Holy Qur'an

Here too, having a grasp of the Arabic Alphabet is a prerequisite. There is a mechanical engineer by the name of Abduldaem Al-Kaheel who has a site which is dedicated to the mathematical patterns in the Holy Qur'an. He also has a number of free ebooks on the numerical patterns of the Holy Qur'an. Here is his site: http://kaheel7.com/eng/index.php/numeric-miracle

A free ebook among others on the site: Mathematical patterns in the Qur'an

You might also want to have in your personal library the book by Professor Neil Robinson, Discovering the Qur'an, which you can have a look at. I did some research and I found that it is available for free on scribd. If however you like the touch and smell of a paper book it is available on amazon or you can check at your local bookstore. On scribd it is here: https://www.scribd.com/document/4563754 ... l-Robinson

You may have to register to scribd to access the book and the registration is free anyway!

If I can help you in anything else, please do not hesitate. Whatever I know that is of benefit, I will share it with you. It might not be much as I am still learning myself, but at least it might be a good start to get you going, if God wills.

One thing that you can readily appreciate though without understanding the Arabic language is experiencing the recitation of the Holy Qur'an. The Holy Qur'an should be listened to and studied in a clean place, i.e. not in the bathroom. Here are some recitations of the Holy Qur'an. Observe the effect on you.

1. Al-Fatiha (chapter 1):https://www.youtube.com/watch?v=z6r47L-8uf8
2. At-Teen(chapter 95):https://www.youtube.com/watch?v=SK88FsJ3xQM
3. Al-'Asr (chapter 103): https://www.youtube.com/watch?v=b5iElBopeyA
4. Al-Feel(chapter 105): https://www.youtube.com/watch?v=gfBECWykm2o

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Eodnhoj7
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Re: How deep is math?

Post by Eodnhoj7 » Tue Dec 12, 2017 10:05 pm

Thanks for the info, I will look into it when I have the time.

I think it was Ibn Rushd (or Avicenna?) which made the observation the what seperated the prophets from non-prophets was the perspective in which they seen creation. Now you can correct me on that, however it was an obscure quote I heard close to a decade ago when I was studying philosophy in university.

However if there is any degree of truth in it, and it appears to be the case, prophecy is merely an advance form of rationality applied through acknowledgement of a the Creator as "Logos" or "Reason/Divine Plan/etc.". In these respects reason is a divine process that allows us to give and maintain proportional actions that are equivalent in form and function to "justice". Under those premises that application of divinity appears to be the application of reason.

In these respects, with reason as proportion, reason can be observed as both quantitatively (number) and qualitatively (language) mathematical. In these respects, every one is a mathematician to some degree, in the respect of applying dimensions as forms of measurement.

I know Aquinas, a Roman Catholic theologian/philosopher, was influenced to a relatively significant degree by muslim philosophers (I "think" it was Ibn Rushd again). Where any Muslin philosophers affected by Aquinas? Considering Aquinas Aristotelian roots, I do not believe it would much of stretch.

Averroes
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Re: How deep is math?

Post by Averroes » Fri Dec 15, 2017 6:02 pm

Eodnhoj7 wrote:
Tue Dec 12, 2017 10:05 pm
Thanks for the info, I will look into it when I have the time.
You are most welcomed.
Eodnhoj7 wrote:
Tue Dec 12, 2017 10:05 pm
I think it was Ibn Rushd (or Avicenna?) which made the observation the what seperated the prophets from non-prophets was the perspective in which they seen creation. Now you can correct me on that, however it was an obscure quote I heard close to a decade ago when I was studying philosophy in university.
I do not know what they said exactly and in which context about the matter you are referring to, so I cannot judge about that simply because I do not have the facts. But what I can say is that even though these Muslims were great intellectuals and very intelligent, they were still human beings and made mistakes.

A Prophet is a human being who speaks on behalf of God, The Almighty. A Prophet is a human being who was chosen by God, the Almighty to convey His message to other created beings. There were many Prophets who were sent to mankind, but in the Holy Qur'an there are only 25 Prophets (peace be upon them) who are mentioned by name. In chronological order in which these prophets appeared: Prophet Adam, Prophet Idriss, Prophet Noah, Prophet Hud, Prophet Salih, Prophet Ibrahim (Abraham), Prophet Ishmael, Prophet Ishaq (Isaac), Prophet Lut (Lot), Prophet Ya'qub (Jacob), Prophet Yusuf (Joseph), Prophet Shu'aib, Prophet Ayyub (Job), Prophet Musa (Moses), Prophet Harun (Aaron), Prophet Dhul Kifl, Prophet Dawud (David), Prophet Suleiman (Solomon), Prophet Ilyas (Elijah), Prophet Yunus (Jonah), Prophet Zakariya, Prophet Yayah (John), Prophet Isa (Jesus) and Prophet Muhammad (peace be upon them).

In Islam, Muslims believe that Prophet Muhammad (peace and blessings of Allah be upon him) is the last and final messenger of God, the Almighty. And after Prophet Muhammad (pbuh) no one can claim to prophethood.
Eodnhoj7 wrote:
Tue Dec 12, 2017 10:05 pm
However if there is any degree of truth in it, and it appears to be the case, prophecy is merely an advance form of rationality applied through acknowledgement of a the Creator as "Logos" or "Reason/Divine Plan/etc.". In these respects reason is a divine process that allows us to give and maintain proportional actions that are equivalent in form and function to "justice". Under those premises that application of divinity appears to be the application of reason.
In the Holy Qur'an, God, The Almighty explains the process of revelation (interpretation of meaning):

Say, "Whoever is an enemy to Jibreel- for indeed he has brought it (i.e. Quran) down upon your heart (O Muhammad!) by the permission of Allah, confirming what came before it and a guidance and glad tidings for the believers." [Quran, interpretation of meaning 2:98]

"Jibreel" is the Arabic name of Angel Gabriel (peace be upon him).

Prophethood requires receiving revelation from God, the Almighty and only specific human beings have been chosen by God, the Almighty to receive revelations from Him. Whereas our ability to use reason applies to nearly every human being (i.e. applies to those who effectively use reason!).

Now, if one says that our faculty of reason has a divine origin, then I agree by replying that everything was created by God, the Almighty. We were created by God, the Almighty. The heavens and the earth, and anything in between was created by God, the Almighty. Our ability to see, hear, think and reason, to name a few, were given to us by God, the Almighty. However, I will not go in the direction of equating the use of reason with being a prophet or similar to a prophet. What I say is as follows: we should use reason but our use of reason does not make us prophets. The degree of Prophethood is a very high status (the highest for a human being) in Islam, and is not to be equated with a mere increase in degree from the mere use of reason. And after Prophet Muhammad(pbuh), no one can claim prophethood.

These Messengers! We preferred some over others. Among them were those with whom Allah spoke, and He raised some of them in degrees. And We gave Isa(Jesus), son of Maryam, clear proofs and supported him with the Holy Spirit. And if Allah had willed, those succeeding them would not have fought each other after clear proofs had come to them. But they differed, some of them believed and some denied. And if Allah had willed, they would not have fought each other, but Allah does what He intends. [Qur'an, interpretation of meaning 2:253]

This is a very interesting subject (namely revelation and reason) that you have raised here. So, I will address that in a separate post.

Averroes
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Re: How deep is math?

Post by Averroes » Fri Dec 15, 2017 6:07 pm

Revelation and reason.

We can say, as Ibn Rushd has qualified it in the Decisive Treatise, that reason is a tool. And we know that tools are used to craft materials in desired forms. If we have the tools but no materials to work on, then the tools will not be very useful to us.

The works on logic that Aristotle pioneered are now available in a collection of six books known as the Organon. Aristotle’s logic was refined by 19th century mathematicians and philosophers, as we have already addressed in this thread. In essence, whether it be the ancient logic of Aristotle or the modern Frege-Russell logic, logical reasoning involves drawing conclusions from a set of established or accepted premises. A valid argument is one for which the truth of the premises guarantees the truth of the conclusion. This means that if the premises of a valid logical argument are accepted as true, then one is bound to accept the truth of what follows from these premises, i.e. the conclusion. So far, so good.

But how do we establish the truth of our premises?!! So, with the analogy of logic as a tool, true premises would be the materials on which that tool is to operate. And the conclusion would be the end product or finished goods.

In Islam, we believe that revelation from God, the Almighty is already the truth which is being given to us through the Prophet (peace and blessing of Allah be upon him), and there is no mistake at all in the Words of God, the Almighty. Whereas with reason, for a valid argument, the truth of our conclusion depends on the truth of our premises! Our argument may exhibit flawless logic, but yet still be false in its conclusion because the premises were false! So the problem is not with the tool but with the materials on which the tool operate. Examples of such erroneous use of reason is abundant in the history of philosophy, especially with the ancients themselves; we will see some notorious examples shortly if God wills. So reason can never be on the same level as Revelation from God, the Almighty to His Prophet (pbuh). But this does not mean that we are to reject reason altogether, but on the contrary we must use reason with uttermost diligence, humility and prudence. Reasoning is hard work! Whereas revelation from God, the Almighty is a great mercy to mankind. As a matter of fact in Islam, it is a religious requirement to use the intellect. God, the Almighty says in the Holy Qur’an (interpretation of meaning):

Indeed, worst of the living creatures in the sight of Allah are the deaf and the dumb, those who do not use their intellect. [Holy Qur’an, interpretation of meaning 8:22]

Here, I propose to present some of the notorious examples where Aristotle erred in reasoning, because he had proceeded from false premises, which would seem to many (including ourselves) to be intuitively true, if later scientific empirical observations had not shown us otherwise! And of course, Aquinas, Ibn Rushd and Avicenna and nearly all other Aristotelians fell into these errors also.

1. Aristotle argued that the Earth is at rest.

There are many arguments which he proposed in his book On the heavens. Here is one:
Aristotle wrote:It is clear, then, that the earth must be at the centre and immovable, not only for the reasons already given, but also because heavy bodies forcibly thrown quite straight upward return to the point from which they started, even if they are thrown to an infinite distance. From these considerations then it is clear that the earth does not move and does not lie elsewhere than at the centre. [On the Heavens Book 2, part 14]
His arguments can be expressed as follows:

1. If the Earth moved, then heavy bodies thrown straight up would land elsewhere.
2. Heavy bodies thrown straight up land at where they were thrown.
3. Therefore, the Earth does not move.

The above argument is a valid argument (but not sound). This type of argument is called a modus tollens:
If P then Q.
Not Q.
Therefore, not P.

But obviously, the conclusion is known to be false nowadays. So what is wrong with the argument of Aristotle? Obviously, it is the premises, as the argument is valid. The first premise in now known to be false. This would follow from an application of the principle of inertia, which Galileo discovered and was sentenced to death for that discovery! But Newton took up the idea and came up with his equations of motion!

Aristotle thought that once an applied force were removed from an object that object would stop moving, whereas the law of inertia states that an object continues in its state of rest or uniform motion unless compelled by a force.

If you have difficulty imagining the consequences of this principle, then do not worry. We can do an experiment to verify this. The following YouTube video is about a cyclist going at about 18 miles an hour, and then drops a ping pong ball. On Aristotle’s account, the ball should land behind the biker, while on Galileo’s account, the ball should land next to the biker. Who is right? Well, maybe you can tell me! :) Inertia experiment: https://www.youtube.com/watch?v=gJcbb6EynDk

So based on his ignorance of the law of inertia, Aristotle denied that the Earth was rotating on its axis for example.
Aristotle wrote:Let us first decide the question whether the earth moves or is at rest. For, as we said, there are some who make it one of the stars, and others who, setting it at the centre, suppose it to be 'rolled' and in motion about the pole as axis. That both views are untenable will be clear if we take as our starting-point the fact that the earth's motion, whether the earth be at the centre or away from it, must needs be a constrained motion. [On the heavens, Book 2, part 14]
This is an example which shows that mere logic is not sufficient to reach the truth, but logic itself needs to proceed from the truth (i.e. true premises) to reach other truths (true conclusions).

2. Aristotle thought that heavy bodies fall faster than lighter bodies.
Aristotle wrote:The difficulty must have occurred to every one. It would indeed be a complacent mind that felt no surprise that, while a little bit of earth, let loose in mid-air moves and will not stay still, and more there is of it the faster it moves, the whole earth, free in midair, should show no movement at all. Yet here is this great weight of earth, and it is at rest. And again, from beneath one of these moving fragments of earth, before it falls, take away the earth, and it will continue its downward movement with nothing to stop it. The difficulty then, has naturally passed into a common place of philosophy; and one may well wonder that the solutions offered are not seen to involve greater absurdities than the problem itself. [On the Heavens, Book 2 part 13]
Now, we know in physics that whatever the mass of a body, they all fall with the same speed in a vacuum. Physicist Brian Cox does the experiment of Galileo at NASA in the following video: https://www.youtube.com/watch?v=QyeF-_QPSbk

3. Aristotle argued that the earth is at the center of the Universe and also the universe was eternal.

I think it was in his Metaphysics or Physics that he argued for that. But any way, in the book we are at present considering, i.e. On the Heavens, he clearly states his belief in an eternal universe thus:
Aristotle wrote:But the order of the universe is eternal. [On the Heavens Book 2, part 14]
These are now clearly known to be false premises/conclusions with the advance of science. And now we know that the earth is not at the center of the universe; and also that the universe is expanding and had a beginning in the Big Bang and therefore is not eternal. And here as elsewhere, it must be said that science is lagging behind the knowledge given by God, the Almighty in the Holy Qur’an:

Have not considered those who disbelieved that the heavens and the earth were a joined entity, then We parted them and made every living thing from water? Then will they not believe. [Holy Qur’an, interpretation of meaning, 21:30]

And We constructed the heaven with strength, and indeed, We are (its) Expanders. [Qur’an, interpretation of meaning, 51:47]
________________

And there are other things as well that Aristotle got wrong. This is not to belittle Aristotle efforts. Far from it. As I clearly stated before, I have much respect for Aristotle’s works: as the work of a human being who labored hard with the means that he had, to search for the truth. And, he has had great insights. But in the end he was a normal human being who was prone to make mistakes.

God, the Almighty does not make mistakes. And that is why, reason must always be subordinated to the Words of God, the Almighty and never be considered to be an alternative to revelation. When reason is subordinated to the Revelation of God, the Almighty then it becomes a very powerful tool for then it can safely proceed from true premises to derive true conclusions.

In the Holy Qur’an God, the Almighty says (interpretation of meaning):

Then do they not ponder on the Qur’an? If it had been from other than Allah, surely they would have found much contradiction in it. [Qur’an 4:82]

In science, reason is subordinated to empirical observations, i.e. the premises are given through empirical observations. That too is good, for the heavens and the earth and anything in between were created by God, the Almighty. And we are also informed in the Holy Qur’an, how God, the Almighty creates:

His Command, when He intends a thing, is only that He says to it, “Be,” and it is. [Qur’an 36:82]

So it is then obvious for us that the Word of God, the Almighty always precedes the empirical or witnessed world. So in addition to reason, even the empirical world is subordinated to the Word of God, the Almighty.

In Islam, revelation, reason and science do not contradict each other but in fact agree with each other. In the Holy Qur'an, God the Almighty commands us to use reason, and also to look around and investigate the world that He created. And in turn modern science has time and time again been confirming Qur'anic statements/verses revealed more than 1400 years ago, when these new scientific discoveries could not have been known by scientific investigations!
Last edited by Averroes on Fri Dec 15, 2017 6:23 pm, edited 1 time in total.

Averroes
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Re: How deep is math?

Post by Averroes » Fri Dec 15, 2017 6:16 pm

Eodnhoj7 wrote:
Tue Dec 12, 2017 10:05 pm
In these respects, with reason as proportion, reason can be observed as both quantitatively (number) and qualitatively (language) mathematical. In these respects, every one is a mathematician to some degree, in the respect of applying dimensions as forms of measurement.
That statement seems to me to be a perfectly reasonable statement for one to make; as most human beings need to go to the grocery store or to the shop every so often! And if one were to follow this interesting line of reasoning through, this would make mathematics a universal language. :)

So, I agree that everyone is a mathematician to some degree, but I also believe that not everyone enjoys mathematics to the same degree! And this applies to other activities as well, such that each of us has a speciality which he/she enjoys doing and is good at.
Eodnhoj7 wrote:
Tue Dec 12, 2017 10:05 pm
I know Aquinas, a Roman Catholic theologian/philosopher, was influenced to a relatively significant degree by muslim philosophers (I "think" it was Ibn Rushd again).
You are right, it was Ibn Rushd (i.e. the real Averroes). Aquinas referred to Ibn Rushd by the title "The Commentator", i.e. the commentator of Aristotle. Ibn Rushd influenced a whole lot of people from the Scholastic to the Renaissance thinkers. As I already mentioned in a previous post, nowadays Ibn Rushd is known as the father of philosophy of the modern West. Ibn Rushd was from what is now called Spain (Al-Andalus at that time), so he was a Westerner too. And Today as well Ibn Rushd continues to influence a lot of people, namely in France. For example, I know that former Minister of Education of France and Philosophy Professor Luke Ferry is an admirer of Ibn Rushd. For those who understand French (I know there are some here) the following YouTube is Prof Luke Ferry expressing his admiration for and praising Ibn Rushd and Islam: https://www.youtube.com/watch?v=dS1Y6LLavnY

And there were also other Muslim philosophers who had significant influence on the scholastics and later the Renaissance thinkers as well. For example, you also mentioned Avicenna (Ibn Sina). Avicenna had considerable influence on the Western thinkers and scientists as well. Avicenna was also a medical practitioner like Ibn Rushd and he was a prodigy as well. By the age of 10 or so, it is said that he had memorized the whole Holy Qur'an. In the West, his books on medicine called the Canon of medicine were used as textbooks until well into the 18th century.

In philosophy, we all know of the hyperbolic doubt method of Descartes which lead him into the famous statement, "Je pense donc je suis" ( "Cogito ergo sum" in Latin or "I think therefore I am" in English). Descartes was French. But the method of Descartes and the philosophy which emanated from it, has strikingly strong similarities with the "flying/floating man" argument of Avicenna, written more than 500 years before Descartes! I read the Meditations of Descartes a long time ago, but I do not remember Descartes ever mentioning Avicenna! ;)
Any way you can check this article on Wikipedia for a refresher on this: https://en.wikipedia.org/wiki/Floating_man

We can also mention the great Al-Ghazzali as well. Today, in modal logic we make use of the possible world semantics which was introduced by Saul Kripke at around the late 1950s. But, Al-Ghazzali had already expressed this idea in his masterpiece The Incoherence of the Philosophers in the 11th century CE, which was a refutation of some of the positions of Avicenna. This is the earliest exposition of these ideas from Western sources. Wikipedia says that it is only implicitly stated in Al-Ghazzali's work but when I read Al-Ghazzali I found it to be fully and clearly expressed! :) But any way, even Wikipedia recognizes that Al-Ghazzali was the first one who developed that idea!
Wikipedia wrote:Scholars have found implicit earlier traces of the idea of possible worlds in the works of René Descartes, a major influence on Leibniz, Al-Ghazali (The Incoherence of the Philosophers), Averroes (The Incoherence of the Incoherence), Fakhr al-Din al-Razi (Matalib al-'Aliya) and John Duns Scotus. The modern philosophical use of the notion was pioneered by David Lewis and Saul Kripke. Wikipedia
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Eodnhoj7 wrote:
Tue Dec 12, 2017 10:05 pm
Where any Muslin philosophers affected by Aquinas? Considering Aquinas Aristotelian roots, I do not believe it would much of stretch.
I do not know any Muslim philosopher who was affected or influenced by Aquinas. But I am not a scholar, so I cannot say. Of course Muslims reading Aquinas would not be a stretch, as I myself have read some of the translations of his commentaries on Aristotle. But being affected or influenced by Aquinas, that I do not know. I am thinking that I should also say now that I have read many of the translations of the commentaries (short, medium and long) of Ibn Rushd on Aristotle before I read that of Aquinas. What I can say on that now is that I totally understand how Ibn Rushd came to be so influential on Aquinas and others!

However now, what I know is that, at that time (i.e. as from the 12-13th century), knowledge were flowing from the Muslim civilization into the West, not the other way around. According to Nietzsche for example, the sophistication of the Muslim civilization at that time were several hundreds of years ahead of its time, whereas the West in general (i.e. except for Spain which was Muslim at that time) was in the Dark Ages.

If we were to drill into this subject, we could consider the Spain of Ibn Rushd, which is not very far in time from Aquinas. At that time, 12-13th century CE, the libraries in Cordoba, Spain were the best libraries in the world. Historians say that the caliphal library in Cordoba housed some more than 400,000 volumes while in the rest of Europe the best libraries housed no more than 400 manuscripts. And moreover, the caliphal library was itself only one of the 70 libraries in Cordoba! And there were also many private collections! The English historian Edward Gibbon admired these Spanish Muslims for their love and respect of books and knowledge and he contrasted these Muslims with the anti-book attitude of the rest of medieval Europe. Same as Nietzsche, Gibbon recognized the superiority of the Muslims in the sciences at the time, compared to the rest of Europe.

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Re: How deep is math?

Post by Eodnhoj7 » Mon Jan 01, 2018 7:37 pm

Philosophy Explorer wrote:
Mon Nov 27, 2017 4:40 pm
Or how involving is math? By deep I mean that in its broadest sense so feel free to interpret it your particular way.

I'm an explorer of math and run into situations that make me wonder. For example, with factorials, I've checked the internet and found no patterns with them listed there.
Here's a case I've discovered:

1! = 1
3! = 2•3
5! = 4•5•6
7! = 7•8•9•10

There is no method that would have led to this pattern. You can see this is a four-line pattern that doesn't go any further. Some would say you can prove anything in math which isn't true and patterns are special - you just can't come up with any pattern.

I chose this particular example as it's easy to understand and follow for many people. Yet it's part of the math mysteries which isn't easy to explain.

So how deep is math for you?

PhilX 🇺🇸

Considering it is four numbers, with four being synonymous to the square and approximates of it, under the Pythagorean perspective, it may equate to foundational matrix of primes for the base 10 numbers from which all number stems. The square, as dual duals, equates to a matrix of sorts.

In simple terms, these four primes may act as a "womb" from which the other numbers are "birthed". Forgive the non-mathematical metaphor.

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Re: How deep is math?

Post by Science Fan » Mon Feb 26, 2018 5:20 pm

I'm not really sure what you mean by the word "deep," but in my opinion, I would describe mathematics as being broad as opposed to deep. Because everything in mathematics is based on an abstract object, whether we are referring to an element, set, number, function, relation, circle, infinity, etc., these are all abstractions. And since mathematics deals with abstractions, these abstractions can be broadly applied. For example, you mentioned factorials. When I think of factorials I think of probability, I think of such things as combinations and probabilities. This is why I am unconvinced that there is some intelligence behind the universe that made it consistent with mathematics, because one would expect mathematics to have broad applicability, to physics, chemistry, biology, economics, etc., just due to the fact that abstractions can be applied to all sorts of concrete situations, by the very nature of abstraction. So, I see people often overlooking the fact that math deals with abstractions, which explains why it is so useful in describing such a large variety of topics, and instead start speculating on its applicability as being evidence for some God at work.

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Re: How deep is math?

Post by Philosophy Explorer » Fri Mar 09, 2018 4:59 am

Philosophy Explorer wrote:
Tue Nov 28, 2017 3:26 pm
Here is another case I've just discovered with factorials:

3! + 2 = 2⁳
4! + 3 = 3⁳
5! + 5 = 5⁳
6! + 9 = 9⁳

Something to think about.

PhilX 🇺🇸
I just discovered this:

y = 2ⁿ + 1

Let n = 0 to 3 and y becomes 2, 3, 5 and 9. Compare with the 2nd column and you will see they are the same.
Coincidence?

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Re: How deep is math?

Post by wtf » Fri Mar 09, 2018 5:47 am

Philosophy Explorer wrote:
Fri Mar 09, 2018 4:59 am
Let n = 0 to 3 and y becomes 2, 3, 5 and 9. Compare with the 2nd column and you will see they are the same.
Coincidence?
A sample size of four data points doesn't count for much.

Of course no finite set of data points can ever be sufficient. There are many numerical patterns that go on for millions of datapoints then fail.

But even if we can never be certain a pattern holds for all integers when we can only test finitely many, if there are a LOT of datapoints, at least it makes for an interesting story. T

With four data points, you haven't got anything compelling. There's no story there. Nobody's going to sit up and go, Wow that's amazing!

I would like to illustrate this point with a mathematical anecdote. You don't have to follow every detail but there is a point at the end.

Let Primes(n) be the number of primes less than or including n. For example Primes(3) = 2, because 2 and 3 are the only primes less than or equal to 3. Likewise Primes(6) = 3, because the smaller primes are 2, 3, and 5. Note that the Primes() function gives us a sensible answer for any positive integers, prime or composite.

Now it's easy to calculate Primes(n) for a given n. You just count them! You could write a program. The problem is that it would be a SLOW program. As n gets large, the computation becomes intractable.

It would be good if we could somehow approximate Primes(n) with a more tractable function that grows more slowly. In the early 1900's people found such a function to approximate Primes(). The function is called the logarithmic interval. Nevermind what Li() is, that's not important. What's important that as n gets large, P(n) and Li(n) get as close together as you like. Li() approximates Primes().

It was noted that Primes() always seems to be less than Li() no matter how many values of n they could calculate at the time. Everyone believed that Primes(n) < Li(n) for every n.

Then in 1914, Littlewood (the guy played by Toby Jones in The Man Who Knew Infinity) proved that there must be SOME n for which the number of primes was greater than the output of the formula for that n. In other words the difference switches signs. But he had no idea how large such an n must be.

In 1933 one of Littlewood's former students, Skewes, proved that (if you assume the Riemann hypothesis) the n that Littlewood predicted could be no larger than a certain specific number. There's a big number, called Skewes' number, such that there is SOME value of n less than Skewes' number with Primes(n) > Li(n). The inequality flips.

The value of Skewes number is 10^10^10^34.

How big is this number? Exponentiation associates from right to left. Reading from the right, 10^34 is 1 followed by 34 zeros. And 10 to the power of that, is 1 followed by 10^34 zeros, And finally there's one more level up. Skewes' number is a 1 followed by 10^10^34 zeros.

That's how big this number is. We can't imagine such a number.

And the point of all this is, that for all we know, the claim "Primes() is always smaller than Li() holds up for every single value of n less than Skewes' number.

Now there have been much sharper lower bounds since then, but the point remains. A proposition can hold for a huge, unimaginably large number of datapoints; and then fail.

Four datapoints ... that just ain't gonna cut it.

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Re: How deep is math?

Post by Philosophy Explorer » Fri Mar 09, 2018 4:47 pm

wtf wrote:
Fri Mar 09, 2018 5:47 am
Philosophy Explorer wrote:
Fri Mar 09, 2018 4:59 am
Let n = 0 to 3 and y becomes 2, 3, 5 and 9. Compare with the 2nd column and you will see they are the same.
Coincidence?
A sample size of four data points doesn't count for much.

Of course no finite set of data points can ever be sufficient. There are many numerical patterns that go on for millions of datapoints then fail.

But even if we can never be certain a pattern holds for all integers when we can only test finitely many, if there are a LOT of datapoints, at least it makes for an interesting story. T

With four data points, you haven't got anything compelling. There's no story there. Nobody's going to sit up and go, Wow that's amazing!

I would like to illustrate this point with a mathematical anecdote. You don't have to follow every detail but there is a point at the end.

Let Primes(n) be the number of primes less than or including n. For example Primes(3) = 2, because 2 and 3 are the only primes less than or equal to 3. Likewise Primes(6) = 3, because the smaller primes are 2, 3, and 5. Note that the Primes() function gives us a sensible answer for any positive integers, prime or composite.

Now it's easy to calculate Primes(n) for a given n. You just count them! You could write a program. The problem is that it would be a SLOW program. As n gets large, the computation becomes intractable.

It would be good if we could somehow approximate Primes(n) with a more tractable function that grows more slowly. In the early 1900's people found such a function to approximate Primes(). The function is called the logarithmic interval. Nevermind what Li() is, that's not important. What's important that as n gets large, P(n) and Li(n) get as close together as you like. Li() approximates Primes().

It was noted that Primes() always seems to be less than Li() no matter how many values of n they could calculate at the time. Everyone believed that Primes(n) < Li(n) for every n.

Then in 1914, Littlewood (the guy played by Toby Jones in The Man Who Knew Infinity) proved that there must be SOME n for which the number of primes was greater than the output of the formula for that n. In other words the difference switches signs. But he had no idea how large such an n must be.

In 1933 one of Littlewood's former students, Skewes, proved that (if you assume the Riemann hypothesis) the n that Littlewood predicted could be no larger than a certain specific number. There's a big number, called Skewes' number, such that there is SOME value of n less than Skewes' number with Primes(n) > Li(n). The inequality flips.

The value of Skewes number is 10^10^10^34.

How big is this number? Exponentiation associates from right to left. Reading from the right, 10^34 is 1 followed by 34 zeros. And 10 to the power of that, is 1 followed by 10^34 zeros, And finally there's one more level up. Skewes' number is a 1 followed by 10^10^34 zeros.

That's how big this number is. We can't imagine such a number.

And the point of all this is, that for all we know, the claim "Primes() is always smaller than Li() holds up for every single value of n less than Skewes' number.

Now there have been much sharper lower bounds since then, but the point remains. A proposition can hold for a huge, unimaginably large number of datapoints; and then fail.

Four datapoints ... that just ain't gonna cut it.
Let's make it more interesting. Can you find something with more than four consecutive datapoints that matches up with my example - perhaps an algebraic equation?

PhilX 🇺🇸
Last edited by Philosophy Explorer on Fri Mar 09, 2018 4:53 pm, edited 1 time in total.

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Re: How deep is math?

Post by Plato's Rock » Fri Mar 09, 2018 4:53 pm

easy, and trite response; Math is just codified logic that shares a codified system of symbols.

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Re: How deep is math?

Post by Philosophy Explorer » Sat Mar 10, 2018 5:12 pm

A specific example would help clarify this thread.

There was a 300+ year-old math puzzle called Fermat's Last Theorem (FLT) which defied many attempts to solve it. There are two branches of math: elliptic equations and modular forms that many in the math community felt were completely unrelated. Back in the 50's, a Japanese mathematician started finding examples from each branch that were related. Finally Andrew Wiles proved the two branches were the same proving FLT (for more details, I refer you to the hardcover book, Fermat's Enigma by Simon Singh starting on page 181).

So relating two branches of math solved the most famous math puzzle and has ripple effects throughout math.

PhilX 🇺🇸

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Re: How deep is math?

Post by Philosophy Explorer » Sun Mar 11, 2018 10:30 pm

Here's another example many would take for granted at face value and not think further about:

8 1 6
3 5 7
4 9 2

This is the simplest magic square, a 3 x 3 without the boxes or cells, whose magic sum is 15 for its rows, columns and diagonals.

That's the way you may see it. I see much more, many multigrades (specifically bigrades).

Here are two multigrades:

8⁲ + 1⁲ + 6⁲ = 4⁲ + 9⁲ + 2⁲ = 101

8⁲ + 3⁲ + 4⁲ = 6⁲ + 7⁲ + 2⁲ = 89

So the first and third rows share a special relationship as well as the first and third columns.

Then we have this by reversing the concatenated numbers:

816⁲ + 357⁲ + 492⁲ = 618⁲ + 753⁲ + 294⁲ = 1,035,369

Next I came up with this one:

8⁲ + 3⁲ + 4⁲ + 16⁲ + 57⁲ + 92⁲ = 6⁲ + 7⁲ + 2⁲ + 18⁲ + 53⁲ + 94⁲ = 12,058

This is just a sample from this magic square which seems to readily yield multigrades which other magic squares wouldn't (different magic squares have special properties).

If you want to explore this further, check the internet and get ahold of Before Sudoku: The World Of Magic Squares by Block and Tavares (over 200 pages, besides recreational considerations, it shows practical uses for magic squares and it also covers Sudoku).

PhilX 🇺🇸

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