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Is it true that at the bare-bones of all advanced math, its all just mostly arithmetic?

In computer programming languages they are mostly constrained to arithmetic operators. I suppose this is because computer CPUs compute numbers using the Arithmetic Logic Unit. Computers can do all advanced math with just the ALU, so I'm guessing all math is basically arithmetic.

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    $\begingroup$ Computers can do all advanced math - It is not a true. Only arithmetics and analysis part (and if we are not talking about infinity). Many of this things are approximating with some approximation error. For example try explain this: example Why it is not zero for computer? For us it is easy. Computers also can't proof any difficult theorem (authomatic proofs are nor workings for it) but it is most part of math - the proofs. $\endgroup$ – Piotr Wasilewicz Jul 24 '16 at 20:32
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    $\begingroup$ @Piotr, please don't. You are using a feature of W|A that is by design to illustrate unrelated issues. 1.1 is notation for inexact numbers, and W|A behaves as expected. There isn't anything that is "obvious" for "us" what "computers" do "not get". If you meant to write exact $11/10$, write it. $\endgroup$ – dbanet Jul 24 '16 at 20:37
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    $\begingroup$ This sounds like a lazy formulation of Turing's thesis, essentially that any human computation can be performed by the basic instructions of a CPU and is thus reducible to recursive arithmetic. But this isn't a question about math: you might as well ask whether all poetry is just advanced arithmetic. What we know how to achieve by automated process tends to be very small compared to the full richness of any human endeavour, including mathematics. $\endgroup$ – Erick Wong Jul 24 '16 at 20:47
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    $\begingroup$ Your erroneous assertion that computers can do advanced math is the clearest indication of where you're coming from. Let's take an example: "Twin primes" are prime numbers like $101$ and $103$ that differ by $2$. Nobody knows whether there are infinitely many of them, although Euclid showed in about 300 BC that there are infinitely many primes. Suppose you ask a computer: Are there infinitely many twin primes? A computer can't answer that any more than a computer can tell you whether Purgatory exists. But you may use a computer in your efforts to answer those questions. $\qquad$ $\endgroup$ – Michael Hardy Jul 25 '16 at 0:31
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    $\begingroup$ @ErickWong : I learned that as "Church's thesis", and later saw it being called the "Church--Turing thesis". Do you prefer attributing it only to Turing? $\qquad$ $\endgroup$ – Michael Hardy Jul 25 '16 at 0:45
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In a certain (deep and amazing) sense, yes, all mathematics does seem to be 'equivalent' to "advanced arithmetic", tho there are some subtleties about which you seem to be confused or unaware.

The details of exactly how any particular mathematical theorem is equivalent to a theorem in arithmetic can be incredibly complex. I don't think your current intuition is doing as much work as you might expect based on your question. Analogously, all chemistry is "basically" physics, tho describing even the simplest chemical reaction in the language (or, rather, one of the languages) of physics would be beyond tedious. It would be similar to describing the history of a country by describing the daily actions of each of its inhabitants or visitors. And in a very real sense, doing so misses the point and does its job poorly and (extremely) inefficiently.

There's also a cultural component that you're missing. 'Mathematics' doesn't (just) cover an ideal Platonic reality consisting of all possible 'mathematic objects'. It is also a human endeavor, thus it consists not only of what we know (or accept, or believe) to be true, 'mathematically', but also of what is suspected or imagined. It encompasses a way of thinking – really many ways of thinking – and it consists of many current and historical philosophies about what exactly (or inexactly) mathematics 'is', essentially or in practice. Further more, it is a process or activity by which people engage in these ways of thinking, argue about the relevant philosophy, and share their thoughts and results with one another.

Altho there are many strong reasons to suspect or believe that human thought is 'fundamentally' equivalent or reducible to computation, currently only (some) humans are capable of participating in and contributing to the ongoing endeavor that is 'mathematics'. And that process, while possibly 'equivalent' to some (Vast) arithmetical calculation, is arguably more richly and productively described exactly as it is now by those in the mathematics community, e.g. talks given, papers published, theorems proved, new fields created (or discovered).

So in many certain (and also deep and amazing) senses, no, mathematics is very much not "basically arithmetic".

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Your erroneous assertion that computers can do advanced math is the clearest indication of where you're coming from. Let's take an example: "Twin primes" are prime numbers like $101$ and $103$ that differ by $2.$ Nobody knows whether there are infinitely many of them, although Euclid showed in about 300 BC that there are infinitely many primes. Suppose you ask a computer: Are there infinitely many twin primes? A computer can't answer that any more than a computer can tell you whether Purgatory exists. But you may use a computer in your efforts to answer those questions.

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  • $\begingroup$ My guess is that there will be programs proving questions about an infinite numbers of some sort. And besides, I think a lot of mathematics evolves towards simple classifications suitable for being represented mainly by arithmetic. Example: commutative compositions of one-to-one functions on a set evolved to the structure theorem of Abelian groups. $\endgroup$ – Lehs Aug 5 '16 at 9:51
  • $\begingroup$ @Lehs : Computers are used in doing mathematics, just as things done with chalk on a blackboard are. But they cannot be so used unless there is a mathematician who understands. That mathematician is the one who does mathematics; the chalk and the blackboard are not doing mathematics. $\endgroup$ – Michael Hardy Dec 21 '16 at 22:28

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