What do you regard as the most basic operations in Mathematics ?

[studiot]

The title says it all.

What operations do you need to carry out before you can do any sort of mathematics ?

[Intoscience]

Recognising numbers and the order they go: counting

The functions I learned in primary school: +  -  /  x

But I'm not a mathematician so not qualified to answer with any authority.

 

[iNow]

Division, but more in a conceptual sense than an operational one.

Specifically, there must be more than one concept or object or entity to compare and transact to perform maths. You can do little with just ONE object/entity/concept/etc., but the moment you have two (aka more than one), mathematical operation only then becomes possible. 

Ergo, in order to carry out any sort of mathematics, I suggest that the operation of division is prerequisite. 

[zapatos]

I don't know if this is correct, but I've read that all mathematics are fundamentally different versions of counting.

[Intoscience]

2 hours ago, iNow said:

in order to carry out any sort of mathematics, I suggest that the operation of division is prerequisite

Thinking back, I heard/read something similar to this before.

Good point

[MigL]

Is any operation valid without identity ?

[studiot]

3 minutes ago, MigL said:

Is any operation valid without identity ?

Since I don't understand what you mean, I can't answer the question.

But I don't doubt you will have a worthwhile opinion on the OP, when you explain.

[Sensei]

Bitwise operators.

https://en.wikipedia.org/wiki/Bitwise_operation

NOT, AND, NAND, XOR, OR. XNOR.

(if you have AND and NOT, you can make NAND)

(if you have OR and NOT, you can make NOR)

(if you have XOR and constant %(1), you can make NOT)

(and a couple similar equivalences)

If you have bitwise operators, you can implement addition, subtraction, multiplication, division, modulo, and so on..

 

[studiot]

43 minutes ago, Sensei said:

Bitwise operators.

https://en.wikipedia.org/wiki/Bitwise_operation

NOT, AND, NAND, XOR, OR. XNOR.

(if you have AND and NOT, you can make NAND)

(if you have OR and NOT, you can make NOR)

(if you have XOR and constant %(1), you can make NOT)

(and a couple similar equivalences)

If you have bitwise operators, you can implement addition, subtraction, multiplication, division, modulo, and so on..

 

So how would you establish the equivalence relations greater than and less than in this system ?

[MigL]

3 hours ago, studiot said:

Since I don't understand what you mean, I can't answer the question.

Identity function is a process, additive, multiplicative, etc., that leaves the original value unchanged.
If the value returned cannot be the same as the original, it has no 'identity', and no other operations are valid.

Sorry I'm not using 'flowery' mathematical language, but identity function is usually used in set theory.

Or maybe I'm just not remembering correctly.

[Sensei]

1 hour ago, studiot said:

So how would you establish the equivalence relations greater than and less than in this system ?

CPU has internal flags, Z-Zero, N-Negative , to implement greater than, greater than or equal, equal, not-equal, less than or equal, and less than.

It is done by fake subtraction (CMP instruction), which result is lost, just sets/clears CPU flags. If a-b == 0 then Z flag is set. If a-b != 0 then Z flag is cleared. If a>b then a-b is positive so N flag is cleared. If a<b then a-b is negative number therefore N flag is set.

Z flag can be implemented as series of OR operators. n-th bit OR n-th-1 OR [....] OR 1st bit.

N flag is the highest bit of the number with specified number of digits. 8 bits number has 7th bit sign, 16 bits number has 15th bit sign, 32 bits number has 31th bit sign etc.

 

 

Edited August 9, 2022 by Sensei

[studiot]

36 minutes ago, Sensei said:

CPU has internal flags, Z-Zero, N-Negative , to implement greater than, greater than or equal, equal, not-equal, less than or equal, and less than.

It is done by fake subtraction (CMP instruction), which result is lost, just sets/clears CPU flags. If a-b == 0 then Z flag is set. If a-b != 0 then Z flag is cleared. If a>b then a-b is positive so N flag is cleared. If a<b then a-b is negative number therefore N flag is set.

Z flag can be implemented as series of OR operators. n-th bit OR n-th-1 OR [....] OR 1st bit.

N flag is the highest bit of the number with specified number of digits. 8 bits number has 7th bit sign, 16 bits number has 15th bit sign, 32 bits number has 31th bit sign etc.

 

 

So you can't do mathematics without a CPU ?

 

52 minutes ago, MigL said:

Identity function is a process, additive, multiplicative, etc., that leaves the original value unchanged.
If the value returned cannot be the same as the original, it has no 'identity', and no other operations are valid.

Sorry I'm not using 'flowery' mathematical language, but identity function is usually used in set theory.

Or maybe I'm just not remembering correctly.

Thanks for the clarification.

I still think that is too high a level of operation.

 

Inow has a similar idea to mine, with 'division'.  Mibne is still more basic though.

 

The two most basic operations in maths I can think of are are

1) Distinguishing between two 'objects', say A and B.

2) Putting 'objects' into correspondence with each other.

These two operations may have also been the start of language as well as mathematics.

 

All the proposed operations above (by others) rest on these two but may be that someone can think of some more.

 

This is a genuine discussion topic folks.

[Ghideon]

">"

I think this was one of the first "mathematics-related" things I grasped* before counting, numbers, addition or other mathematics. That there could be more (or fewer) of something and more or fewer of different things. Understanding for instance "there are more apples than there are oranges in this picture" came before counting items in the picture. I have no formal or mathematical argument, this is more of a personal reflection.

I note that @iNow's argument probably could be used here to put divison "/" before comparison ">"; first one realises that there are apples and oranges rather than one singe pile of fruit. Then comparisons (or other operations) can take place.

 

edit (just saw Studiot's post):

54 minutes ago, studiot said:

1) Distinguishing between two 'objects', say A and B.

I note that my example above starts with this, I did not think of that as an operation.

 

*) This was early in life so my recollection may be incorrect. I did of course not know or understand the specific notation or the mathematical implications at that time.

Edited August 9, 2022 by Ghideon
x-post

[Sensei]

1 hour ago, studiot said:

So you can't do mathematics without a CPU ?

Sorry, I don't understand your statement..

What I described is the built-in hardware of the processor in the form of transistors and other electronic components.

If engineers could do it with just logic gates, that I mentioned, then analyzing the inner workings of the processor will give you clues on how to do it yourself. Reverse engineer the most basic CPU internals and rewrite as mathematical equation.

[studiot]

1 hour ago, Ghideon said:

">"

I think this was one of the first "mathematics-related" things I grasped* before counting, numbers, addition or other mathematics. That there could be more (or fewer) of something and more or fewer of different things. Understanding for instance "there are more apples than there are oranges in this picture" came before counting items in the picture. I have no formal or mathematical argument, this is more of a personal reflection.

I note that @iNow's argument probably could be used here to put divison "/" before comparison ">"; first one realises that there are apples and oranges rather than one singe pile of fruit. Then comparisons (or other operations) can take place.

 

edit (just saw Studiot's post):

I note that my example above starts with this, I did not think of that as an operation.

 

*) This was early in life so my recollection may be incorrect. I did of course not know or understand the specific notation or the mathematical implications at that time.

I like it. +1

But correspondence is more primitive than counting - which requires numbers.

Consider the following.

The Australian Aborigenes had a number system    one, two, many.       3 numbers only.

So they could distinguish but not count.

 

Marked tally bones (there may have been tally sticks as well but these will not have lasted as well as bones) for 30,000 - 50,000 years ago have been found.

What were they for ?

Well perhaps the head of the tribe sent scouts out to mark a tally stick by scoring marks onto the tally in one-to-one correspondence with either goats they were herding or hunting or the war aprty from another tribe or whatever.
 

Perhaps the headman could count but the scouts ?  A matter of conjectures.

These marks predate known cave writing, though not pictures.

So the scouts were distinguishing and tallying.

 

1 hour ago, Sensei said:

Sorry, I don't understand your statement..

What I described is the built-in hardware of the processor in the form of transistors and other electronic components.

If engineers could do it with just logic gates, that I mentioned, then analyzing the inner workings of the processor will give you clues on how to do it yourself. Reverse engineer the most basic CPU internals and rewrite as mathematical equation.

Mathematics is about a whole lot more than machine calculation and numbers, geometry for instance.

How would your machine tell the diffeence between a square and a rectangle  or discuss the symmetries of a snowflake ?

[iNow]

38 minutes ago, studiot said:

How would your machine tell the diffeence between a square and a rectangle  or discuss the symmetries of a snowflake ?

Your mention of A and B and mine of one vs greater than one makes the answer here to the quoted question obvious…

With binary!

I’m in town all week, folks. Try the schnitzel. 

[joigus]

Perhaps the most primitive mathematical "operator" is that of a relation?: Any well-defined connection between two elements of a set.

aRb = a is related to b by means of relation R within a set A

and a relation within a set is any binary pairing, that is, any subset R of AxA

So a is related to b by relation R if (a,b) is in R (we write \( aRb \) )

and a is no related to b by R if (a,b) is not in R (we write \( a{\not}R b \)

For that you kind of must have set theory first, so... Perhaps "belongs", as an external operation (between elements and sets).

If you think about it, equivalence relations are just a particular example of relations in general.

If aRb, then bRa

If aRb and bRc, then aRc

and aRa always

And equivalence relations are at the basis of our categorical thinking. 

But this is so abstract that my head hurts.

So I guess what I'm saying is the Cartesian product.

Edited August 10, 2022 by joigus
correction

[studiot]

On 8/10/2022 at 1:12 AM, joigus said:

Perhaps the most primitive mathematical "operator" is that of a relation?: Any well-defined connection between two elements of a set.

....etc

Seems rather less primitive than I was thinking.

Surely you can only establish a connection if you can 'distinguish' the two elements ?

And many so called basic operations such as counting can be mimiced by not counting at all but simply placing in one to one correspondence.

[joigus]

OK I have to come clean at this point and confess I do not know what "being more primitive" means in mathematics.

I think it was Poincaré who tried to base everything mathematics in terms of group theory.

Another attempt of basing maths on something "primitive" was Felix Klein's Erlangen program to unify geometry.

Category theory seems to be another attempt at building a really primitive branch of maths. "Primitive" meaning something like "least number of assumptions." [?]

Perhaps "primitive" means theory A can be based on concepts derived from theory B, but not the other way around, and therefore theory B being more "primitive" than theory A?

I'm not sure of what mathematicians mean when they say they're trying to refer things to something more primitive.

[studiot]

4 hours ago, joigus said:

Perhaps "primitive" means theory A can be based on concepts derived from theory B, but not the other way around, and therefore theory B being more "primitive" than theory A?

Yes I think that's a good way to put it.  +1

But wider still, not only can theory A be based on theory B but also theories C, D, E....   and still not the other way round.

 

Thanks for that insight.

[Ghideon]

On 8/10/2022 at 12:53 AM, studiot said:

But correspondence is more primitive than counting - which requires numbers.

Thanks for your reply. I agree to your points and the example provided is a good illustration worthy some thoughts. I considered the ability to distinguish between things a skill necessary to survive rather than an example of primitive/basic mathematic operators. An example is the ability or required knowledge to sort out edible berries and reject poisonous ones.

This means that I could interpret "primitive"/"basic" in different ways*:
1 Early use of mathematics in human history. 
2 The first kind of primitive operator I grasped.
3 Primitive operators defined in contemporary mathematics.

I would probably apply different lines of thought regarding the title's question depending on which one of the alternative interpretations I choose.

 

*) There are others, these are the ones I see from my participation and comments so far in this topic.