6 ms·

The title of this sounded very promising. Something that actually describes math from scratch needs to exist, but this ain't it. It runs off the rails from th

by lisper 5y ago

The title of this sounded very promising. Something that actually describes math from scratch needs to exist, but this ain't it. It runs off the rails from the very first sentence, where it asks "What is math?" but doesn't actually answer the question except to tell you that it's easy. I stopped reading when I got to "Numbers are objects you build to act like the concept of a number." No no no no no. No! Math is the act of inventing sets of rules for manipulating symbols that produce interesting or useful results, or exploring the behavior of sets of rules invented by others. One of the earliest example of such interesting or useful results is rules for manipulating symbols so that the results correspond to the behavior of physical objects (like sheep, or baskets of grain, or plots of land), and in particular, to their quantity so that by manipulating the right symbols according to the right rules allowed you to make reliable predictions about the behavior of and interactions between quantities of sheep and grain and land. The symbols that are manipulated according to these rules are called "numerals" and the quantities that they correspond to are called "numbers". But you can invent other rules for other symbols that produce other useful and interesting behavior, like "sets" and "vectors" and "manifolds" and "fields" and "elliptic curves." But it all boils down to inventing sets of rules for manipulating symbols.

_448 5y ago
> One of the earliest example of such interesting or useful results is rules for manipulating symbols so that the results correspond to the behavior of physical objects (like sheep, or baskets of grain, or plots of land)... Or was it the other way round? First came the objects and then people saw patterns in the transactions and then abstracted it away with symbols? One of the reasons children find maths difficult is because we tend to describe maths in this reverse order i.e. from symbols to physical objects(the way you described above). But explaining it in the actual order i.e. from physical objects to symbols will make maths accessible to many more kids.
- lisper 5y ago
  The evolution of math came by way of things like tally marks on sticks and pebbles in pots. The line between symbol and physical system is fuzzy. I think one of the reasons that people find math difficult is that the emphasis is on the symbol-manipulation rules and not on the reasons that certain sets of symbol-manipulation rules are useful. Kids are taught to count "1 2 3 4..." as if these symbols were handed down by God rather than being totally arbitrary. They should be taught to count "*, **, ***, ****, ..." and then, when they get to "***********" or so, they should be encouraged to invent shorter ways of writing these unwieldly strings. (Note, BTW, that when you use the "*, **, ***, ****, ..." numeral system, addition and multiplication become a whole lot easier!)
bawolff 5y ago
I don't get your objection "Numbers are objects you build to act like the concept of a number." So, numbers (i read that as a colloquialism for "numerals") are symbols that represent the abstract idea of a "number". You can't have (non-trivial) rules for manipulating symbols without symbols, so i don't understand your objection.
- lisper 5y ago
  The definition as stated is circular. It has the same information content as "A foo is an object that you build to act like the concept of a foo." Also, the symbols that stand for numbers have a name. They are called numerals. So: a number is an abstraction of a quantity, which is a number plus a unit. "Seven sheep" is a quantity of sheep. "Seven bushels of grain" is a quantity of grain. "Seven acres of land" is a quantity of land. "Seven" is the abstract property that these things have in common, and the symbol "7" is the numeral that denotes this property.
  - bawolff 5y ago
    I don't think its circular, just really informal. The two usages of the word "number" are refering to two different meanings of the word. If your original objection was a lack of formality, i'd say its a matter of taste, audience and author intent, but ultimately a fair objection. But this seems very far afield from your original complaint.
    - deleted 5y ago
      [deleted]
AnimalMuppet 5y ago
That "definition" rubbed me the wrong way, too. A bit recursive...
- Smaug123 5y ago
  I am quite happy with that "definition", not least because it's obviously not intended to be formal. But anyway, a very standard thing for a mathematics lecturer to say is "This is the only thing it could be", or "you know what this concept is already; we are just making some formal symbols to capture it", or words to that effect. It's pedagogically important to distinguish between genuinely new things that the reader is going to have to put some effort into rearchitecting their worldview around, versus things the reader already knows but might not have seen in this form before. Since everybody in this target audience already knows how to count, the treatment of the natural numbers is surely going to be in the latter category.
  - lisper 5y ago
    It's worse than non-formal, it's circular, and hence vacuous. It has the same information content as "A foo is an object that you build to act like the concept of a foo."
    - Smaug123 5y ago
      No! But I am not going to re-explain it; I have already said precisely why the information content is different, in the comment to which you replied. I shall merely quote it again. > It's pedagogically important to distinguish between genuinely new things that the reader is going to have to put some effort into rearchitecting their worldview around, versus things the reader already knows but might not have seen in this form before. Since it is common knowledge that nobody already knows what a "foo" is, but it is common knowledge that everyone already has a pretty uniform idea of what a "number" means in everyday life (really, how many ways are there you can count from 1 to 10), my assertion applies.
      - lisper 5y ago
chobytes 5y ago
Based on my reading you dont seem to fundamentally disagree with the author. I think hes just playing with the ambiguity of the word number for some effect. So: (Some mathematical formalism for) numbers are objects you build to act like the (informal) concept of a number.
- lisper 5y ago
  No! There is nothing informal about the mathematical concept of a number. And numbers can be defined without resorting to circular definitions that include the word "number" in the definition. It just so happens that the symbols that denote numbers behave like certain things in the real world when manipulated according to the rules that are customarily associated with the symbols. Those things in the real world have properties that correspond to the informal idea of numbers. But that has absolutely nothing to do with the math (except insofar as it's one of the things that makes math worth doing).
  - threatofrain 5y ago
    "Number" is an informal concept in mathematics. The real numbers are a specific concept with multiple formalizations. A model of a number system is an assembly of behaviors, rules, or objects which behave like the numbers you want. I don't see the circularity here, and given that they are tackling an informal concept of number, I don't see any loss of clarity either.
    - lisper 5y ago
      > A number is an informal concept in mathematics. No, it isn't. You're just wrong about that. Now, it is true that there are different kinds of numbers in math, but each different kind of number is formally defined. All of those definitions are interrelated, and collectively they define what it means to be a number. There is nothing informal about it.
      - threatofrain 5y ago
        > All of those definitions are interrelated, and collectively they define what it means to be a number. There is nothing informal about it. Then you could simply point to the formalism which unites various mathematical activity under some universal model of number. I will reiterate the claim again — "number" is an informal concept among mathematicians and they are fine with that.
Smaug123 5y ago
I disagree, and I believe you have made a category error: you are mistaking the description of a thing for the thing itself. I believe your error is the same as that of the one who asserts "the natural numbers are the finite transitive sets". In fact, the finite ordinals are only one possible implementation of the natural numbers in set theory, and what the natural numbers really are is "the abstract notion of the smallest structure you can do induction on". Similarly, any given symbolic representation of a thing is an implementation of that thing in mathematics, but it need not be the thing itself. Perhaps our best (or even only) hope of automatic mathematics verification is to find a symbolic representation of a thing, and there are some areas where we have in some sense "definitely found the right symbols" which cleave very tightly to the thing they represent; and it's probably always true that if you can phrase something in the symbolic language, then you will be more rigorously constrained to say correct things by construction. But when an actual human actually does mathematics, they may very well not be manipulating symbols at all until they come to the point where they must transmit their thoughts to someone else. In fact, in some cases it may even be possible to transmit nontrivial mathematical thoughts without symbols; one possible method is simply "analogy". To revisit the example at the start, by the way: If you choose the language of category theory, of course, you can formalise "the abstract notion of the smallest structure you can do induction on" into the symbolic representation whose name is "natural numbers object", and thereby claim that once more we are in the world of rules for manipulating symbols. But I do not believe the implied claim that "unless you think of the naturals as their category-theoretic representation, then you are not doing mathematics"!
- lisper 5y ago
  > I disagree With what? > you are mistaking the description of a thing for the thing itself How? > what the natural numbers really are is "the abstract notion of the smallest structure you can do induction on". And who granted you the authority to define what the natural numbers "really are"? The last time I checked, numbers were not actually part of physical reality, they were a kind of concept or idea. People are free to attach whatever labels they like to concepts and ideas. They can even overload terms to mean different things in different contexts. So just because I use the word "number" in some colloquial context to stand for a different idea than you do in some other context doesn't make it wrong.
kmill 5y ago
Ah, I see you're a mathematical formalist. That's one philosophy of what math is, but it's not the only one. What's going on in my mind when I visualize 3-dimensional manifolds and manipulate them? Cutting pieces apart, reattaching them elsewhere, stereographically reprojecting objects embedded in a 3-sphere. I don't see where the symbols are exactly here -- it seems like I'm (at least trying to) manipulate "actual" mental objects, ones my visual cortex are able to help me get glimpses of. Everything I'm doing can be turned into symbolic reasoning, but it's a fairly painful process. It seems closer to the basis for the philosophy of intuitionism, that math derives from mental constructions. I think it's rather interesting that many creatures (including ourselves) have a natural ability to at a glance tell how many things there are, up to about five objects or so. Maybe it's useful evolutionarily for counting young, or for detecting when a berry has gone missing, etc. I think it's also rather interesting that many creatures have episodic memory. Sometimes it seems to me that the invention of symbolic number derives from these two capabilities: through our experience that things can happen in sequence, we extend our natural ability to count while believing these "numbers" are meaningful. Each number is a story that plays out for what's been accumulated.
- lisper 5y ago
  The boundary where abstract noodling around with mental images becomes math lies precisely at the point where you can render those images symbolically. That is the thing that distinguishes math from all other forms of human mental endeavor.
  - WhitneyLand 5y ago
    So spending say 5 years, noodling around with abstract mental concepts and ending with symbols that turn out to be manipulatable and useful in a mathematical way means, you haven’t done math until year 6 starts? I’m trying to see the line as clearly drawn as your comment suggests. The exact boundary doesn’t seem as simple as, stops here / starts here.
    - lisper 5y ago
      > So spending say 5 years, noodling around with abstract mental concepts and ending with symbols that turn out to be manipulatable and useful in a mathematical way means, you haven’t done math until year 6 starts? Yes, that's right. Until you write down the symbols you quite literally haven't done the math.
deleted 5y ago
[deleted]
dwohnitmok 5y ago
"the quantities that they correspond to are called 'numbers'." is rather circular here no? You haven't really defined quantities or numbers except in terms of each other and yet at least one of these seems important enough to be called a mathematical concept that is distinct from a numeral. Put another way, we clearly can distinguish many different numeral systems as referring to the "same thing." We can encode that "same thing" symbolically via a set of logical symbols to form axioms, but even there the same issue arises. Encoding the natural numbers via an embedding of second-order Peano Axioms in first-order ZFC or via a straight FOL Peano Axioms definition also seems to fall flat, as we still have not been able to capture the idea that "these are all really the same thing" since these are now different sets of symbols and rules. And we haven't even touched the thorny issue of what "encoding" and more generally "mapping" really means if everything is just symbols and rules. But perhaps you are fine with infinite egress (which is perfectly defensible). The natural numbers have a more distressing problem when it comes to Godel's Incompleteness Theorems. There is presumably only one "true" set of natural numbers in our universe because the natural numbers have physical ramifications. That is for any symbolic, FOL statement of the natural numbers, we can create a corresponding physical machine whose observable behavior is dependent on whether that statement is true or false. And yet by Godel's incompleteness theorem we can never hope to fully capture the rules and symbols that determine these "natural numbers," and yet presumably most people would agree that these natural numbers exist and are a valid object of mathematical study. So how does that fit in? What do we call the subject that studies these objects? What do we even call these objects if not mathematical numbers? (As an aside I'm perhaps more of a formalist than I let on, but I find the realist side a fun playground to explore in.)
- lisper 5y ago
  > "the quantities that they correspond to are called 'numbers'." is rather circular here no? No. "Quantity" is different from "number". A quantity is a number plus a unit. "Two" is a number, not a quantity. "Two sheep" is a quantity. This is definitional progress because I can actually show you two sheep in order to explain to you what that means, whereas I cannot show you "two". > infinite egress It is not an infinite regress, though explaining why is much too long for an HN comment. It also makes an interesting little puzzle to figure out where and how the recursion bottoms out, but here's a hint: how do you know what a "sheep" is?

by lisper 5y ago

_448 5y ago

> One of the earliest example of such interesting or useful results is rules for manipulating symbols so that the results correspond to the behavior of physical objects (like sheep, or baskets of grain, or plots of land)... Or was it the other way round? First came the objects and then people saw patterns in the transactions and then abstracted it away with symbols? One of the reasons children find maths difficult is because we tend to describe maths in this reverse order i.e. from symbols to physical objects(the way you described above). But explaining it in the actual order i.e. from physical objects to symbols will make maths accessible to many more kids.

lisper 5y ago
The evolution of math came by way of things like tally marks on sticks and pebbles in pots. The line between symbol and physical system is fuzzy. I think one of the reasons that people find math difficult is that the emphasis is on the symbol-manipulation rules and not on the reasons that certain sets of symbol-manipulation rules are useful. Kids are taught to count "1 2 3 4..." as if these symbols were handed down by God rather than being totally arbitrary. They should be taught to count "*, **, ***, ****, ..." and then, when they get to "***********" or so, they should be encouraged to invent shorter ways of writing these unwieldly strings. (Note, BTW, that when you use the "*, **, ***, ****, ..." numeral system, addition and multiplication become a whole lot easier!)

bawolff 5y ago

I don't get your objection "Numbers are objects you build to act like the concept of a number." So, numbers (i read that as a colloquialism for "numerals") are symbols that represent the abstract idea of a "number". You can't have (non-trivial) rules for manipulating symbols without symbols, so i don't understand your objection.

lisper 5y ago
The definition as stated is circular. It has the same information content as "A foo is an object that you build to act like the concept of a foo." Also, the symbols that stand for numbers have a name. They are called numerals. So: a number is an abstraction of a quantity, which is a number plus a unit. "Seven sheep" is a quantity of sheep. "Seven bushels of grain" is a quantity of grain. "Seven acres of land" is a quantity of land. "Seven" is the abstract property that these things have in common, and the symbol "7" is the numeral that denotes this property.
- bawolff 5y ago
  I don't think its circular, just really informal. The two usages of the word "number" are refering to two different meanings of the word. If your original objection was a lack of formality, i'd say its a matter of taste, audience and author intent, but ultimately a fair objection. But this seems very far afield from your original complaint.
  - deleted 5y ago
    [deleted]

AnimalMuppet 5y ago

That "definition" rubbed me the wrong way, too. A bit recursive...

Smaug123 5y ago
I am quite happy with that "definition", not least because it's obviously not intended to be formal. But anyway, a very standard thing for a mathematics lecturer to say is "This is the only thing it could be", or "you know what this concept is already; we are just making some formal symbols to capture it", or words to that effect. It's pedagogically important to distinguish between genuinely new things that the reader is going to have to put some effort into rearchitecting their worldview around, versus things the reader already knows but might not have seen in this form before. Since everybody in this target audience already knows how to count, the treatment of the natural numbers is surely going to be in the latter category.
- lisper 5y ago
  It's worse than non-formal, it's circular, and hence vacuous. It has the same information content as "A foo is an object that you build to act like the concept of a foo."
  - Smaug123 5y ago
    No! But I am not going to re-explain it; I have already said precisely why the information content is different, in the comment to which you replied. I shall merely quote it again. > It's pedagogically important to distinguish between genuinely new things that the reader is going to have to put some effort into rearchitecting their worldview around, versus things the reader already knows but might not have seen in this form before. Since it is common knowledge that nobody already knows what a "foo" is, but it is common knowledge that everyone already has a pretty uniform idea of what a "number" means in everyday life (really, how many ways are there you can count from 1 to 10), my assertion applies.
    - lisper 5y ago

chobytes 5y ago

Based on my reading you dont seem to fundamentally disagree with the author. I think hes just playing with the ambiguity of the word number for some effect. So: (Some mathematical formalism for) numbers are objects you build to act like the (informal) concept of a number.

lisper 5y ago
No! There is nothing informal about the mathematical concept of a number. And numbers can be defined without resorting to circular definitions that include the word "number" in the definition. It just so happens that the symbols that denote numbers behave like certain things in the real world when manipulated according to the rules that are customarily associated with the symbols. Those things in the real world have properties that correspond to the informal idea of numbers. But that has absolutely nothing to do with the math (except insofar as it's one of the things that makes math worth doing).
- threatofrain 5y ago
  "Number" is an informal concept in mathematics. The real numbers are a specific concept with multiple formalizations. A model of a number system is an assembly of behaviors, rules, or objects which behave like the numbers you want. I don't see the circularity here, and given that they are tackling an informal concept of number, I don't see any loss of clarity either.
  - lisper 5y ago
    > A number is an informal concept in mathematics. No, it isn't. You're just wrong about that. Now, it is true that there are different kinds of numbers in math, but each different kind of number is formally defined. All of those definitions are interrelated, and collectively they define what it means to be a number. There is nothing informal about it.
    - threatofrain 5y ago
      > All of those definitions are interrelated, and collectively they define what it means to be a number. There is nothing informal about it. Then you could simply point to the formalism which unites various mathematical activity under some universal model of number. I will reiterate the claim again — "number" is an informal concept among mathematicians and they are fine with that.

Smaug123 5y ago

I disagree, and I believe you have made a category error: you are mistaking the description of a thing for the thing itself. I believe your error is the same as that of the one who asserts "the natural numbers are the finite transitive sets". In fact, the finite ordinals are only one possible implementation of the natural numbers in set theory, and what the natural numbers really are is "the abstract notion of the smallest structure you can do induction on". Similarly, any given symbolic representation of a thing is an implementation of that thing in mathematics, but it need not be the thing itself. Perhaps our best (or even only) hope of automatic mathematics verification is to find a symbolic representation of a thing, and there are some areas where we have in some sense "definitely found the right symbols" which cleave very tightly to the thing they represent; and it's probably always true that if you can phrase something in the symbolic language, then you will be more rigorously constrained to say correct things by construction. But when an actual human actually does mathematics, they may very well not be manipulating symbols at all until they come to the point where they must transmit their thoughts to someone else. In fact, in some cases it may even be possible to transmit nontrivial mathematical thoughts without symbols; one possible method is simply "analogy". To revisit the example at the start, by the way: If you choose the language of category theory, of course, you can formalise "the abstract notion of the smallest structure you can do induction on" into the symbolic representation whose name is "natural numbers object", and thereby claim that once more we are in the world of rules for manipulating symbols. But I do not believe the implied claim that "unless you think of the naturals as their category-theoretic representation, then you are not doing mathematics"!

lisper 5y ago
> I disagree With what? > you are mistaking the description of a thing for the thing itself How? > what the natural numbers really are is "the abstract notion of the smallest structure you can do induction on". And who granted you the authority to define what the natural numbers "really are"? The last time I checked, numbers were not actually part of physical reality, they were a kind of concept or idea. People are free to attach whatever labels they like to concepts and ideas. They can even overload terms to mean different things in different contexts. So just because I use the word "number" in some colloquial context to stand for a different idea than you do in some other context doesn't make it wrong.

kmill 5y ago

Ah, I see you're a mathematical formalist. That's one philosophy of what math is, but it's not the only one. What's going on in my mind when I visualize 3-dimensional manifolds and manipulate them? Cutting pieces apart, reattaching them elsewhere, stereographically reprojecting objects embedded in a 3-sphere. I don't see where the symbols are exactly here -- it seems like I'm (at least trying to) manipulate "actual" mental objects, ones my visual cortex are able to help me get glimpses of. Everything I'm doing can be turned into symbolic reasoning, but it's a fairly painful process. It seems closer to the basis for the philosophy of intuitionism, that math derives from mental constructions. I think it's rather interesting that many creatures (including ourselves) have a natural ability to at a glance tell how many things there are, up to about five objects or so. Maybe it's useful evolutionarily for counting young, or for detecting when a berry has gone missing, etc. I think it's also rather interesting that many creatures have episodic memory. Sometimes it seems to me that the invention of symbolic number derives from these two capabilities: through our experience that things can happen in sequence, we extend our natural ability to count while believing these "numbers" are meaningful. Each number is a story that plays out for what's been accumulated.

lisper 5y ago
The boundary where abstract noodling around with mental images becomes math lies precisely at the point where you can render those images symbolically. That is the thing that distinguishes math from all other forms of human mental endeavor.
- WhitneyLand 5y ago
  So spending say 5 years, noodling around with abstract mental concepts and ending with symbols that turn out to be manipulatable and useful in a mathematical way means, you haven’t done math until year 6 starts? I’m trying to see the line as clearly drawn as your comment suggests. The exact boundary doesn’t seem as simple as, stops here / starts here.
  - lisper 5y ago
    > So spending say 5 years, noodling around with abstract mental concepts and ending with symbols that turn out to be manipulatable and useful in a mathematical way means, you haven’t done math until year 6 starts? Yes, that's right. Until you write down the symbols you quite literally haven't done the math.

deleted 5y ago

[deleted]

dwohnitmok 5y ago

"the quantities that they correspond to are called 'numbers'." is rather circular here no? You haven't really defined quantities or numbers except in terms of each other and yet at least one of these seems important enough to be called a mathematical concept that is distinct from a numeral. Put another way, we clearly can distinguish many different numeral systems as referring to the "same thing." We can encode that "same thing" symbolically via a set of logical symbols to form axioms, but even there the same issue arises. Encoding the natural numbers via an embedding of second-order Peano Axioms in first-order ZFC or via a straight FOL Peano Axioms definition also seems to fall flat, as we still have not been able to capture the idea that "these are all really the same thing" since these are now different sets of symbols and rules. And we haven't even touched the thorny issue of what "encoding" and more generally "mapping" really means if everything is just symbols and rules. But perhaps you are fine with infinite egress (which is perfectly defensible). The natural numbers have a more distressing problem when it comes to Godel's Incompleteness Theorems. There is presumably only one "true" set of natural numbers in our universe because the natural numbers have physical ramifications. That is for any symbolic, FOL statement of the natural numbers, we can create a corresponding physical machine whose observable behavior is dependent on whether that statement is true or false. And yet by Godel's incompleteness theorem we can never hope to fully capture the rules and symbols that determine these "natural numbers," and yet presumably most people would agree that these natural numbers exist and are a valid object of mathematical study. So how does that fit in? What do we call the subject that studies these objects? What do we even call these objects if not mathematical numbers? (As an aside I'm perhaps more of a formalist than I let on, but I find the realist side a fun playground to explore in.)

lisper 5y ago
> "the quantities that they correspond to are called 'numbers'." is rather circular here no? No. "Quantity" is different from "number". A quantity is a number plus a unit. "Two" is a number, not a quantity. "Two sheep" is a quantity. This is definitional progress because I can actually show you two sheep in order to explain to you what that means, whereas I cannot show you "two". > infinite egress It is not an infinite regress, though explaining why is much too long for an HN comment. It also makes an interesting little puzzle to figure out where and how the recursion bottoms out, but here's a hint: how do you know what a "sheep" is?