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What I understand:

A number is an abstract concept. We use numerals such as names (i.e. cuatro), symbols (i.e. Arabic "4"), and numerical expressions (i.e. "9 - 5") to represent numbers.

Where I get tripped up:

"Value" is defined as the number a numeral represents, where "4" is a value of "3 + 1". Doesn't this contradict the author's earlier definition of "4" being a symbol?

Is "4" a number or a numeral?

Bonus Question

"4" is then defined as a numerical expression. Is a numerical expression a type of numeral, or are they synonymous to one another?

I value your insight!

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    $\begingroup$ At a certain point, it doesn't matter. Whether by $4$ we mean the symbol used, the natural number $4$, the integer $4$, the rational number $4$, the real number $4$, etc... they are all used interchangeably. The fact that we can rigorously define as whichever of these and can define how we can rigorously go from one to another is important to keep in the back of our mind in case anyone ever presses us on it, however in practice we don't bother including the details in everyday mathematics as it would needlessly complicate otherwise trivial matters. $\endgroup$ – JMoravitz Jan 26 at 13:33
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    $\begingroup$ There is no contradiction, just a shortcoming of language when we try to compare a thing to its name. When the author says 'Thus, $4$ is the value "$3 + 1$",' he means 'Thus, [the value represented by] $4$ is the value [represented by] "$3 + 1$", not 'Thus, [the numeral] $4$ is the value "$3 + 1$". $\endgroup$ – David Diaz Jan 26 at 13:46
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    $\begingroup$ Separate from mathematics, whether 4 is a symbol or a value strikes me as a question of the use-mention distinction, which is a matter of context and/or quotation marks. $\endgroup$ – Mark S. Jan 26 at 13:46
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    $\begingroup$ When you use quotes you are transforming an expression (that usually names a thing) into a name for an expression. Thus "the number $4$" vs "the numeral (expression) "$4$" " $\endgroup$ – Mauro ALLEGRANZA Jan 26 at 13:52
  • $\begingroup$ What is the source of the quoted excerpt? $\endgroup$ – Barry Cipra Jan 26 at 15:45
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I think this is mostly a semantic issue. If someone asked me who I was, I would say 'I'm Joe'; I wouldn't say 'I am the person represented by the name Joe'. It's good that the author initially draws a distinction between how we represent a number and what that number actually is. This is because some people overlook the fact that there is no difference whatsoever between $1/2$, $\frac{1}{2}$, $0.5$, $0.4\overline{9}$, and $0.4999...$—they all represent the same number in a kind of platonic sense. In practice, however, we would just say that they're the same number, and avoid quibbling over details. Mathematics would be far too cumbersome if every statement we make had to be $100\%$ precise.

So, as David has already mentioned in the comments, really we should say 'the number represented by $3+1$ is the same as the number represented by $4$', but no one bothers. And it is perfectly correct to write $3+1=4$, since everyone understands the meaning of those symbols, and it would be impossible to write mathematics without any shared notation.

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Just as "four" and "quatro" and "$4$" are symbols for the value $4$, the words "one hundred and one" is a symbol for the value $101$. It has more "words" but it is just a symbol. Here, "three plus one" might be called "three and one" (think "four and twenty" blackbirds baked in a pie) and it is just another symbol as is "$3+1$" so the conflict is resolved.

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The document is using the convention that "4" is a numeral, whereas $4$ [without quotations] represents a value. The problem with abstract ideas is that you need an abstract keyboard to type it out.

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