Let me see if I understand this. You’re making a distinction between the legitimate use of the word ‘is’ to make a statement about objective reality vs the use of the word ‘is’ to state a subjective preference, and your only concern here is with confusions between the two contexts that result in a subjective use of ‘is’ appearing to be an objective use? — Joshs

Correct.But I'll add that I consider many apparently objective statements to be subjective.
Example: "The cat is on the table" -> "I see the cat on the table"

As a mathematician I must object to your example though. Saying 'two plus two is four' rather than the more formal 'two plus two equals four' will often lead to confusion. We just don't need 'is' in that context and it causes trouble if we do use it. The word 'equals' in mathematics conveys a relationship with a precise meaning that differs from that usually attributed to the dreaded verb 'is'. — andrewk

Good point and good response overall. Do you have a better example of a truly objective statement?
What about "There is no largest prime number"?
I consider that as a genuine objective statement (that is true).

Using it to express category membership (attributing properties) also seems harmless to me, and shorter than the e-prime alternative. Only the 'identity' and 'existence' uses cause serious trouble. — andrewk

I think attributing properties can be problematic, too, as in "That is a good movie"
But maybe attributing to myself is OK. "I am feeling happy"

I have worked on minimising my use of the the verb 'to be' over the past few years and find it a really helpful discipline, with profound benefits. — andrewk

Yes. I think the book I wrote in mostly E-Prime is a better book than it would have been otherwise.
But sometimes E-Prime seems awkward. For instance, the last sentence could have been
Yes. I regard the book I wrote in mostly E-Prime as superior to what I might have written otherwise.

Open question: Does the E-Prime attitude better accord with quantum mechanics in that, under the Copenhagen Interpretation, QM tells us what we will see if we measure rather than what IS happening when we aren't measuring. (On the other hand, Bohmian Mechanics does tell us what is happening.)

... if you kept all 4 donuts, that would be different from sharing them out 3 for you and 1 for me. — Real Gone Cat

You have completely missed the point. It is not about the math. It is about the word 'is'. The sum of 2+2 is 4, the sum of 3+1 is 4, but 2+2 is not 3+1.

If 2+2 is not 3+1 simply because they represent different partitions of 4, then 2+2 is not 4 either.

If 2+2 is 4 because they have the same numeric value, then 2+2 is 3+1.

The only way I can make any sense of what you're saying is to assume that you are thinking of "4" as the name of a set to which distinct elements 2+2 and 3+1 belong. But does 4 belong to "4"? Then 2+2 is distinct from 4 and clearly 2+2 is not 4.

then 2+2 is not 4 either. — Real Gone Cat

You are catching on. The sum of 2+2 is (equal to ) 4.

If 2+2 is 4 because they have the same numeric value, then 2+2 is 3+1. — Real Gone Cat

Here we get into the question of number theory. The most important contemporary work on this is Jacob Klein's "Greek Mathematical Thought and the Origins of Algebra". Numbers are often treated as abstract entities, but for the Greeks a number tells us how many. It is always a number of something, of some unit, the unit of the count.

Klein worked with Husserl on this. It is not simply a historical study of an outmoded way of thinking about numbers. The claim is that something is lost when we treat numbers symbolically.

When you shift from thinking about numbers as abstract entities to counting then it becomes clear why 2+2 and 3+1 are not the same. Any child who learns math using manipulatives knows this. If I have 3 units, donuts or dollars and you have 1, that is not the same as each of us having 2. If I have 10 dollars and you have 10 cents, we each have 10 of something but not the same thing. The numerical value is the same but 10 dollars is not 10 cents.

You have stated, over and over, that "2+2 is 4" and "3+1 is 4". Without qualifying the "is". Go back and check.

Now it's some great revelation that 2+2 is NOT 4 ?

In math, we call what you're referring to partitions. But unless you and your audience already know that you are talking about partitions, no one - NO ONE - would say "2+2 is not 3+1". Especially after having claimed "3+1 is 4".

Except the mystics on TPF. You're always searching for the woo.

So from now on, when discussing numbers, we know that "is" refers to partitions. Got it.

Oh, and your last paragraph? A total non sequitur.

Without qualifying the "is" — Real Gone Cat

At the risk of sounding like Bill Clinton, the question is what is is. It is the OP that stated 2+2 is 4. What I said in my first response was:

"is" as used here is short for "is equal to". — Fooloso4

and in the next:

This is commonly understood to mean two plus two equals four and not two plus two is the same thing as four. 3+1 "is" 4 in the sense of equals 4 but not that 3+1 and 2+2 are the same thing. We could do without "is": 2+2=4, 3+1=4, 2+2=3+1. — Fooloso4

Now it's some great revelation that 2+2 is NOT 4 ? — Real Gone Cat

It is not a revelation, it is a clarification on what it means to say that 2+2 is 4. The OP contrasts mathematics and "the material world". But this is to treat numbers or arithmetic (Greek ἀριθμός - arithmós, meaning number) as an abstraction. While there are certainly advantages to this, we should not lose sight of the fact that a number still retains its original meaning, that is, it tells us how many of something. And what that something is is not first or foremost abstract units.

In math, we call what you're referring to partitions. — Real Gone Cat

You seem to have no idea what I am referring to. Let me try one more time. If I ask how many, in order to answer you will have to know how many of what. You have to know what it is that is being counted. If you are to count how many apples, the oranges do not count. If you are counting pieces of fruit the fruit flies do not count.

Once again, the division the OP makes is problematic. Our concern is not simply with numbers as abstractions, but with the question of how many of something. Knowing that 2+2=4 is of limited interest unless we are talking about 2+2 of something or other, that is, we are still within the material world. You cannot make an apple pie with oranges. Although two plus two equal four, two apples plus two apples do not equal four oranges

So you want to take math back to pebble counting. Okay, let's try a thought experiment. If you hold a donut and someone hands you another donut, do you have 1+1 or 2 donuts? Does holding them in one hand or in separate hands matter?

You're using "is" to refer to the partitioning of sets. And now that I know, I'm fine with it. But we could have avoided any confusion if you had said from the beginning, "2+2 and 3+1 are different because they break up the number 4 in two distinct ways".

So you want to take math back to pebble counting. — Real Gone Cat

Nope.

It is a matter of ontology.

Okay, let's try a thought experiment. If you hold a donut and someone hands you another donut, do you have 1+1 or 2 donuts? — Real Gone Cat

Can you count? Maybe you do need pebbles or some other manipulative.

Okay, let's try a thought experiment. If you hold a donut and someone hands you a dollar do you have 1+1 or 2?

You're using "is" to refer to the partitioning of sets. — Real Gone Cat

Do we need to go over this again? I am using "is" as it is typically used, short for is equal to.

"2+2 and 3+1 are different because they break up the number 4 in two distinct ways". — Real Gone Cat

That is one way of looking at it, but you are still treating numbers as abstractions, as symbolic entities. If I have 3 of something and you have 1 this is not breaking up the number, it is breaking up whatever it is we are counting.

This might help you see what is at issue: It is a review of Klein and Husserl's work on mathematics: https://ndpr.nd.edu/reviews/the-origin-of-the-logic-of-symbolic-mathematics-edmund-husserl-and-jacob-klein/

It begins:

This much needed book should go a long way both toward correcting the under-appreciation of Jacob Klein's brilliant work on the nature and historical origin of modern symbolic mathematics, and toward eliciting due attention to the significance of that work for our interpretation of the modern scientific view of the world.

A bit further on:

Specifically at issue is Husserl's expressed concern over the loss of an "original intuition" to ground symbolic mathematical science, and the consequent breakdown of meaning in that science. For the Husserl of Crisis, the history of this breakdown consists of two stages. First is the geometrical idealization of the world via what he terms "Galilean science" (taken as a kind of collective noun). Second is the formalization of that science by means of symbolic algebra, which latter surreptitiously substitutes symbolic mathematical abstractions for the directly intuited realities of the real world ("life-world"). In the face of such loss of meaning, which fundamentally determines (and threatens) modern western civilization in the modern scientific age, the urgent task of philosophy is to bring to light or to "desediment" (so Hopkins) the historically accreted, and by now almost entirely occluded, original meaning constituents of the concepts of modern mathematical science, so as to recover and reactivate the authentic sense of these concepts./quote]

Husserl and Klein want to take math back to pebble counting. And you have apparently joined in. Good for you. I'm not an intuitionist and have no interest.

You object to my 1+1 vs. 2 example. I assume you think you're holding 2 donuts. But why does handing me one turn it into 1+1? What if we are holding the donuts so that they are touching?

If 4 people each hold a donut, you would say that that is different from 1 person holding all four (1+1+1+1 is not 4). But what if 2 of to them live in Paris an 2 in New York? Isn't that 2+2?

If I'm holding a donut and a dollar, the cardinality of the set of objects I'm holding is 2. You seem incapable of accepting a set made up of disparate objects.

Husserl and Klein want to take math back to pebble counting. And you have apparently joined in. — Real Gone Cat

You clearly have not understood them or more likely did not even take the time to read the review.

Instead of snide remarks that make you feel superior because you can like any competent school child, look up who Husserl and Klein are and the importance of their contributions.

This is a philosophy forum. Ontology is of central concern. Adding is not.

From the blurb you provided

Specifically at issue is Husserl's expressed concern over the loss of an "original intuition" to ground symbolic mathematical science ...

and

For the Husserl of Crisis, the history of this breakdown consists of ... symbolic algebra, which latter surreptitiously substitutes symbolic mathematical abstractions for the directly intuited realities of the real world ("life-world") ...

What are "directly intuited realities"? Could it include pebble counting? Apparently, abstraction is the devil's work (and thus Kronecker hounds poor Cantor into depression).

If you're not too furious, please check out the next post.

I have been giving this some thought. Our debate has nothing to do with the word "is", it's with the word "plus".

I realized I have no idea what you mean by the + symbol. It could indicate the addition of numbers as in arithmetic (this seems unlikely given your rejection of "is" meaning "equal to"). It could indicate the cardinalities involved in the union of disjoint subsets (although you seem to recoil at the notion of partitions). What is your definiton of "plus"?

Then questions follow

• Is your "plus" commutative? (i.e., are 3+1 and 1+3 the same or different?)
• Does your "plus" have an identity element? (i.e., if you have all the donuts and I have none, is that 4+0 or just 4?)
• Are negatives defined for your "plus"? (i.e., does -1 + 5 have meaning for you? How do you count -1 donut?)

Let me know. Hope you're not too angry.

A math joke to lighten the mood ...

A biologist, a physicist, and a mathematician are sitting on a bench across from an apartment building. They observe two people enter the building. Five minutes later, three walk out. How does each react?

• The biologist : "They must have reproduced."
• The physicist : "My initial measurement must have been wrong."
• The mathematician : "Now if one person enters the building, it will be empty again."

Could it include pebble counting? — Real Gone Cat

It is more far reaching. A count it related to the idea of giving an account as well as the question of what is to count, that is, not what it is to count but what counts. There is also a connection with logos in its original sense of gathering together. There is also the question of the 'one' and the 'one and the many', which plays out in various ways in Plato and Aristotle.

Aristotle says that two is the first number. One is not a number, it is the unit (the one) of the count. We count "ones". This is why the question of how many must address the question of how many of what. We can still see this in that when we say that there is a number of things we don't mean one thing.

Plato says that the Forms are each one. Each is distinct and unique.

I have been giving this some thought. Our debate has nothing to do with the word "is", it's with the word "plus". — Real Gone Cat

Well, it started with "is", but in order to see why I would say the 3+1 is not 2+2 I raised the question of what a number is. As abstract entities 3+1 and 2+2 might be regarded as the same since both equal 4, but when we shift to the "material world" other things come into consideration.

given your rejection of "is" meaning "equal to") — Real Gone Cat

No, just the opposite. What I said, several times and from the beginning is that:

"is" as used here is short for "is equal to". — Fooloso4

Is means equal to.

Hope you're not too angry. — Real Gone Cat

Not at all.

Your joke kind of points to what I am getting at.

Gotcha. But we've moved on from the ancient Greeks' weird take on numbers. Now we accept that one is a number, and so is zero. So are negatives, and irrationals, and imaginaries, and transcendentals. (By the way, "number of things" is a metaphor.)

Math left the Greeks behind a long time ago. Their contributions were incredibly important but eulogizing their achievements sometimes led to misunderstandings or wrong turnings. Look at how clinging to Euclid's fifth postulate held back geometry.

Not every pronouncement by Plato and Aristotle should be held up as exalted. Aristotle thought women had fewer teeth than men.

Finally, I believe very few mathematicians today belong to the intuitionist school of thought.

So "is" means equal to. Unless it doesn't. I'm sorry, but that's incoherent. If "is" means equal to, then 3+1 is 2+2. If "is" doesn't mean equal to then you need to define it as more than "looks like".

And I ask again, what is your definition of "plus"? Is it commutative? Does it have an identity element? Does it allow for inverses (i.e., negatives)? Is it mathematical at all?

You've evaded many of my questions (or shrugged them off with a "that's obvious!" argument). I want to go back to an earlier question. If you are holding a donut in each hand, is it 1+1 or 2? Why does handing me a donut matter?

So "is" means equal to. Unless it doesn't. — Real Gone Cat

Right.

I'm sorry, but that's incoherent. — Real Gone Cat

The sum of 2+2 is equal to the sum of 3+1. This much we agree on. But sum totals are not the only thing at issue.

If I have 3 dollars and you have 1 dollar that is not equal to me having 2 dollars and you having 2 dollars. In that case we do not each have an equal amount of dollars. In that case 3+1 is not 2+2.

This is so basic I am surprised you do not understand it. Most children would immediately recognize that one person having more and the other less is not an equal amount.

The problem is your definition of "plus", and you won't answer me. To be generous, I think you mean something like "3 things over here and 1 thing over there" when you say "3+1". But that's called partitioning in math.

What we've arrived at is this: sometimes "is" means "is equal to", and other times "is" means "is the same partition as". When you say "3+1 is 4", you mean "3+1 is equal to 4". When you say "3+1 is not 2+2", you mean "3+1 is not the same partition of 4 as 2+2".

My contention - stated in an earlier comment - is you can't switch between meanings in the same sentence. You can't say, "3+1 is 4, but 3+1 is not 2+2", without sowing confusion. No one will understand you. I don't know why you can't see this. It's like saying, "Bill cans peas, but Sally cannot peas". It's nonsense.

You still don't get it. Time for me to move on.

So "is" means equal to. Unless it doesn't. — Real Gone Cat

There's yet another issue with taking "is" as something like "is the same as" in the most general sense.

"3 + 1" and "4" are obviously different expressions. So, to say that "3 + 1" is "4" must not mean they're the same expression, only that they have the same value. You'll agree with that, I assume. So our equals sign doesn't mean "is the same as" but only "has the same value as".

The reason that's interesting, to some philosophers, is because it means that "3 + 1 = 4" can be informative. "4 = 4" might also be informative, but what "4 = 4" tells you, that 4 is equal to itself, is different from what "3 + 1 = 4" tells you. But in mathematics we are authorized to substitute equals for equals anywhere and always: "3 + 1 = 4" is also a substitution rule, so anywhere you see "3 + 1" you can substitute "4" without changing the truth-value of your equations.

But even though you're not changing the truth value of the equation, you're changing something, else "3 + 1 = 4" would say the same thing as "4 = 4", and there's clearly a sense in which it doesn't. Frege's solution to this little puzzle was, roughly, that "3 + 1" and "4", seen as expressions rather than as values, have a sense as well as a reference: they both refer to the same value, 4, but in different ways. On such a scheme, "3 + 1 = 4" informs you that these two different expressions have the same value, and you have to be told that because expressions have a sense as well as a reference, and the sense of "3 + 1" is different from the sense of "4".

If you don't have some such scheme, you have to have some other explanation for what we're doing when we teach someone mathematics. How is that someone could know that "3233 = 3233" but not know that "53 * 61 = 3233"?

Again, the problem I have with Foolos4 is switching between meanings of "is" in a single sentence. You shouldn't say, "3+1 is 4" AND "3+1 is not 2+2" in the same sentence. Either they're both "is" or they're both "is not".

Taking it a bit further : I concede that 3+1 and 2+2 can have different meanings. We teach children that 3+1 and 2+2 are different ways to arrive at 4 (called partitions when we get to advanced math). But once you're above the age of 8 (or so), to hear someone say "3+1 is not 2+2" is going to be problematic. The speaker is then going to have to explain, "Oh, I meant splitting 4 things into 3 and 1 is different from 2 and 2". The speaker can't just say, "Come on, it's obvious!" (or assume the listener will have Frege's notions of sense and reference instantly leap to mind). And the reason is we learn to associate "plus" with addition, and "is" with equal-to when numbers are being used. A better sentence would be, "3 and 1 is not the same partitioning as 2 and 2".

I once wrote a book mostly in E-Prime. It’s available for free reading and download at ScienceAsNaturalTheology.org — Art48

This looks like an interesting read. I will give it a go. Thank you for the link, and the effort it must have taken to put this together.

Again, the problem I have with Foolos4 is switching between meanings of "is" in a single sentence. You shouldn't say, "3+1 is 4" AND "3+1 is not 2+2" — Real Gone Cat

The point is that it should not be taught that 2+2 "is" 4. That is the point of my seemingly contradictory or paradoxical statement. 3+1 "is" 4 is generally unproblematic when it is understood that what is meant is "is equal to", but when it is taken to mean something like "the same as" or "one and the same" confusion can arise. 3+1 is not the same as 2+2.

My second post, which was a response to you:

This is commonly understood to mean two plus two equals four and not two plus two is the same thing as four. 3+1 "is" 4 in the sense of equals 4 but not that 3+1 and 2+2 are the same thing. We could do without "is": 2+2=4, 3+1=4, 2+2=3+1. — Fooloso4

The speaker is then going to have to explain, "Oh, I meant splitting 4 things into 3 and 1 is different from 2 and 2". — Real Gone Cat

You mean like when I said?:

If we are given 4 donuts and I take 3 and give you one, you might complain that is not fair. Would you be satisfied if I defended this by saying that since 2+2 is 4 and 3+1 is 4 then 3+1 is 2+2? Or would you say, as I did above that:

3+1 "is" 4 but 3+1 "is not" 2+2 — Fooloso4

I suspect that what is really at issue can be found in remarks such as the following:

Wow. I encounter so many people on TPF who do not know basic math, it's striking. — Real Gone Cat

And:

You want to find mysticism here. — Real Gone Cat

And again to someone else:

If you still want to introduce mysticism into math — Real Gone Cat

And yet again:

Except the mystics on TPF. You're always searching for the woo. — Real Gone Cat

At least with regard to this discussion you seem to see what is not there and fail to see what "is".

Again, an overreaction. — Banno

Only to someone in a position of power.
For everyone else, might makes right, and one must hold as true whatever the person says who holds more power than oneself. Or else, face socioeconomic consequences.

There are three senses of "is"

1. The predicative sense is of the form "x is F" where F is a property that x bears.
Example: The apple is red.
Logical form: Ra

2. The identity sense is of the form "x is y" where x and y are identical, they're the same thing.
Example: Superman is Clark.
Logical form: s=c
This also means that when counting, we'd not count Superman & Clark as two different people, for they're one and the same, and that they have the same properties.

3. The existential sense is of the form "There is x" which we just assert the existence of something.
Example: There exists an apple.
Logical form: \Existential-quantifier x x=a
This is philosophically controversial: certain Meinongians as well as proponents of free logic alternatively propose an existence predicate, though this comes with its own set of nasty problems.

Just to add my two cents, I think most native English speakers would agree that 3+1 is 4 but that 4 is not 3 + 1.

It is interesting to consider why. It appears that one of the senses of ‘to be’ tells us the WORD ‘4’ is sufficient to tell us about 3+1 and 3 plus 1 things are sufficient to give you 4 things. Perhaps that’s all one of the uses of ‘to be’ is.

Aristotle would equate this to the formal cause (I’ve written about this elsewhere). I suspect the four causes of Aristotle are a relationship between the WORD for something and then the thing in the world, rather than just between two things. In this case ‘4’ is the cause and 3+1 is the effect.

Right! 3 + 1 is the cause and 4 is the effect (the result of an operation, here addition).

Well that’s right, although I was talking about one of Aristotle’s four causes which don’t map neatly onto ‘cause’ in English.
There appear to be four permutations that words can map onto things in a sufficient/not sufficient way and I think that’s all the four causes are.

Two are ‘to be’ in English,
One is ‘to mean’ in English.
One is ‘to cause’ in English.

It’s a bit of a pet theory, there’s more in that other thread: https://thephilosophyforum.com/discussion/13583/is-causation-linguistic-rather-than-in-the-world

a) 4 is IIII
b) 4 is even

The Temperature Paradox

1. The temperature is 90

2. The temperature is rising

Ergo,

3. 90 is rising

:snicker: