I don’t see the connection between question and answer here. Could you unpack it for me?
That diagram suggests that there are many ways we could “carve nature at the joints”. And yet at the perceptual level, judging from everyday experience, we all (and even animals) seem to carve it up the same way, which suggests to me that what is being carved tightly constrains how it can be sensibly carved.
I get what you are trying to say, but also :
0 is an even number. It is “represented” by 64 consecutive bits perhaps in an off state. It’s a position on the number line, the origin. It’s the additive identity. And so on.
Yes. I agree. But what are concepts ? In this context, I suggest that a number is a way of treating numerals. Following Sellars, we might say that a number like 0 is a role that a mark can play to be a numeral. Not the only approach, but one rarely considered.
I like this issue. We can also say that wheel was “there” as “possible.” Once we have the wheel, the past itself has changed. “The past isn’t what it used to be.”
I get the appeal of a “fixed past” —a convenient assumption for a jury — but it often involves something like a configuration of atoms and void that was ( determinately ) present. From “no” perspective. Just as that grid of 10000 pixels is stripped of reference to perspective. The singular aperspectival true history of atoms and void or bits in a grid…
Yes indeed, Kronecker. The positive integers are about as “intuitive” and “shared” as it gets, though even here there’s disagreement about how they exist or whether “all of them” exist. Is there “already” a (completed) infinity ? What do we mean by statements about all positive integers ?
What does this disagreement suggest about the ground of logic ?
The wheel was there as previously undiscovered possibility, so once discovered our account of the past changes. Yet was the possibility not always there? If so, what has changed apart from our account?
I’m not exactly settled on this issue. Mostly I “assume the fixed past” like everyone in practical life.
But what is possibility exactly ? It “could” have been discovered/invented.
Then why wasn’t it ?
“If I had a time machine, then …”
We play with the initial conditions of a mathematical model, or start the video game over and try a different strategy.
Is our account a kind of “cream” that floats on top of a substrate, not really touching it ? This substrate is conceived perhaps as the evolving configuration of “atoms and void.”
Yet “atoms and void” is a relatively recent “wheel.” I mean “quarks” are like wheels and zeros, something we project backward as discovered rather than invented.
And the invention is all the more powerful because we do this. When phenomenology “foregrounds,” this metaphor implies that what is foregrounded was lurking already in the background. It is invented as having already been there.
And yet its “revelation” is crucial, or why do philosophy ? Hence “remembering.” The “same” insight re-occurs. The “same enough” pattern in sayings and doings.
I was asked what I mean by ‘necessary structures of consciousness’. I’m trying to argue that this is what such things as numbers and logical principles are. They are (as Frege says) the ‘laws of logic’. If I ask you what two plus two equals, you are obliged by the laws of arithmetic to answer ‘four’. But I don’t want to say that, therefore, these rules are subjective, in the sense of pertaining only to an individual mind. They are the rules for any mind capable of counting - and they’re real. Hence, ‘structures in consciousness’. And the same can be said for all manner of intelligible objects (although I’m very wary about the use of the term ‘object’ in this context, as they’re more acts than objects i.e. the ‘act of counting’.)
At issue in this is the nature of the reality of such intelligible acts. Platonists (including Aristotle) accept that intelligibles such as numbers and logical principles are real as the constituents of rational thought. Fine, most will say, but that means they’re ‘in the mind’. That is how we’re almost bound to understand it. But they are something more than that, because they also give us massive purchase over the external world (through mathematical physics and other sciences.) They are regularities in our structured experience-of-the-world. So They’re neither ‘in the mind’ nor ‘in the world’. They oblige us to recognise something fundamental about the structure of rational cognition.
I don’t deny that number is encountered in the world. It is everywhere. the world is inconceivably diverse, differentiated and structured. I was questioning the idea that consciousness per se can sensibly be said to be structured per se.
We might say it is structured by perceptual experience, but perceptions are structured the way they are―we have no say in that apart from our capacity to pay attention to different aspects that present themselves.
I “feel the truth” of 2 + 2 = 4. So I get the “obligation.” The metaphor “law” is fascinating. In extreme cases, they will lock you up, if you write a bad check perhaps because you don’t add numbers as others do.
The obligation feels “internal” but it is at least also social. Grammar school, report cards, parents called in if the student “won’t get it.”
Is this a truism though ? To be capable of counting is to reliably follow the rules. I can’t “peer at 3” through your intuition, nor you through mine. But we do the same thing with numerals.
Elsewhere I suggest “generalized numerals.” If I count 65 beans in a jar, then that jar of beans “becomes” or “is discovered as” a (generalized) numeral for 65.
A Platonist should perhaps agree with this, for the typical mark “65” is arbitrary, merely conventionally somehow used to “point at” the number that gives it life.
Why do numbers give us power ? More precise descriptions. Measurements. Where number “connects” to perception.
I guess the issue is whether cognition is separable from the manifestation of the world. Wheels and numbers are “there in the world we share.” So the “laws of logic” are the “laws of being.” But for me these “laws” evolve like language. And language for me is not internal meaning stuff but like the cries of other animals. Except that our cries and marks are richly self-referential.
Suppose in the world I see four rocks. Suppose I added another rock and every time the result was three rocks.
Because of Enactivism, where the mind emerges from its environment, within our numbering and linguistic practices it would be logical that 4 + 1 = 3.
Anyone who behaved as if 4 +1 = 5 would be sent back to school.
That’s just it. The way we’ve been trained, we don’t know how to suppose such a thing.
We can talk about this in the first place because we react to little squiggles in mostly the same way.
What I will readily grant Wayf is that most cultures do something “equivalant” when it comes to positive integers. So simple math is relatively independent of particular human communities. But if there were no communities, it’s not clear what would be left. And we wouldn’t be here to ask the question anyway. And what is the “truth” on this matter if a reality is not being “shared” ?
Your chin-rub emoji is hard to interpret. Perhaps you can share ?
Was it the jar of beans as a “generalized numeral” ? I am serious about that. Recently read a history of math book, and I was especially interested in the simplest math. Hands and fingers. I hold up three fingers. I could also hold up a sign with a “3” on it. If you just want to buy the right number of tickets, it doesn’t matter whether it’s fingers or a “3.” Both empirical events/gestures play the same role, and that role is the number that makes the numerals numerals. That is the claim I am floating, to see what others can make of it.
The disagreement doesn’t shake the ground.
The ground holds.
What it permits isn’t fully determined.
We are already in everything, because there is no out.
Novelty isn’t something that comes from outside — it’s what the field hasn’t foreclosed yet.
A boundary isn’t a limit on access to reality. It’s what reality looks like from the only place anything can be. It’s what I am.
But are there layers of ground ? Is there a ground between you and me ? You and others ? A particular situational ground, more or less articulated ?
I am talking with a stranger and it’s going well, but then they imply that their friend was recently resurrected from the dead, after a month in the grave.
I feel the ground between us give way. And yet I still trust that we both speak English. That level of the ground is intact.
Yes, the giving way is real, it is just not all the way down.
No, that’s missing the concept. The point is that enactivism accounts for how numerical concepts are formed through practice. It’s not that reality itself becomes mathematically arbitrary. Arithmetic norms remain stable because only certain patterns of coordination succeed in our shared engagement with the world. Treating 4 + 1 = 3 would therefore reflect a failure to follow the rules, not a rival mathematical framework produced through enactivism.
Yes, 4 + 1 = 5 is probably agreed by all human communities, because adding one rock to four rocks results in five rocks. . This becomes the basis of human logic, because all human communities have evolved in this particular world.
However, in a different world, where adding one rock to four rocks results in three rocks, then 4 + 1 = 3 would become the basis of the logic of an alien community.
I like your imagination. I guess the issue is whether we can know what we mean by such a world. Can we really imagine it ?
If I imagine it, then ( for instance) 2 rocks somehow vanish. I would look for an empirical cause. "Something eats rocks on this planet. An invisible-to-humans rockmuncher ? "
If you add -1 rock to 4 rocks, then it becomes 3 rocks. Is it then the logic of an alien community?
Wikipedia - Enactivism
Enactivism is a position in cognitive science that argues that cognition arises through interaction between an acting organism and its environment.
Yes, Enactivism shows us that 4 +1 = 5 is the basis for human logic because humans have evolved in a particular environment, this particular world, where adding one rock to four rocks results in five rocks.
But Enactivism also shows us that if humans had evolved in a different environment, where adding one rock to four rocks resulted in three rocks, then 4 + 1 = 3 would then become the basis of human logic.
It is not so much the case that the reality an organism lives in is arbitrary, but rather which reality the organism lives in is arbitrary.
Yes. Alien to us.
We know about destructive interference, where 1 + 1 = 0
Wikipedia - Wave interference
Out of phase: (here by 180 degrees), the two lower waves combine (right panel), resulting in a wave of zero amplitude (destructive interference).