Does a dog see a computer as a single entity? — Joshs

:up: Point well made.

And yet:
https://core.ac.uk/reader/224990548
https://www.jstor.org/stable/1422405

Yes; I'm well aware of such studies. Hence my note above, in which I made it clear that I think pattern recognition has a part to play.

But it is not the whole story.

Notice that the study I cited used both canonical and non-canonical spacial disposition.

Thank you for citing something with a bit of empirical content... :wink:

The debate is still going on I'm afraid. I've not kept up with this thread, but as I understand it, the debate is about automaticity of number recognition, yes? The conflict between various stroop-like experiments manipulating visuo-spatial indicators of magnitude with actual numerical representations (a big number '7' vs a tiny number '9' - that sort of thing). Cohen Kadosh pretty much put that to bed in 2008, so papers from before then would have to be viewed in the light of a then open debate which is now considered less so. As any follower of developments in neuroscience will have come to expect, the matter turns out to be much more complex. Elements of visuo-spatial signals (size, pattern) are taken into account alongside priors expectations from things like ordinal and magnitude judgements in context. In fact Cohen Kadosh found that the specific instructions by the experimenters lead to different patterns of activation. Ultimately there's a significant degree of non-abstract numerical representation - numbers are represented in the brain not as the concept 'three' but as a combination of patterns, language, magnitude, numerosity, and even more synaesthetic relations. We know from something as simple as digit recognition that the inferotemporal layer can make 'best guesses' with ambiguous signals from the v4 regions (higher order perceptual features like overall shape, texture etc). It's likely that the final behaviour (assigning a number word, performing a calculation, weighing magnitudes...) is both dependant on, and influences (by backward acting suppression) the balance of collected 'evidence' in the v4 region.

Basically, a range of evidence is gathered and which evidence takes priority is dependant on the task at hand. (don't know why I didn't just say that at the beginning, still...it's written now.)

What I would add though, is that the often touted studies on infant number recognition are being misused to support mathematical realism. Infant studies done thus far just about all support the prevailing magnitude-estimation hypothesis, they are not about infants recognising 'the eternal number three' or any such. Magnitude estimation and numerosity are two different processing streams and shouldn't be confused. One can estimate the relative magnitude of two pages full of dots without having to, or even being able to, count them. Fine-scale magnitude recognition is both granular (ie based on individuating objects) and scalar. It's this granular magnitude recognition that's often misquoted as support for innate number recognition, but experiments such as this one https://pubmed.ncbi.nlm.nih.gov/11814309/ (and other subsequently) demonstrate that it is granular magnitude-recognition that underlies these infant abilities.

Cheers. Just thought you might be interested, it being your area.

In fact Cohen Kadosh found that the specific instructions by the experimenters lead to different patterns of activation. — Isaac

Complexity increasing with yet another loop.

...numbers are represented in the brain not as the concept 'three' but as a combination of patterns, language, magnitude, numerosity, and even more synaesthetic relations. — Isaac

Which in my ignorant head harmonises with the Wittgensteinian rejection of concepts as pieces of furniture in minds. If we are to look to use instead of meaning, we would not expect to find a "spot" in the brain for each number, but instead to see something reflecting the range of stuff we can do with numbers.

Magnitude estimation and numerosity are two different processing streams and shouldn't be confused. — Isaac

Interesting then that Subitising seems, according to the study I mentioned, to use the same networks as counting - is that the same as numerosity?

For those interested in how the abilities of counting or estimating magnitude pan out in other species than our own, Octolab has a series of fun experiments on octopuses, including on such topics as object permanence and numeration.

https://octolab.tv/can-an-octopus-count-viewer-request/

Which in my ignorant head harmonises with the Wittgensteinian rejection of concepts as pieces of furniture in minds. If we are to look to use instead of meaning, we would not expect to find a "spot" in the brain for each number, but instead to see something reflecting the range of stuff we can do with numbers. — Banno

Yes, that's it. The prescience of some of Wittgenstein's ideas still surprises me.

Interesting then that Subitising seems, according to the study I mentioned, to use the same networks as counting - is that the same as numerosity? — Banno

Not quite. As the authors say "we show that left-lateralized parietal activation is modulated by numerosity and is not involved in subitizing 1– 4 dots". The controversy of the paper is over the involvement of the right-lateralized parietal area, which is involved in this granular magnitude estimation I referred to earlier, but had not been postulated to be active in previous studies like this. I should be clear though, that this area is involved in a lot of fine-grained attentional shifting activities, of which counting is only one.

The study seems really interesting and raises some serious questions about previous models. My gut feeling is that we're seeing the involvement of multiple related processes because of the experimental set-up, a set of three dots 'means' 3, which, when observed is going to have triggering effects in the parietal areas anyway, and possibly some backward acting suppression on whatever pattern recognition was being employed - imagine someone shows you symbolic picture of a train, you might trigger linguistic areas ("train") and little else, but a train enthusiast might engage regions involved in more detailed edge-discernment, simply because they're expecting to see some feature you and I wouldn't even know ought to be there. In this sense, we're all number-enthusiasts.

It makes it difficult to study. we want to see what parts of the brain recognise three dots and our brains, like enthusiastic five-year-olds are excitedly telling us everything we know about 'three'. I'd love to see such a study done on people who are innumerate, like very young children, who would be less likely to have enumeration priors for such an experimental setup.

The relation is as real as the things. — unenlightened

Well, the relating is as real as the things. The pointing a relation word at them. There just aren't any relations being pointed at, like there are things being pointed at. Unless you're Plato.

The science will settle the issue. Subitizing is not just pattern recognition, but involves counting.

Edit: Just to make my point clear, your claim was that subitising is just pattern recognition. In this study it was shown that pattern recognition showed up in groups of four or less, and also in groups of more than four. That is, pattern recognition was found in both subitising and counting. It has a part to play, but is not the whole of the story. — Banno

You can throw together brain imaging studies of cognitive tasks , add to this reaction time measurements, recordings from groups of neurons and other such quantitative readings associated with particular behaviors, and get consensus of a large group of psychologists over the accuracy of these results. But raw empirical data and interpretation are two different things. Mirror neuron studies and theories of empathy is a good example. Almost all psychologists involved in this area agree on the specific neurological findings , but there are at least four distinct theoretical camps in the explanation of how mirror neurons contribute to human and animal empathy. These camps differentiate themselves along philosophical lines.

So the raw data will give us guidelines concerning the meaning of , and difference between, enumeration and subtizing, but the science won’t settle the issue of what counting , subsitizing and their distinction is without you picking philosophical sides first. I will say this. There seems to be more and more convergence these days among Wittgensteinian, phenomenological and neuropsychological approaches to cognition in general.

Which leads me to wonder what we’re debating here, if anything. I read Isaac’s posts and found them very helpful. I don’t see that anything in it contradicts Husserl’s analysis. Of course , his is conducted at a different level from the neuroscientific studies. But I think there is agreement that effortful enumeration is one category ( or maybe a series of related
categories) of mnemonic process , and subitizing can reasonably be linked to a different class of processing that bypasses the intense demands on working memory by drawing from learned patterns in long term memory and using them to fill in ( Isaac would say pattern match).

Let me throw in a little more detail foe the heck of it. Described at a phenomenological level, effortful enumeration involves separating out, identifying and counting each individual dot in a pattern of dots. Each isolated individual must then be assigned a numeric value. To do this one relies upon a pre-learned mnemonic sequence that one must keep in mind during every stage of the counting. This menomiic allows us to quickly recall any number name by its learned association with a lower number. the name ‘two’ cues ‘three’ and three cues four , and so on. This is similar to the way we remember the sequence of letters
of the alphabet , or song lyrics or melodies. The effort comes in when we have to make sure that we are not recounting previously counted dots.
This means we have to use a strategy ( finger pointing, blocking off segments of the pattern) to remember what we’ve already counted as we go along.

Subitizing frees us of this short term memory effort by bringing up from long term memory an almost identical version of the shape of the whole pattern we are trying to count. We may still have to begin by enumerating a small number of dots before we can access the pattern as a whole. This may be analogous to reading words. Our prior expectations allow us to rapidly fill in rather than having to process each letter individually. But we likely begin with a rudimentary analysis of lines and curves before letters pop into view. I wouldnt be surprised if this preliminary sequential processing of simple , isolated lines didnt make use of the same area of the brain as enumeration.

I’m reminded of George Miller’s famous paper of the 1950’s , 7+_2. He wrote that we can only keep in short term memory around 7 unrelated items( which is why such things as phone numbers are about that length). In order to recall any larger number of items we have to ‘chunk’ them, find a way to link them together as a unity that can be recalled all at once. The key is they have to be associate meaningfully. There are all kinds of well known mnemonic techniques that achieve this, such as the pegword method. Take a list of unrelated words, like a grocery list , associate each one with some object on a well trod path of yours, such as walking through the rooms of your house. Imagine each word in some humorous, shocking or ridiculous concrete way with the refrigerator, the front door , etc.

Notice that our number system is constructed for easier recall Take Roman numerals. Rather than just an increasing series of vertical lines, the sequence of numbers is ‘chunked’ at regular intervals. III becomes IV instead of IIII. Similarly , 0 through 9 becomes chunked at 10 , and then regularly thereafter. So we don’t actually do a lot of counting. We mostly use shortcuts to avoid having to count.

Do you have any objections to the assertion that both effortful enumeration and subitizing are constructive cognitive activities that are common to a range of phenomena( assigning letters of the alphabet
to individual dots) of which number is just one example?

You seem to be comfortable, as I am , in determining number in Wittgensteinian terms as a wide variety of use-dependent senses. Furthermore , there is nothing ‘pure’ about number , either from a realist or a platonic perspective.

There just aren't any relations being pointed at, like there are things being pointed at. Unless you're Plato. — bongo fury

I must be Plato, then. But no, When the cat is on the mat, the real cat is really on the real mat. But there is no eternal realm containing the form of 'on' nor the forms of cat and mat. Relations are real. This is really a response to your post and not really unrelated to it. But mathematics is the study of relations in the abstract, and that is why counting works for beans and spoons and sheep. To put it another way, mathematics is the study of possible relations, and of course not everything possible is actual. but when relations are actual, one can indeed point to them, and it is an ambiguity of pointing that Mr W. alludes to somewhere, that one cannot exactly tell whether one is pointing at the left pant, the right pant or the whole pair, or to its lurid pink colour, or something else, because pointing fails to specify its units.

Which leads me to wonder what we’re debating here. — Joshs

I don't think we are.

Thanks again.

In a previous life I taught teachers and parents the importance of subitising. The thinking at the time was based on studies in which it was found that an inability to subitise was an early sign of dyscalculia, and hence there was a need to ensure kids could subitise. One of the tasks then was to convince folk that subitising was more than recognising the patterns on die. Indeed, it was going beyond the recognition of the iconic patterns that was thought important.

I think of groups of objects of any number as kinds of patterns regardless of their precise configurations. So a group of three objects is a kind of repetition pattern however the elements are arranged.

My own experience is that if groups of more than about four objects are arranged "geometrically" I can recognize the number of objects immediately. If they are placed randomly I may have to count them; that is, the number of the group is not instantly recognizable. Others' experience may be different, obviously.

I tend to think the recognition of numerosity and the beginnings of numeracy do begin with pattern recognition. If counting begins with noticing multiples of similar objects then apprehending similarity of conformation (which is a kind of pattern recognition) would be important

And if you don't reference it or contemplate it, then there's no subject of discussion. — Wayfarer

Here's the thing: our not talking about something does not make it disappear.

When you put the spoons back in the draw, they do not cease to exist. You do not go to the shop to buy new spoons because you surmise that sine can't see the ones in the draw they no longer exist.

That is, what you have said here seems to be in error. The subject is still there even when we do not talk about it.

The usual idealist reply is to ask for proof that the spoons in the draw still exist, as if this were an empirical claim. Of course it is not. But you and I will both go to the draw for the spoons, and not order new ones online. That is, the idealist indulges in the philosophical pretence that we cannot know if the spoons are really in the draw, while for all other intents and purposes behaving as if they are.

I am questioning that they exist 'objectively and mind-independendently', — Wayfarer

I'll join you in questioning if absent things exist "objectively", since as I've argued the term is next to useless. The 'mind-independent' bit is where we differ. It seems you think that a mind is needed in order that what you think of as the unknowable 'whatever' of the nomenon can be imbued with "spoonness", or something along those lines. What I've pointed out is that the assumed unknowable whatever is, by that assumption, unknowable, and so cannot form part of our analysis - we cant say anything about it. This is to say that the world is indeed interpreted, put into the context of our thoughts and actions, and to take seriously the point that nevertheless, there is a world that is so interpreted. That we can't say anything about the supposed nomenon renders it outside of our considerations, but importantly this does not mean we cannot talk about stuff.

The only alternative would be to suppose that since we cannot say anything about the nomenon, we cannot say anything about spoons, tables and draws. This is of course Stove's Gem, and there are folk hereabouts who take this seriously. Don't ask them for a cup of tea.

I've been rabbiting on about direction of fit. It fits here, too. What I'm advocating is that treating the world as "mind-independent" is what we in fact do. The arguments tend to be oddly passive, as if all we ever do is absorb phenomena. But of course we pick spoons out of the draw, we scoop sugar, we stir the tea; we interact, moving from world-to-word to word-to-world intentions freely, indeed blithely.

And that's what realism is. Not the conclusion of an argument but the very presumption on which what we do sits.

It takes a philosopher to decide otherwise.

You seem to think your view gains purchase from considerations of relativity and quantum magic. That would require some extensive and clear argument.

Relativity suggests that the laws of physics are true for any observer. This is a clearer way to think about these laws than the supposed "god's eye view", that the laws are true for every observer.

And Quantum mechanics' supporting your contention of the centrality of the observer depends on what counts as an observer: a mind or a measurement. This is an issue of contention in physics.

Cheers. I think this subitising discussion is off-topic; it might seem to some that how we recognise numbers is central to what numbers are, but it is clear that the salient feature of numbers is the purpose to which they are put and not their supposed phenomenology.

Cheers. I think this subitising discussion is off-topic; — Banno

Perhaps, but on the other hand if numeracy can be explained by recognizing similarity, difference, repetition and pattern if real objects, then there would be no need to appeal to the reality of universals, a kind of realism which, ironically, is really idealism.

but on the other hand... — Janus

Not only recognizing similarity, difference, repetition and pattern, but use.

hnece,

"1" has the superficial grammar of a noun, but this is misleading.

Rather "1" is to be understood through its role in the process of counting. It is understood in learning how to count, not in pointing to individuals.

And of course this goes for other mathematical entities, too. They are things we do, not things we find. — Banno

When you put the spoons back in the draw, they do not cease to exist — Banno

Kind of like the beetle in the box? We can of course say the spoons still exist outside of our interaction with them, but saying so is meaningless outside of some use such an utterance serves relative to our pragmatic purposes. Notice that claiming use-independent existence is meaningless is quite different from claiming that the thing-in-itself is unknowable. Kantian Idealism isnt denying the existence of a world of use-independent givens. Wittgenstein and phenomenology, however , are arguing that such assumptions are pointless and don’t do anything for us, except to the extent that they emerge as relevant out of some ongoing project. It’s interesting to note that a spoon can be meant as a cultural object as well as a physical object. When Im hungry and look for my spoon I’m searching for an implement to feed myself. When I look to see if there is object permanence to a physical object called a spoon, my aim is different. Or I could be looking for the last entry in the book that I am writing. When I find it I can say that it existed when I wasn’t there. But does it exist for others, too? Something certainly exists for them if they locate my words on a laptop. But I know they don’t understand the words the way I intend them. So what they find is not the same for them as what I find.
And when I find my previous words , I notice that they seem a little different than I remember them when I typed them. Just the act of going back to them changed their sense in some subtle way. Is this experience so different from that of locating physical objects that I have put away?
Is there no interpretive change in the sense of what the object is for me as I return to use it day after day? So yes, I could say that the spoon continues to exist without me , but now I’m realizing its existence WITH me is one of a continually contextual shift in sense over time. So if that’s what underlies the so-called self-identical persistence of the object when I’m using it , then it seems beside the point to posit persisting self-identically foe the objects that are not currently being used by me.

We can of course say the spoons still exist outside of our interaction with them, but saying so is meaningless outside of some use such an utterance serves relative to our pragmatic purposes. — Joshs

Yes! Our utterances and acts are what constitute meaning, in so far as meaning is anything at all. And so of course that meaning is in a state of flux induced by our changing acts and wants.

...it seems beside the point to posit persisting self-identically foe (for?) the objects that are not currently being used by me. — Joshs

And yet it is what you and I do; here, you in looking for the reply you now read; me in writing it with an expectation that it reach you. Beside the point? What could be more salient here, now than your reading this?

Indeed, it was going beyond the recognition of the iconic patterns that was thought important. — Banno

Yeah. Some of my wife's early work was with the development of children's writing recognition. A similar (I think) thing is found there. The recognition that this 'G' is a letter G no matter what font it is written in, is a skill that is a layer above the basic pattern recognition and into the more Bayesian modelling (though we didn't think of it in those terms then). It's as much about suppressing extraneous data as it is about processing relevant data. I think people (universalists?) too often think of this process as leaving behind some kind of platonic 'essence' of 'G', but it's not, the data discarded in some contexts overlaps with the data included in other contexts. It's about applying fast heuristic hypothesis testing, leaping forward in the word, bringing in context, sentence meaning, location in the word... My guess is that subitising is the same.

She had children who had to learn to read one font at a time...

But, you're right, we're very far off topic.

Not only recognizing similarity, difference, repetition and pattern, but use. — Banno

All those have to be in place before there can be any use. But sure, use is obviously important; we, like all other species, are basically pragmatists.

The recognition that this 'G' is a letter G no matter what font it is written in, is a skill that is a layer above the basic pattern recognition and into the more Bayesian modelling (though we didn't think of it in those terms then). It's as much about suppressing extraneous data as it is about processing relevant data. — Isaac

Gs written in most fonts are still recognizable as a particular configuration or pattern.

When you put the spoons back in the draw, they do not cease to exist. You do not go to the shop to buy new spoons because you surmise that since can't see the ones in the draw they no longer exist. — Banno

Thanks for the considered reply, although I think it's still rather 'argumentum ad lapidem'. But I will leave off for now and think about it some more.

...it seems beside the point to posit persisting self-identically for the objects that are not currently being used by me.
— Joshs

And yet it is what you and I do; here, you in looking for the reply you now read; me in writing it with an expectation that it reach you. Beside the point? What could be more salient here, now than your reading this? — Banno

Yes, our anticipating into the next moment is absolutely central to what salience, mattering and pragmatic
use are all about. Maybe I’ve been reading too much phenomenology , but when I think about the spoons being where I look for them , I don’t have in mind the persistence of a thing , but an expected new variation in a an ongoing performance. The performance is the enacting of a body-environment interactive cycle.
The spoon isn’t an independent element that just happens to participate in the performance. It ‘drops out of’ the performance as a derivative biproduct. If a ‘spoon’ is only a slot in an ongoing narrative and body-world performance, then the looking for and finding of a spoon just demonstrates that as self and world feed back into and modify each other moment to moment , there is a referential continuity to this creative
becoming that gives our experience ce a thematic consistency and predictive utility. One could say the objects of our experience self-persist as returning to themselves and to us differently but recognizable in relation to what we want to do with them. The world talks back to us , but only in response to our formulations. It’s feedback changes
those formulations , which then trigger a newly modified talking back from the world. The aim of all this back and forth between formulations and the changing feedback it triggers from the world is to coordinate the interaction in more and more intimate and intricate ways, choreographing the dance between changing self and changing world in the direction of seamless movement through new events.

Maybe I’ve been reading too much phenomenology — Joshs

I think so. It's not good for your mental health, you know. You will find yourself writing long, convolute sentences that say very simple things. Yep, the way the spoon looks changes over time.

What changes?

The way the spoon looks.

Therefore there is a spoon.

Therefore there is a spoon. — Banno

Of course there is a spoon. It's the implications of there being a spoon that phenomenologists and metaphysicians are interested in. You don't have to be interested, though, if the basic spoon is good enough for you. It's perfectly adequate for baby food. :wink:

...phrenologists... — Janus

Yeah, them. :wink:

Seems as some folk think that phenomenology has something to say in regard to the OP - that maths is not made up. Issue here is conceptual clarification - what is it that phenomenology tells us about maths, and if nothing, why are they on this thread?

You don't have to be interested, though — Janus

They came here; I'm not chasing them, just asking them to explain themselves.

...phrenologists... — Janus


Yeah, them. :wink: — Banno

The spectres of predictive text...

Yep. You'd have to have lumps on your head.

Therefore there is a spoon. — Banno

I see you've progressed from cups. Impressive.