We already discussed a similar issue here: On Gödel's Philosophy of Mathematics

where is the domain of natural numbers? It doesn’t exist anywhere, but it includes some numbers, and excludes others. The square root of minus one is used in all kinds of mathematical operations, but it is ‘outside’ that domain.

And the problem with saying that it’s ‘merely’ an invention of the human mind, is that it doesn’t allow for the unreasonable effectiveness of mathematics in the natural sciences. Maths is predictive, through it you can discern facts about nature which you would have no way of finding otherwise.

But what did Plato mean?

Yoou could call it as well the reasonable effectiveness. The reason being that Natural processes are patterned inherently structured.

And the problem with saying that it’s ‘merely’ an invention of the human mind, is that it doesn’t allow for the unreasonable effectiveness of mathematics in the natural sciences. — Wayfarer

"the unreasonable effectiveness" is also a product of human mind? which is synthetic analytic judgement?

"the unreasonable effectiveness" is also a product of human mind? "

:heart:

If this is what Platonists believe, then where do they think that these objectcs exist? If it's not inside our physical realm then in what realm do these objects exist and do they move inside of it? — Prishon

Good question.

I think that mathematical objects such as geometric shapes are related to Plato’s Ideas or Forms.
When we perceive something in visual cognition, for example, we really see shape, size, color, number, etc.

Therefore Shape would be something similar (though not identical) to an universal that awareness or consciousness uses to organize itself in order to generate determinate cognition.

As such, the Ideas or Forms seem to exist in latent or potential form within indeterminate forms of consciousness from where, on becoming activated, they emerge and generate particular objects of determinate cognition.

If we consider the following aspects or levels of intelligence,

1. The Good or the One;

2. Nous or "intellect" proper;

3. Logistikon, "intellectual" or "thinking" aspect;

4. Thymos or "emotional" aspect;

5. Epithymetikon or "sensual aspect" (relating to sense-perception),

then the Ideas or Forms are objects of the nous and the mathematical objects are objects of the logistikon. But the ultimate source of the Forms seems to be the Good or the One that may be described as a form of superordinate or universal consciousness.

If this is what Platonists believe, then where do they think that these objectcs exist? If it's not inside our physical realm then in what realm do these objects exist and do they move inside of it? — Prishon

And where does our physical realm exist? If it is embedded in a larger space, where does the larger space exist?

Space is just one of many mathematical objects and all mathematical objects exist in virtue of being logically consistent and in mutual relations, of which spatial relations are just a special kind of relation. And by the way, time is just a special kind of space, at least according to theory of relativity.

We can distinguish two kinds of mathematical objects: concrete and abstract. For example, there are concrete triangles (like concrete "give way" road signs) and one abstract triangle, which is a property instantiated in all concrete triangles. The Platonist objects are the abstract ones. Some people think that the abstract objects don't "really exist", that they are just words or ideas in our heads. Yet these words or ideas express an objective similarity between concrete objects, so the abstract objects can also be understood as being in a sense "dispersed" in concrete objects.

the problem with saying that it’s ‘merely’ an invention of the human mind, is that it doesn’t allow for the unreasonable effectiveness of mathematics in the natural sciences. Maths is predictive, through it you can discern facts about nature which you would have no way of finding otherwise. — Wayfarer

But mathematics is predictive only in the sense that physicists have assumed (as the Pythagoreans Heraclitus Plato Galileo had) that the unknowable natural world is mathematically orderly. This pragmatic assumption has sent mathematical physicists scouring through all maths in search of hypotheses to fit physical observations.

As of today, I doubt that there is any maths left that has not been incorporated in some physics. New maths is spurred on both from discovery-invention within pure mathematics and from mathematical physics in search of logical justification for some fanciful ideas.

Intuitionists" believe that mathematics is just a creation of the human mind. In that sense you can argue that mathematics is invented by humans. Any mathematical object exists only in our mind and doesn't as such have an existence.

"Platonists", on the other hand, argue that any mathematical object exists and we can only "see" them through our mind. Hence in some sense Platonists would vote that mathematics was discovered. — Prishon

One should not have to choose between these extremes. Mathematics was discovered through technological trial and error to be reasonably but not perfectly predictive of sounds coming out of musical instruments and from observations of the day and night sky. It was obvious from the first that mathematics is the guide to a hidden world that lies beneath the appearances that we take for granted as the reality of speech and action.

What 'exists' does not belong. Existence is a construct needed to describe fixed objects in a supposedly timeless reality.

We can distinguish two kinds of mathematical objects: concrete and abstract. For example, there are concrete triangles (like concrete "give way" road signs) and one abstract triangle, which is a property instantiated in all concrete triangles. The Platonist objects are the abstract ones. Some people think that the abstract objects don't "really exist", that they are just words or ideas in our heads. Yet these words or ideas express an objective similarity between concrete objects, so the abstract objects can also be understood as being in a sense "dispersed" in concrete objects. — litewave

Correct. But we must not forget the Forms.

There are (1) concrete or perceptible mathematical objects, (2) abstract or ideal ones, and (3) Forms.

For example, if we hand-draw a triangle on a peace of paper or in the sand, we have a perceptible triangle. But our thinking faculty tells us that this triangle is less than ideal. In doing so, we form the concept of an ideal triangle in our mind. This is the ideal object. However, we can only form an ideal object in our mind by referring to something like a universal form or pattern of which we can only have an innate intuition. This universal form or pattern is the "Form of the triangle".

The Forms are at once "dispersed in concrete (and ideal) objects" and transcendent in relation to them. This means that the Forms themselves are outside time and space, though their imperceptible properties are approximately perceptible in concrete objects like reflections of the sun in water.

I agree whole-heartedly with the Forms. All Forms can be described by math. An arbitrary periodic form can be translated (is that the right word?) into base sine forms. Like the epicylces did for planetary motion. In fact ALL FORMS can be translated or described. Not all can be reduced (important for the non-perturbative approach in quantum field theory). This idea comes closest to Plato. The math. forms are indeed not part of the physical world. But neither in an unaccessible metaphysical realm. Unless you think that the world of ideas (themselves being Forms too) IS this world. Accesible to us obviously.

The math. forms are indeed not part of the physical world. But neither in an unaccessible metaphysical realm. — Prishon

Plato is a very complex writer and it is important to read him carefully and on his own terms. But I think that a first step in the right direction would be to bear in mind that the Forms are not the same as ideal objects.

An ideal object, e.g., an ideal triangle, is something that I form in my mind. But my ideal triangle is not the same as your ideal triangle; it is multiple as it exists in many minds; it is subject to time as it is not permanently fixed in the mind, etc.

In contrast, the Form of Triangle is one, unchanging, and eternal. It is beyond space and time and cannot be expressed in language.

The other peculiarity of the Forms is that they are at once (1) present in particulars through their properties and, therefore, immanent and (2) other than each and all particulars and, therefore, transcendent to them.

Acquainting ourselves with the concept of ideal objects is a necessary step toward understanding the Forms. But, eventually, we must go beyond the level of ideal objects in order to “attain to the knowledge of reality” as Socrates puts it in the Phaedo (66a).

Seems more complex than at first sight! Like many first sights. It sounds even religious. Though the Greek had a mountain full already. Damned! this is how a forum should be!

In contrast, the Form of Triangle is one, unchanging, and eternal. It is beyond space and time and cannot be expressed in language. — Apollodorus

If something is beyond space and time, then where could it be?

It sounds even religious. — Prishon

It may sound "religious" to the modern mind. But Plato's primary concern is never religion per se which is based on belief (pistis), but knowledge (noesis or gnosis) which is based on experience.

Religion, in so far as it plays a role in the acquisition of knowledge, is just an intellectual framework or ladder that leads to an actual experience that transcends both belief and reason.

If something is beyond space and time, then where could it be? — Corvus

Somewhere beyond space and time? I.e., within a form of awareness or consciousness where experience of time and space has not yet emerged.

You need to have some cognitive elements, visual or auditory, etc. in order to perceive space and time. Prior to this, there is no time and space. The Forms being unchanging, eternal, etc., cannot be anywhere else.

All determinate experience, including time and space, begins with the Forms. This is why Plato is actually serious about the Forms. It isn't just literary licence.

Plato's Forms and their corresponding Name are similar to the nama-rupa ("name and form") concept of Indian philosophy.

Yet these words or ideas express an objective similarity between concrete objects, so the abstract objects can also be understood as being in a sense "dispersed" in concrete objects.

But for Plato, this isn't exhaustive, he routinely distinguished between:

- the object in which a property is instantiated (the apple, the yield sign)
- the property concrete instantiations share (the redness of an apple, the triangle-ness of a yield sign)
- the Form in which these objects participate by sharing a given property (the Form of Apple, the Form of Triangle)

most later and certainly contemporary realists dispense with the 3rd one, which was sort of Plato's signature, and may have been more motivated by other concerns (aesthetic, religious, cultural, etc) than strictly philosophical or logical ones

What I find very thought-provoking is that, all things considered, mathematical objects are mental objects. So, if they exist like say a chair does, they must do so in a mind and immediately God becomes a possibility we have to take seriously.

“I believe that the only way to make sense of mathematics is to believe that there are objective mathematical facts, and that they are discovered by mathematicians,” says James Robert Brown, a philosopher of science recently retired from the University of Toronto. “Working mathematicians overwhelmingly are Platonists. They don't always call themselves Platonists, but if you ask them relevant questions, it’s always the Platonistic answer that they give you.”

Other scholars—especially those working in other branches of science—view Platonism with skepticism. Scientists tend to be empiricists; they imagine the universe to be made up of things we can touch and taste and so on; things we can learn about through observation and experiment. The idea of something existing “outside of space and time” makes empiricists nervous: It sounds embarrassingly like the way religious believers talk about God, and God was banished from respectable scientific discourse a long time ago. — Smithsonian Magazine, What is Math?

Is God A Mathematician? — Many have asked

Mathematics is the language in which God has written the universe. — Galileo Galilei

the Form of Apple, the Form of Triangle) — Seppo

I think the idea that there is a Form for every conceivable thing under the sun is unwarranted. Different Forms would be perfectly capable to combine to form virtually any perceptible object.

Some of Plato's statements cannot be taken literally and are simply presented to make a point or illustrate an argument in order to make it easier for the reader to understand certain concepts. At the end of the day, readers need to exercise their own judgement. Plato simply shows the way ....

Yeah but for the materialist, these mental objects are located in the brain. There is a model for explaining how concepts like God or math can spread across billions of human minds, memes, parasitic reproducing bits of idea. They have a physical being in the neurons of their hosts. God is just a very effective meme.

For memes undergoing natural selection pressures as they reproduce in their hosts it only makes sense that ones that explain the world well and have predictive power would come out on top. Mathematical ideas are memes they are successful because they have utility for their host.

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To get back to the original post, from an idealist perspective , Hegel, you have the universal (forms) producing the specific, since we can only understand the world through ideas (universals). Since the true is the actual, and the truth is the whole, it follows that it is these ideas that give rise to the world of experience, the only world we can speak of directly. It also develops that world to become more complete, to reach a higher stage of truth.

It's a take I find appealing. More than I do Neoplatonist or Gnostic versions, which have the forms living in a kind of magical soul dimension of pure mind and pure ideas. The problem there, is that, as Aristotle showed, and Plato acknowledged in the Parmenides, the world would be filled with various infinitely regressing forms- a whole dimension of reductio ad absurdum infinities.

I think the idea that there is a Form for every conceivable thing under the sun is unwarranted. Different Forms would be perfectly capable to combine to form virtually any perceptible object.

Plato agreed. In the Parmenides, he disavows the idea that there are Forms for low or gross things (I forget the specifics examples, but iirc "dirt" or "mud" may have been given), he tended to think that there were only forms for things like Truth, Beauty, Justice and so on. The problem is, his theory didn't really provide any basis for such a distinction (once again this seemed more motivated by non-logical or philosophical considerations, like aesthetic or religious ones), and so this certainly was a problem/inconsistency with his picture.

And as far as the reducibility/redundancy of certain Forms, I thinks that's also a very valid objection- where do we draw the line? Is there a Form for Square apart from the Form of Rectangle? Maybe Forms for geometrical shapes or objects all reduce to more fundamental concepts like the Form of Line Segment or Angle? This seems somewhat arbitrary and subjective, and contingent on our particular purposes or context or what sort of conceptual schema we happen to be using, which undermines the notion of a separate, independent, objective realm wherein these Forms exist/are located.

Yeah but for the materialist, these mental objects are located in the brain. — Count Timothy von Icarus

The physicality of the mind isn't as cut-and-dried as is necessary to matter. Too, God being material is absolutely fine by me.

There is a model for explaining how concepts like God — Count Timothy von Icarus

Petitio principii.

Not really. I'm not saying it's the case, it's just a model that explains the forms and how they could arise from material processes.

Not really. I'm not saying it's the case, it's just a model that explains the forms and how they could arise from material processes. — Count Timothy von Icarus

Yes, I get that.

Maybe Forms for geometrical shapes or objects all reduce to more fundamental concepts like the Form of Line Segment or Angle? — Seppo

That may be one way of looking at it. Plato certainly follows the reductivist tendency already found in Greek philosophy, and in natural science in general, that sought to reduce the number of fundamental principles of explanation to the absolute minimum, hence the “first principle” or arche of the earliest Greek philosophers.

It seems to follow the inner logic of Plato’s explanatory framework which is hierarchical and necessarily leads from the many to the One.

In any case, all objects of knowledge and, in particular, the Forms need to be considered in the light of the Good (= the One) which is their ultimate source. The Forms merely serve as a ladder to ultimate reality. They can be reached only by transcending reason and they in turn need to be transcended in order to reach the highest.

The Platonic method is the Upward Way, Ano Odos, a process of vertical progress that takes the philosopher through a hierarchy of realities ranging from human experience to ultimate truth,

They have a physical being in the neurons of their hosts. — Count Timothy von Icarus

I think this to mean that bits of matter somehow represent ideas, in the same way that codes represent objects in, say, computer systems. It seems natural, even obvious.

The problem is that even very simple mental operations can generate enormously divergent patterns of neural activity. Very simple stimulus and response patterns in mice are subject to what is called 'representational drift' - the same stimulus evokes responses in very different regions in the brain over time. Same thing happens with humans, albeit even more complicated. I read that long neurological studies attempted to trace characteristic patterns of activity in human brains when learning simple tasks, like memorising a new word, but that the activities were so divergent that researchers could detect no consistent pattern despite years of studies (see Why Us?, James le Fanu.)

Furthermore, consider the way in which a divergence of symbolic forms can be used to convey the same idea. A number can be represented by a variety of symbols, but they all specify the same idea. So the meaning of the idea is in some sense separable from its physical form. The mind, of course, can recognise such equivalences and translate one form to another - but again, can that be understood as a physical process? I think rather that it's a pretty cogent argument for dualism.

The problem there, is that, as Aristotle showed, and Plato acknowledged in the Parmenides, the world would be filled with various infinitely regressing forms- a whole dimension of reductio ad absurdum infinities. — Count Timothy von Icarus

Forms are not shapes, or necessarily even entities or things. They are more like principles.

...Among all the kinds of forms which can be signified by terms, according to Aquinas, there is no one uniform way in which they exist. The existence of the form “sight,” by which the eye sees, may be some positive presence in the nature of things (which biologists can describe in terms of the qualities of a healthy eye that gives it the power to see), but the existence of the form 'blindness' in the blind eye need be nothing more than the nonexistence of sight ‒ the 'form' of blindness is just the privation of the form of sight and so not really an additional form at all.

In general, distinguishing and qualifying the different ways there can “be” a form present in a thing goes a long way toward alleviating the apparent profligacy of the realist account of words signifying forms. ....

Aquinas’s famous thesis of the unicity of substantial forms is an example of another strategy: linguistically I may posit diverse forms (humanity, animality, bodiliness) to account for Socrates being a man, an animal, and a body, but according to Aquinas there is in reality just one substantial form (Socrates’ soul) which is responsible for causing Socrates to be a man, an animal, and a body. In this and other cases, ontological commitment can be reduced by identifying in reality what, on the semantic level, are treated as diverse forms. As Boethius had seen, what the mind is capable of logically distinguishing need not be actually distinct in the nature of things.

In principle, any number of strategies for reducing overall ontological commitment are available within the framework of realist semantics, so that in general, the kind of form that fulfills the required semantic function did not need to be the kind of form that has a distinct and positive metaphysical presence in the nature of things. — Joshua Hothschild, Whats Wrong with Ockham?'