Beautiful
"This feeling of life brought about by our intuition of an indeterminate
object with form is the feeling of pleasure that Kant associates with
the beautiful. The feeling of pleasure we get when we encounter a
beautiful object, says Kant, is nothing other than “a feeling of the
free play of the powers of representation in a given representation for
cognition in general.” (Kant, p. 102) When we judge an object in a
merely reflecting way, we feel pleased in the awareness of the
“animation of both faculties (the imagination and the understanding).”
(Kant, p. 103) This animation is what makes us judge the object to be
beautiful. For Kant, an object is considered beautiful if and only if
it is an indeterminate self-enclosed object, which allows itself to be
contemplated freely, without any constraint from empirical or ethical
concepts.
Thus the beautiful, for Kant, is neither a feeling of agreeableness
in the senses (the empiricist model), nor the perception of perfection
in an object (the rationalist model). The beautiful, for Kant, is the
result of a merely reflecting judgment which, in refusing to attribute
any determinate content to the object at hand, enables a pleasurable
mode of thinking, where the imagination and the understanding are in a
free and harmonious relation to one another." -
http://courses.washington.edu/asthetik/thiti_beautiful.html
A persons imagination and what they see of something through that imagination is what allows things to be seen as beautiful. It is not about the perception of an object or how it actually looks but more about what an individual persons mind and imagination allows them to see. This is what allows us to see and determine that things are beautiful. But everyones view of what is beautiful will be different due to what their imagination allows them to see, again no two people will be able to imagine the same thing because imagination if fueled by personality and experiences that have taken place.
Sublime
"Like the judgment of the beautiful, the judgment of the sublime, for
Kant, is a type of merely reflecting judgment. As such, these two forms
of judgment share one important feature with one another: they both
give rise to pure aesthetic pleasure – i.e. pleasure not grounded in
any determinate empirical or ethical concept. Just as the pleasure of
the beautiful is the result of a certain mode of thinking, rather than
any particular attribute of the intuited object, so is the pleasure of
the sublime the result of a certain process of thought, which is
unrelated to any objective feature of the experienced phenomenon."
"the sublime is to be found only in a phenomenon of experience that is
boundless, without any discernible shape within its spatial or temporal
structure"
http://courses.washington.edu/asthetik/thiti_sublime.html
In Kant's theory the sublime and the beautiful are similar in respect to how his judgments differ.
As each one person has a different view of life which is shaped by experiences that they have had what they see as sublime will be different from the next person. But the general feeling of the sublime is about things being exceptional and going above what could have ever been though to be possible. Things being great and beyond what is see as normal, the unexpected things in life. I think what Kan't is getting at with the term sublime is things being normal but things that we don't really have to think of as being normal, the everyday things that we pay no attention too as they are familiar and a way of life, he wants to go av=o
Mathamatical Sublime
" Now that we have a general sense of what Kant means by the sublime, it
is time to discuss the more specific notion of the mathematical
sublime. The mathematical sublime is a feeling of the sublime which we
experience when we encounter something overwhelming in
size.
Thus, to understand what gives rise to the feeling of the mathematical
sublime, we need to understand how Kant conceives of the mind to be
capable of judging sizes.
Kant makes a distinction between two ways of estimating size:
aesthetically and mathematically. An aesthetic estimation of size, says
Kant, is something that occurs “in mere intuition (measured by eye).”
(p. 134) When we run into a tall person on our way to the office, for
example, we judge the person to be tall, not through the use of a
ruler, but through what we consider to be the norm of a person’s size –
i.e. through “the average magnitude of the people known to us.” (p.
133) An aesthetic estimation of size is thus something that we do every
day, intuitively, without any attempt at calculation.
A mathematical estimation of size, on the other hand, is something that
requires us to calculate (i.e. brings things into numerical relations
with one another). When we want to judge whether a person’s height has
broken the world record, for example, we do not simply eyeball the
person’s height: we measure the person’s height quantitatively, “by
means of numerical concepts.” (p. 134) Thus, in a mathematical
estimation of size, instead of using the norms of intuition as our
guide, we use the norms of reason (i.e. numerical relation) for our
estimation. A mathematical estimation of size is thus an estimation of
size based on quantitative reasoning, which requires us to compare the
size of things based on their numerical relations to one another,
rather than judging merely on the basis of our intuitive norms" -
http://courses.washington.edu/asthetik/thiti_mathematical.html
Mathematical sublime differs from the sublime as it needs to be thought about more, it goes above the normal every day things that are seen and when we have to think about certain things and try and calculate their significance and what they mean.
