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1. A refutation of Matteo Plebani
Matteo Plebani is a scholarly level writer from the university of Turin. This paper seeks to refute him on his heavy
advocation of Stewart Shapiro, and assertions he very much vocalized in his paper “Mathematical platonism meets
Ontological pluralism?”. This paper will argue against his argument of plural platonism answering the
epistemological question of platonism. Although in the previous section he had made solid and concrete assertions
and carefully and perfectly illustrated the ideas he wanted to argue for, I am bothered by the later responses to
problems that were risen in platonism. He answered the epistemological question with plenitudinous platonism.
Plenitudinous platonism is the Platonist account that is quite literally and can only be presented in the form of
ontological and alethic pluralism. This Platonist account has an epistemological thesis, a metaphysical thesis, and
an ontological thesis that it derives from its property of ontological pluralism. These are what the theses say: the
epistemological thesis is that consistency is a form of validity for different mathematical sentences in mathematical
domain of discourses. Consistency is a way of finding truth in mathematics. The ontological thesis: abstracta and
concreta are different modes of existence. The metaphysical thesis: all possible abstracta exist, and not all possible
concreta exist (logical pluralism may be combined with this). This paper will also argue against his advocation for
Shapiro’s acceptance of only intuitionistic logic and having it as the only subsidiary of logical pluralism.
1. The Refutation
Plebani says that Balaguer presents his form of plentinduous platonism
“that every consistent mathematical theory is true of some abstract
mathematical objects”. This entails and is backed up by one of the theses
we had in the introduction. The metaphysical thesis had said that “All
possible abstracta exist”. If there are consistent mathematical theories,
then there are abstract mathematical objects, therefore if mathematical
theories exist there exist abstracta in accordance with each individual
mathematical theory. He then forms the proposition “A theory might be
inconsistent if a certain notion of logical consequence is adopted, but
consistent if another notion of logical consequence is adopted.” But, with
this he accepted and only proposed intuitionistic logic as a valid way of
deducting logical antecedents and consequences. You cannot have
multiple ways of having criterion for the truth of mathematical theories,

2. whereas one of the criterions make a logical consequence of propositions
invalid. You can only have concrete criterions for the validity of logical
consequences and mathematical theories, because if there were criterions
that were used solely invalidity in logical consequences and mathematical
theories, that would be absurd.1
2. A contradiction
There is another type of Platonist account that plebani offers, it is plural
platonism. Plural platonism is combined with ontological pluralism,
alethic pluralism, and plenitudinous platonism. Ontological pluralism is
that there are multiple ontological structures of the world, different modes
of existence. Typically, being the modes abstracta and concreta. Alethic
pluralism is “the view that sentences belonging to the mathematical
domain of discourse are true in virtue of one property, say coherence or
consistency, whereas sentences belonging to the empirical domain of
discourse are true in virtue of another property, say correspondence” (See
Plebani, section 4)2
, and plenitudinous platonism, the view that all possible
1
For more clarification, I would like to say that these criterions would not be of pointing
out the falsity in logical consequences and mathematical theories, but rather it is that they
should not exist as criterions because they exist when there is something wrong about a
mathematical theory or consequence. When you find falsity in a logical consequence, you
find yourself using a criterion that is false.
2 I argue that this definition of alethic pluralism is false. I say this because Plebani offered
two different types of properties to be added to the mathematical sentences in the
mathematical domain of discourse. In set theory, when a predicate symbol is predicating
a variable, there can only be one variable. Therefore, when we are predicating a sentence
in a domain of discourse with properties it can only be one property, thus it is rather

3. abstracta exist, so in accordance with all possible mathematical theory. We
arrive at a problem for this position, Plebani sees this to. His definition of
alethic pluralism adheres to the fact that there are mathematical theories
and sentences that exist in an empirical domain of discourse. Plebani says
that we should assume that an empirical domain of discourse is referring
to empirical mathematical objects, rather than abstracta. But, we have
another position that is fused with alethic pluralism and ontological
pluralism, plenitudinous platonism. Which is the thesis that ascribes all
possible abstracta to mathematical sentences. Although there is not a
complete definition for concreta, we shall assume (just as Plebani did) that
concreta are objects of that can be perceived by phenomena, so particularly
the opposite of abstracta. Plebani named the same three theses I did in the
memoire. The metaphysical thesis said that all possible mathematical
theories had according an according all possible abstracta in existence, but
not all possible concreta. Since we identified concreta being in accordance
with what empirical domains of discourse are, the contradiction is clear.3
monistic then pluralistic. His “alethic pluralism” is rather in tension with ontological
pluralism and is indubitably inclined towards ontological monism.
3
There being two types of domains of discourse, being applied mathematical domain of
discourse and a empirical domain of discourse would be a self-defeater to the Platonist.
“friends of the indispensability argument hold that the mathematical and the empirical
domains of discourse cannot be separated into two well-defined domains”. The Platonist
that has an account of alethic pluralism

4. 3. His response to the epistemological objection
to platonism
In section 4 of “Mathematical platonism meets ontological pluralism?”,
Plebani proposes a plenitudinous Platonist response to the famous
epistemological objection to platonism. “Plenitudinous Platonists reply to
this objection appealing to the principle that consistency is the criterion
for truth and existence” is what Plebani says for the first step of the
objection. Although this may be elementary to some readers, for the ones
reading this who do not know of the epistemological objection to
platonism it goes like this: (i) platonism states that there is a domain of
mathematical abstracta, (ii) these mathematical abstracta are causally
isolated mind-independent entities and are transcendent of the faculty of
phenomenal intention. (iii) from this, to know about mathematical
abstracta, we must be able to form reliable mathematical beliefs on them.
(iiii) but like stated before, these mathematical abstracta are mind
independent, meaning they exist without the faculty of phenomenal
intention or essence and are transcendent of the faculty of phenomenal
intention. (v) So how is it possible to form reliable beliefs about
mathematical abstracta given the characteristics they hold? Plebani’s
response with plenitudinous platonism is that of a variant of Balaguer’s.

5. (E1) we can form reliable beliefs about the consistency of some
mathematical theories. (From the principle of consistency)
and
(E2) If plenitudinous platonism is true, then if a mathematical theory is
consistent, the objects it quantifies over or refers to exist.
This argument, however, does not give a valid response to the
epistemological objection.
If we epistemic accounts that are based on the faculty of phenomenal
intention, then to try and make beliefs about the consistency of
mathematical theories, that which are aligned with mathematical abstracta,
we are still not able to make reliable beliefs about them. I feel as though
Plebani knew that something bad would happen when coming across this
objection of platonism. He stated earlier in his sections that plenitudinous
platonism need not be presented as a combination of alethic pluralism and
ontological pluralism. Earlier in this paper I had already pointed out the
errors and faultiness with that statement. So, it is the case that
plenitudinous platonism needs to be presented as a combination of
ontological pluralism and alethic pluralism. As we have stated before
earlier in this paper, alethic pluralism is the thesis that mathematical
sentences can be looked over at in empirical domains of discourse and
mathematical domains of discourse (discourses of abstracta). Empirical

6. domains of discourse are with empirical objects. If we are not already
aware of the sudden problem, we will go into I will make you aware.
Mathematical platonism adheres to a platonic account of mathematical
abstracta as transcendent objects. We have alethic pluralism as one of the
supplementary properties of plenitudinous platonism. Alethic pluralism is
the thesis that there can be both empirical domains of discourse and
mathematical domains of discourse. If we have both empirical
mathematical objects and mathematical abstracta, we have coincidentally
contradicted one of our own theses in our position. The metaphysical
thesis in plenitudinous platonism states the following…………………….
all possible abstracta exist, and not all possible concreta exist
clearly things purported by the faculty of phenomenal intention are of
concreta. So as the plenitudinous Platonist having both concreta and
abstracta is contradictory. It is even more contradictory than was before
because we now have all possible concreta existing inside of the empirical
domains of discourse being able to align with the mathematical abstracta.
Wherefore the plenitudinous Platonist is supposed to hold that all
mathematical theories exist in mathematical domains of discourse and
these mathematical theories (mainly mathematical sentences) inside of the
mathematical domains of discourse are the only things to align with
mathematical abstracta.

7. Conclusion
I do believe that the above conceptualization of the plenitudinous Platonist
response to the epistemological objection to platonism can be debunked.
The only faultiness I see in it is the justification of plenitudinous platonism
only being a combination of alethic and ontological pluralism. But I do
believe that where there is a refutation of something there is always a
response to the aforementioned refutation. Although this paper was very,
and may I say, very short, I still do believe it holds some importance to
Plebani’s paper as there has been problems pointed out. But I will be
featuring this paper in my new and very long essay called “A brief history
of metaphysics”. It will be going along with the contemporary era of
metaphysics, as Plebani’s paper does hold a lot of significance in this
contemporary era. Although other people may not see it, I do. He gives a
perfectly coherent and expeditious elucidation of the most important
concepts in the philosophy of mathematics. He was able to articulate
himself in the most scholarly and professional way. I hope this paper will
be able to spread more knowledge and inquiry in the philosophy of
mathematics community.