My answer is No.    

Even if we start with any space, there is the question of whether the space is finite-dimensional or not.    Add to that the need to consider the cardinality of subsets in a space in comparing one set with another set.    If we consider mappings f on one space X into another space Y, we either implicitly or explicitly need to consider the the extent of the image set, i.e., whether f maps X into Y or f maps X onto Y.   And so on.

Numbers form the zeroth step in Mathematics.

Yes you can work with interval arithmetics and you will be more accurate in many times.

My answer is No.    

Even if we start with any space, there is the question of whether the space is finite-dimensional or not.    Add to that the need to consider the cardinality of subsets in a space in comparing one set with another set.    If we consider mappings f on one space X into another space Y, we either implicitly or explicitly need to consider the the extent of the image set, i.e., whether f maps X into Y or f maps X onto Y.   And so on.

I agree with you dear @James: There is no Math without numbers.

The answer is a clear 'YES'. If you have ever taken a serious algebra course, you would know that no numbers are needed here. There is a huge, and very important part of mathematics that are NOT concerned with numbers. Algebra is its name.

There is math without numbers. Q.E.D. :-)

From my opinion, eventually (independent from the initial assumptions and a way to account "objects") we should arrive to the concept "numbers". I will agree with the idea of  Demetris Christopoulos: yes, taking a system for accounting we could simplify the life, but also we can make it complicated

A short comment to Michael´s statment. Not every part of algebra is working without numbers. And my opinion, a number must not be a integer, real or some other kind. A number also can be represented by symbols.

My answer to the original question is a clear one: mathematics needs numbers and is the science of numbers.

My answer is it depends on what we exactly mean by what a number is.  Are we talking about quantities (that are fixed), variables that have quantity, or what precisely?  I mean, you can go a while without doing anything with numbers until you actually consider examples.  If you mean the concept of a number with variable quantities, my answer is no.

 If you mean actual numbers (e.g., 2, 3, 4), my answer is yes.  There are ways of formalizing some systems without arithmetic, but I will admit they are pretty limited and often very specialized to specific problems (e.g., problems often associated with elementary sets that have intractable properties, or some aspects of symbolic computation).  It depends on what kind of universe of axioms you want to adopt.  There of course will be some theorems we can never decidedly resolve, but domains such as universal algebra or theory of computing do offer some consideration to doing computation without numbers, but instead logical sentences or automated theorem proving.  The easy way out is to just not use numbers, but instead label an atomic unit as a symbol (say A or B).  Not an efficient way to think about it, but it is indeed possible to work in systems that do not involve numbers, just it may not be very applicable to many problems.

I am assuming that the questioner is suggesting the first option, so I will ultimately say no.  Arithmetic is a core part of the axioms that we consider in all of mathematical logic that builds the framework of Turing Machines, and foundation systems of logic.  There's going to be an evaluation somewhere in a formal system, which is encoded by a decision and typically this is a yes/no answer (0 or 1).

@Michael Patriksson: There is a huge, and very important part of mathematics that are NOT concerned with numbers. Algebra is its name.

Perhaps you are overlooking a few things common to the study of algebra.

1.  Extension fields.   A field K is an extension of a field F, provided K contains F.   An element $a \in K$ is algebraic over F, provided there exist elements $\alpha_0, …, \alpha_n$ in F, not all 0, such that $\alpha_0 a^n + … + \alpha_n a^n = 0$.

I.N. Herstein, Topics in Algebra:

http://www.drchristiansalas.org.uk/MathsandPhysics/AbstractAlgebra/Herstein.pdf

See chapter 5 Fields, starting on page 167 in my copy.

2.  Subscripts.  Linear combination of vectors (see Herstein, p. xv).

3.  Groups with the classic examples: $(\mathbb{Z}, +), \mathbb{Q}, *)$, Herstein, p. 41.

4.  Order of a finite nonempty subset of a group, Herstein, Lemma 2.3.2, proof, p. 54.

5.  Lagrange's theorem.   If G is a finite group and H is a subgroup of G, then the order of H divides the order of G.    See the proof, Herstein, p. 59.

6.  Theorem 2.4.3   A group of prime order is cyclic.    See proof, Herstein, p. 60.

7.  Polynomial rings, Herstein, starting on page 152.

Mathematics without numbers is a human's thinking and  imagination that is devoid of consciousness. Algebra is a choreographed , well established study of algebraic structures in which combinations of unknown/known quantities and constants are intertwined.

In the formation of the algebraic expression, numbers play vital role in being a coefficient and/or a power. The repetitiveness or replication behaviors of expressions/symbols, which are common traits of algebra,  are described in terms of numbers.

In any mathematical structure  of discourse either algebraic, geometric or analytic, numbers are quasi dense everywhere.

Dear Dr. Ahmed Charifi, 

Mathematics has many constituents and numeracy is one of them and I think without numbers mathematics shall loose its identity and will not be meaningful. 

In order to answer such extra-ordinary questions we have to get out of our existence is constraints and think from scratch: Why do we learn arithmetics at the elementary school? Has this something to do with the way that we could describe our local universe or is it one more of our conveniences?

It is true that in our culture and our perception of the world, numbers are everywhere present and they govern our mathematics. Gauss had well expressed in the following sentence, "The Mathematic is the queen of sciences, and the Arithmetic is the queen of mathematics." But then, if we place ourselves in a perception of the universe where our mathematics needs are other than those based on numbers, that could be our mathematics?